Editorial Type:
Article
Article Type:
Research Article

Non-Gaussian Deterministic Assimilation of Radar-Derived Precipitation Accumulations

Mark Buehner Data Assimilation and Satellite Meteorology Research Section, Environment and Climate Change Canada, Dorval, Quebec, Canada

Search for other papers by Mark Buehner in
Current site
Google Scholar
PubMed Close
and
Dominik Jacques Data Assimilation and Satellite Meteorology Research Section, Environment and Climate Change Canada, Dorval, Quebec, Canada

Search for other papers by Dominik Jacques in
Current site
Google Scholar
PubMed Close
Online Publication:
01 Feb 2020
Print Publication:
01 Feb 2020
DOI:
https://doi.org/10.1175/MWR-D-19-0199.1
Page(s):
783–808
Open access
View license

Abstract

Data assimilation (DA) approaches currently used for operational numerical weather prediction (NWP) generally assume that errors in the background state are Gaussian. At the same time, approaches that make no assumptions regarding the background state probability distribution are gaining attention in research. Most such approaches, including the particle filter, are ensemble DA methods that produce an ensemble of analysis states consistent with the background and observation distributions. The present study instead proposes a non-Gaussian deterministic (NGD) DA method for producing a single deterministic analysis state. Consequently, the usual challenge of maintaining an ensemble with sufficient spread and diversity is avoided. The NGD approach uses background ensembles generated by a standard ensemble Kalman filter. A series of noncycled DA experiments is conducted to evaluate the NGD approach for assimilating precipitation derived from North American weather radars to initialize limited-area deterministic forecasts. The resulting forecasts are compared with those produced using either a local ensemble transform Kalman filter (LETKF) deterministic analysis or latent heat nudging (LHN). The experimental results indicate that, for forecast lead times beyond 1.5 h, the NGD approach improves precipitation forecasts relative to LHN. The NGD approach also leads to better temperature and zonal wind forecasts at lead times up to 12 h when compared to those obtained with either LHN or the LETKF. For precipitation, the NGD and LETKF approaches produce forecasts that are of comparable quality. Finally, simple strategies are demonstrated that combine the NGD approach for assimilating radar-derived precipitation accumulations with the ensemble–variational approach for assimilating all other observations.

Denotes content that is immediately available upon publication as open access.

For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Mark Buehner, mark.buehner@canada.ca

Abstract

Data assimilation (DA) approaches currently used for operational numerical weather prediction (NWP) generally assume that errors in the background state are Gaussian. At the same time, approaches that make no assumptions regarding the background state probability distribution are gaining attention in research. Most such approaches, including the particle filter, are ensemble DA methods that produce an ensemble of analysis states consistent with the background and observation distributions. The present study instead proposes a non-Gaussian deterministic (NGD) DA method for producing a single deterministic analysis state. Consequently, the usual challenge of maintaining an ensemble with sufficient spread and diversity is avoided. The NGD approach uses background ensembles generated by a standard ensemble Kalman filter. A series of noncycled DA experiments is conducted to evaluate the NGD approach for assimilating precipitation derived from North American weather radars to initialize limited-area deterministic forecasts. The resulting forecasts are compared with those produced using either a local ensemble transform Kalman filter (LETKF) deterministic analysis or latent heat nudging (LHN). The experimental results indicate that, for forecast lead times beyond 1.5 h, the NGD approach improves precipitation forecasts relative to LHN. The NGD approach also leads to better temperature and zonal wind forecasts at lead times up to 12 h when compared to those obtained with either LHN or the LETKF. For precipitation, the NGD and LETKF approaches produce forecasts that are of comparable quality. Finally, simple strategies are demonstrated that combine the NGD approach for assimilating radar-derived precipitation accumulations with the ensemble–variational approach for assimilating all other observations.

Denotes content that is immediately available upon publication as open access.

For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Mark Buehner, mark.buehner@canada.ca
Keywords: Bayesian methods; Kalman filters; Numerical weather prediction/forecasting; Short-range prediction

1. Introduction

Providing accurate forecasts of precipitation and especially high-impact weather associated with intense rainfall is an important part of the mandate for operational NWP centers. Research directed at improving such forecasts focuses on developing both the high-resolution forecast models and the assimilation of spatially and temporally dense observations necessary to capture the evolution of small-scale convective weather systems. For this purpose, weather radar measurements can play an essential role by directly observing both the three-dimensional structure of precipitation and winds within precipitating systems at high spatial and temporal resolutions. Most NWP centers currently have operational systems dedicated to high-resolution short-term prediction that assimilate data from weather radars (Gustafsson et al. 2018).

The currently operational NWP systems at Environment and Climate Change Canada (ECCC) do not yet assimilate data from weather radars. For deterministic data assimilation (DA), the 4D ensemble–variational DA approach is used (4D-EnVar), which partially relies on ensembles of short-term forecasts for estimating error covariances of the background state (Lorenc 2003; Buehner 2005; Buehner et al. 2013; Buehner et al. 2015; Caron et al. 2015). For ensemble DA, a lower-resolution ensemble Kalman filter (EnKF; Houtekamer et al. 2019) is used for initializing ensemble forecasts and also for providing the ensembles for the EnVar system. Research activity has recently begun at ECCC to evaluate the possible approaches for using radar-derived precipitation measurements to improve high-resolution short-term weather forecasts.

During the last decade, the assimilation of radar reflectivity measurements (either directly or as precipitation estimated from reflectivity) within rigorous DA approaches has been extensively explored elsewhere, including 3D variational DA (3D-Var; Gao and Stensrud 2012), 4D variational DA (4D-Var; Sun and Crook 1997; Lopez 2011; Kawabata et al. 2011), EnKF (e.g., Tong and Xue 2005; Bick et al. 2016), and EnVar (e.g., Gao et al. 2013; Wang and Wang 2017; Duda et al. 2019). These approaches are all based on the assumption that the error in the background state is Gaussian. In contrast, the non-Gaussian nature of convective processes is partially addressed by the two-step procedure used at Meteo-France (Caumont et al. 2010; Wattrelot et al. 2014). In the first step, a non-Gaussian approach is used to retrieve vertical profiles of relative humidity from radar reflectivity. These retrieved profiles are then assimilated in a standard 3D-Var together with other observations in the second step.

Also in use are a number of diabatic initialization approaches, described by Sun et al. (2014), that rely on empirical relationships between prognostic model variables and reflectivity to influence the model trajectory to improve the agreement between simulated and observed precipitation. Among these methods, latent heat nudging (LHN) is a relatively simple technique employed at several NWP centers (Gustafsson et al. 2018) that operates within the forecast model to adjust the temperature and water vapor profiles to improve precipitation forecasts. Jacques et al. (2018) recently evaluated this approach within the Canadian regional deterministic prediction system (RDPS; Caron et al. 2015) in preparation for a future operational implementation.

The necessary assumptions that underpin most Gaussian DA approaches are particularly problematic when considering the physical processes associated with precipitation, especially for convective precipitation events. Such processes are highly nonlinear and the distribution of many of the key variables involved (e.g., water vapor, cloud condensate) are bounded. The particular situation of using radar reflectivity measurements to improve precipitation forecasts has the added challenge that the observed quantity is related to precipitating hydrometeors which, in many meteorological situations, are not the most important prognostic variables in typical NWP models in terms of their effect on forecasts. Instead, what is required is information on other quantities, including vertical profiles of water vapor, temperature and winds. Therefore, the ability to use reflectivity measurements to improve atmospheric forecasts relies on having an accurate estimate of the statistical and physical relationships between the observed radar reflectivity and these prognostic variables. Due to the strong nonlinearity of the physical processes and bounded nature of several key variables, these relationships cannot be well represented with correlations and therefore may often be associated with a highly non-Gaussian multivariate distribution. Consequently, standard Gaussian DA approaches are expected to be less than optimal for such applications (e.g., Robert et al. 2018).

Fully non-Gaussian ensemble DA approaches, including the localized particle filter (LPF; Poterjoy et al. 2017; Poterjoy et al. 2019), have been shown to be effective for assimilating radar reflectively and other observation types. Other examples of non-Gaussian ensemble DA approaches applied in a realistic NWP context include those by Robert et al. (2018) and Potthast et al. (2019). Recent results demonstrate that such approaches can perform better than standard Gaussian ensemble DA approaches. However, non-Gaussian approaches developed with similar basic assumptions as the LPF, but targeted for application to deterministic prediction, have not yet been fully explored. The goal of this study is to evaluate a new fully non-Gaussian deterministic (NGD) approach in the context of assimilating radar-derived precipitation measurements for improving short-term limited-area weather forecasts of precipitation and other atmospheric variables. We note that the term NGD is general and could be used to refer to other non-Gaussian methods specifically designed for deterministic DA. This study is motivated by the practical requirement for most NWP centers to produce deterministic precipitation forecasts (due to the need for high spatial resolution and the much larger computational cost of ensemble forecasts). The proposed NGD approach is described in the following section. This new approach is compared against the LHN approach, described in section 3, and standard Gaussian DA approaches, described in section 4. Approaches for combining the use of LHN or NGD for assimilating radar-derived precipitation with EnVar for assimilating all other observations are described in section 5. The comparison of DA approaches is performed by both 1) examining in detail the results of applying them to a single case and 2) evaluating the impact on a month of short-term forecasts. Details concerning these numerical experiments are given in section 6 and the results are in section 7. The final section provides some conclusions.

2. Non-Gaussian approach for deterministic data assimilation

a. General approach

The overall strategy for obtaining a deterministic analysis state with the proposed NGD approach is somewhat analogous to how EnVar is currently used operationally for NWP at ECCC (Buehner et al. 2015; Caron et al. 2015). An analysis increment is computed to correct a single deterministic background state by making use of an ensemble of short-term forecasts to characterize the uncertainty in the background state. Information on this background uncertainty can include spatial, temporal and multivariate statistical relationships. The ensemble is obtained from a separate EnKF that usually employs a lower-resolution version of the same model to be computationally affordable. For EnVar, only the background-error covariances are estimated from the ensemble and then used to compute the analysis increment. However, if the background probability distribution is not Gaussian, covariances are insufficient to describe it. To overcome this deficiency, one possible approach is to represent the full background state distribution by a sum of delta functions, each with area equal to 1/Ne and centered at the location in phase space given by each ensemble member (xne):
p⁡(x)≈∑n=1Ne1Neδ⁡(x−xne),
where Ne is the ensemble size. This representation is employed in particle filters (e.g., Poterjoy et al. 2019) and is consistent with the background ensemble being a set of equally likely random draws from the background distribution. The mean of this distribution simply equals the ensemble average.
The posterior (or analysis) distribution p(x|y) that results from assimilating the observations, denoted as y, can then be represented as the same set of delta functions, but with weights modified according to Bayes’s theorem:
p⁡(x|y)≈∑n=1Newnδ⁡(x−xne),
where the weight for the nth ensemble member, denoted as wn, is obtained from the conditional probability of the complete set of observations given the nth ensemble member:
wn=p⁡(y|xne)∑m=1Nep⁡(y|xme).
The denominator in (3) ensures that the weights sum to one over the ensemble and, since the weights cannot be negative, they are also all constrained to have values between zero and one.

The weights essentially measure the degree to which each ensemble member is in agreement with all of the observations. Given the typical case for NWP of having a very small ensemble size relative to the total number of observations and state variables, it is exceedingly unlikely that a single ensemble member will ever agree well with all observations. Therefore, as part of their LPF algorithm, Poterjoy et al. (2019) propose an approach to compute localized ensemble weight fields. This is analogous to the use of spatial covariance localization as initially introduced for the EnKF (Houtekamer and Mitchell 2001; Hamill and Whitaker 2001) and later for EnVar (Lorenc 2003; Buehner 2005). By allowing the weights to vary spatially, different ensemble members may receive large weights in one geographical region as compared with another region. The localized weight field formulation of Poterjoy et al. (2019) is used to compute the analysis state for the NGD approach.

In particle filters, such as the LPF of Poterjoy et al. (2019), the weights are used to reconstruct a new ensemble of Ne equally likely states that approximates the non-Gaussian posterior distribution and is used for initializing the subsequent ensemble forecast step. The generation of this analysis ensemble relies on a sequential importance resampling procedure adapted for very high-dimensional systems. This can be very challenging, especially when the weights “collapse,” that is, only one or a few ensemble members have nonzero weight. To avoid weight collapse, a special procedure must be employed to limit the influence of the observations on the weight calculation in densely observed areas [e.g., Poterjoy et al. (2019) adaptively inflate the observation error variance]. Such weight collapse should not be a problem for deterministic DA since only a single “best” estimate of the state is required to specify the analysis state. While the mode of the posterior distribution would likely be a good choice for the analysis state, it is not straightforward to compute for very high-dimensional multivariate distributions represented by a sum of delta functions. The mean of the distribution is simple to compute since it is the weighted ensemble average and therefore it will be used, in part, for computing the NGD analysis state. However, if very few or no observations are assimilated within a sufficiently large geographical region, the weights may be locally unaffected by the assimilation procedure and therefore all members will have equal weight. In these locations, the weighted ensemble average would equal the background ensemble average, which is known to underestimate both the energy at small scales and the amplitude of intense rainfall events (Surcel et al. 2014). Most deterministic DA approaches would simply leave the background state unchanged in any region that is sufficiently far from any assimilated observations such that the background error covariance between the region and the location of the nearest observation is zero. To achieve the same behavior, a separate procedure is applied wherever the ensemble weights are all nearly equal such that the NGD analysis state smoothly transitions to become equal to the deterministic background state, instead of the background ensemble average.

Further details concerning the calculation of the localized weight fields and the procedure for computing the final analysis state and increment are given in the remainder of this section.

b. Weight calculation

The localized weight fields determine the relative local contribution of each member to the resulting NGD analysis (xNGD), which is given by the weighted ensemble averaged:
xNGD=∑n=1Neωn∘xne,
where the operator “∘” is the element-wise Schur product of the two vectors. The procedure for obtaining ω for each ensemble member is now described.
First, for each observation (i = 1, …, Ny) and ensemble member (n = 1, …, Ne), a quantity that resembles the standard observation component of the variational DA cost function is computed:
Jo⁡(i,n)=⁡[yio−Hi⁡(xne)]22σo2.
These are then used to compute a set of scalar weights for each observation and ensemble member:
wi,n=e−Jo⁡(i,n)∑m=1Nee−Jo⁡(i,m).
This formulation for the scalar weights is equivalent to (3) when the observation error is Gaussian, though it is straightforward to adapt (5) and (6) for skewed or bounded observation error distributions (Poterjoy and Anderson 2016). Following Poterjoy et al. (2019), under the assumption that the observation errors are uncorrelated,1 the localized weight field is computed for each grid point (j = 1, …, Nx) and each ensemble member by combining the scalar weights for all observations located near the grid point:
ωj,n=∏i=1Ny⁡[loc⁡(i,j)⁡(wi,n−1/Ne)+1/Ne]∑m=1Ne∏i=1Ny⁡[loc⁡(i,j)⁡(wi,m−1/Ne)+1/Ne],
where the denominator is simply the numerator summed over all ensemble members to ensure that the weights sum to one over the ensemble. The localization function loc(i, j) is similar to those used for covariance localization in ensemble-based Gaussian DA approaches (e.g., EnKF and EnVar). This function typically has values that monotonically decrease from one to zero as the distance between the observation and grid point increases. For a given observation and grid point where the localized weight field is being computed, if the localization function equals one, the contribution of the observation to the localized weight field equals the scalar weight and therefore is equivalent to the standard Bayesian update given by (3). In contrast, if the localization function equals zero (i.e., because the observation is far from the grid point), the contribution is equal for all members and therefore the observation has no effect on the localized weight field.

It is important to note that, because the localized weights can only take values between zero and one and they sum to one over the ensemble, the resulting weighted ensemble average in (4) cannot be outside the range of values present in the ensemble at that location. This ensures that for nonnegative variables, such as water vapor or cloud condensate, the analysis is guaranteed to also be nonnegative. However, it also prevents the analysis from approaching the observed value if it is greater than the largest value within the ensemble or less than the smallest value. This is in contrast with standard Gaussian DA approaches that only use the ensemble for estimating background error covariances and therefore can produce analysis states with nonphysical values [e.g., negative humidity or accumulated precipitation; Posselt and Bishop (2018)]. However, for highly non-Gaussian distributions, such “extrapolation” outside the range of values within the ensemble may produce inaccurate results.

c. Calculation of the analysis and analysis increment

The weighted ensemble average computed using the localized weight fields described in the previous section must be modified such that the NGD analysis state smoothly transitions to the deterministic background state when the ensemble weights are all nearly equal. A measure of the similarity of the weights at the jth grid point is provided by the “effective sample size,” defined as ⁡(Neff)j=1/∑n=1Neωj,n2 (Kong et al. 1994), that varies between 1 (when only 1 member has a nonzero weight) and Ne (when all members have equal weight). The final NGD analysis takes the value of the weighted ensemble average where Neff is small relative to Ne. However, for values of Neff between 75% and 100% of Ne, we choose to allow the contribution of the weighted ensemble average to linearly decrease from one to zero (and the corresponding weight on the deterministic background state to increase from zero to one). This calculation is performed at each grid point and therefore the final analysis state is
x^NGD=α∘xb+(1−α)∘∑n=1Neωn∘xne,
where 1 is a vector of ones and the value of the spatially varying weight on the background state, denoted as α, at the jth grid point is defined as
αj={⁡[⁡(Neff)j/Ne−0.75]/⁡(1−0.75),if⁡(Neff)j/Ne>0.750,otherwise.
It is convenient to combine the localized weight fields with the scale factor α such that ω^n=(1−α)∘ωn (where 1 is a vector of ones) and the analysis state can be expressed as
x^NGD=α∘xb+∑n=1Neω^n∘xne.
While this approach ensures that the final NGD analysis equals the deterministic background state in areas with no assimilated observations, the summing over the ensemble members in the second term of (10) can produce analysis states that are spatially smoother than what would be obtained by a standard Gaussian DA approach when the observation error variance is high (as will be shown in section 7a).

Depending on how the forecast model is initialized with the resulting NGD analysis state, it may be necessary to also compute the analysis increment by subtracting the deterministic background state from the final NGD analysis in (10).

The following two sections describe the LHN, local ensemble transform Kalman filter (LETKF) and EnVar approaches. The LHN and LETKF approaches are used in this study to assimilate radar-derived precipitation measurements for comparison against the NGD approach. The EnVar approach is included for demonstrating the impact of using either NGD or LHN for assimilating radar-derived precipitation in combination with EnVar for assimilating all other observation types.

3. Latent heat nudging for assimilating radar-derived precipitation accumulations

Latent heat nudging (Jones and Macpherson 1997) is a commonly used diabatic initialization technique for using radar reflectivity measurements or radar-derived precipitation to improve forecasts. It is both relatively simple to implement and requires little additional computational cost. As previously evaluated at ECCC (Jacques et al. 2018), the surface precipitation rates from the model are compared with those estimated from radar reflectivity measurements at every model time step within the assimilation time window. Then, depending on whether or not precipitation is present in the model state and observations, the vertical profile of temperature tendencies due to latent heat release is modified in a way that brings the model precipitation into closer agreement with the observed precipitation. The model’s water vapor mixing ratio is also adjusted such that relative humidity is conserved after applying the LHN temperature changes. The LHN approach does not explicitly rely on specific assumptions regarding the distribution of the background and observation errors.

4. Gaussian data assimilation approaches

a. LETKF for assimilating radar-derived precipitation accumulations

The equation used to compute the ensemble mean analysis within the local version of the ensemble transform Kalman filter (LETKF; Bishop et al. 2001; Hunt et al. 2007) can also be used to update a deterministic background state for computing a deterministic analysis (e.g., Schraff et al. 2016). This assimilation approach, which is based on assuming Gaussian background errors, was chosen for comparison with the NGD approach since the two are similar in several ways. First, they both compute a set of spatially varying weights for each ensemble member for constructing the analysis. Also, both approaches independently compute these weights at each grid point by considering all observations that are within a specified distance, as determined by the localization function. In the LETKF the analysis is computed from the spatially varying weight fields as follows:
xLETKF=xb+∑n=1NeωnLETKF∘(xne−x¯),
where x¯ is the ensemble average: x¯=(1/Ne)∑n=1Nexne. The weights for the LETKF are computed with a modified form of the Kalman analysis equation that operates within the subspace spanned by the ensemble member perturbations, given by
ωnLETKF=P˜a⁡(Yb)TR−1⁡[y−H⁡(xb)],
where R is the observation error covariance matrix, the matrix P˜a is the Ne × Ne analysis error covariance matrix in the ensemble subspace, given by
P˜a=⁡[⁡(Ne−1)I+⁡(Yb)TR−1Yb]−1,
and the nth column of the matrix Yb is the ensemble perturbation of the nth ensemble member projected into observation space:
⁡[Yb]n=H⁡(xne)−1Ne∑m=1NeH⁡(xme).

Spatial covariance localization is an important aspect of the LETKF, as it is for any assimilation approach that uses relatively small ensembles. Because the calculation of the analysis increment is performed within the subspace spanned by the ensemble member perturbations, it is not possible to directly apply localization to the background error covariances. Therefore, instead of reducing the background error covariance for increasingly distant locations, the observation error variance is increased for increasing distance from the grid point being analyzed (Hunt et al. 2007). Though not entirely equivalent, both approaches smoothly reduce the influence of observations as the distance increases between the observation and the analysis state grid point. In practice, the observation errors are usually assumed to be uncorrelated and therefore the matrix R in the above equations is diagonal. As a result, spatial localization is applied to the calculation of the weight fields by simply multiplying the diagonal elements of the matrix R−1 by the same monotonically decreasing localization function, loc(i, j) for observation i and grid point j, as used for the NGD approach.

b. EnVar for assimilating all other observations

The currently operational DA systems used to initialize deterministic NWP forecasts at ECCC rely on the EnVar approach (Buehner et al. 2015; Caron et al. 2015). These systems assimilate a diverse range of conventional and satellite observations, but not precipitation-related observations. The EnVar approach is somewhat similar to the LETKF in the sense that spatially varying weights are computed and applied to the ensemble perturbations to partially obtain the analysis increment. However, in the EnVar approach the weights are computed by iteratively minimizing a single global cost function such that the approach is equivalent to assimilating all observations simultaneously using background error covariances with spatial localization applied directly on the model grid (Buehner et al. 2013). Another difference is that the ensemble-based covariances can easily be combined with a separate full-rank climatological covariance matrix that is static in time and involves assuming the spatial correlations are horizontally homogeneous and isotropic [although a reduced-rank approximation of a climatological covariance matrix could be used in the LETKF, as shown by Kretschmer et al. (2015)]. In practice, the use of such “hybrid” covariances has been shown to result in improved forecasts (Buehner et al. 2013; Lorenc 2017)

5. Combining non-Gaussian radar-derived precipitation assimilation with EnVar

In the present study, precipitation measurements derived from radar reflectivity are the only assimilated observations when using the NGD, LETKF, and LHN approaches. Therefore, strategies are needed for combining these precipitation assimilation approaches with EnVar for assimilating all other observations. Since LHN actively computes and applies modifications to the model variables during the integration of the model, combining it with EnVar is straightforward. The NGD and EnVar approaches can be combined by first applying the NGD approach and using the resulting analysis state (10) as the background state for the EnVar assimilation. It seems reasonable to first apply the NGD approach (that assimilates only precipitation observations) since large corrections to the background state from this step will likely be associated with non-Gaussian errors. Because errors in the NGD analysis state will be smaller, they should also be closer to being Gaussian and therefore EnVar would be more effective making additional corrections from assimilating all other observations.

In the simplest approach considered for combining NGD with EnVar (referred to later as the NGD + EnVar experiment), the EnVar configuration is identical to what it would be when used alone. However, due to the reduction in background error, mostly in areas near precipitation, the background error covariances used within EnVar should be reduced. A simple approach for achieving this is to scale the EnVar background error standard deviation (std dev) such that it is reduced in areas where numerous precipitation observations are assimilated. Consequently, the EnVar analysis increment is reduced in areas where the non-Gaussian assimilation is active and equal to its original amplitude in areas where the non-Gaussian assimilation does not change the original background state. In the first tests of this approach (referred to as the NGD + EnVarBlend experiment), the EnVar background error std dev can be reduced to zero at some locations, leading to EnVar having no impact on the final analysis and allowing the two assimilation procedures to act relatively independently in different regions depending on the spatial distribution of precipitation. Similar to (9), a field of scale factors, β, applied to the EnVar background error std dev at the jth grid point is defined as a linear function of Neff, given by
βj={⁡[⁡(Neff)j−1]/⁡(0.5×Ne−1),if⁡(Neff)j<0.5×Ne1,otherwise.
Therefore, the EnVar background error std dev is unchanged when Neff is greater than half the ensemble size and set to zero when Neff equals one. The specific choice of function for the scale factor likely does not significantly affect the results since Neff commonly has values close to its extremes in most numerical experiments performed in this study. More sophisticated approaches for combining the two assimilation approaches could be considered in a future study.

6. Description of numerical experiments

A series of forecasts are performed to evaluate the proposed NGD approach for precipitation assimilation versus both the LHN technique and the LETKF. Also included for comparison are reference forecasts with no DA and forecasts produced using EnVar to assimilate all operational observations (not including precipitation). Finally, experiments are conducted that combine using either the NGD or LHN approaches for assimilating precipitation measurements with using EnVar for assimilating all other observations.

a. Forecast model configuration

To facilitate this first set of numerical experiments for evaluating the NGD approach, an efficient limited-area configuration of the Global Environmental Multiscale (GEM) model (Girard et al. 2014) is employed for both the deterministic and ensemble forecasts. This model configuration is similar to that of the currently operational continental scale RDPS (Caron et al. 2015) and also used in the previous study to evaluate the LHN approach (Jacques et al. 2018). The model employs a limited-area rotated latitude–longitude grid with 10 km horizontal grid spacing and 80 vertically staggered levels between the surface and 0.1 hPa. However, relative to the RDPS, the horizontal extent of the model domain is much reduced, only spanning the eastern portion of North America that is well covered with radar observations (Fig. 1).

Fig. 1.
Fig. 1.

(a) The model surface height field over the limited-area model domain used in this study. The gray box in this and similar later figures indicates the boundary of the inner domain where the model evolves free of direct influence from the LBCs. (b) The spatial coverage of radar data within the model domain shown in terms of the quality index for the precipitation rate estimated from radar reflectivity at 1200 UTC 15 Aug 2016.

Citation: Monthly Weather Review 148, 2; 10.1175/MWR-D-19-0199.1

Each numerical experiment conducted for this study consists of a series of noncycled analyses and forecasts [similar to Jacques et al. (2018) and Ban et al. (2017)] every 6 h during the entire month of August 2016. First, a common set of background states is generated by integrating the limited-area model starting from the RDPS background state and hourly analysis increments 9 h prior to the analysis time by applying the incremental analysis update approach (IAU: Bloom et al. 1996) over a 6-h window (Fig. 2 illustrates the procedure used to produce the forecasts). Based on these background states, for all experiments utilizing the NGD or LETKF approach, analysis increments are computed by assimilating precipitation observations within the 1-h window centered on the analysis time. For the experiments that involve EnVar, all other observations are assimilated within the 6-h window centered on the analysis time. The analysis increments from the NGD, LETKF, EnVar, or combined approaches are then used to initialize 12-h model forecasts with the IAU approach. Unlike in the RDPS, IAU is only applied between −0.5 and +0.5 h relative to the analysis time using a single analysis increment. The increments of horizontal wind, temperature, water vapor mixing ratio, total cloud condensate mixing ratio and surface pressure are used in the IAU procedure. In contrast, the LHN procedure is applied directly within the forecast model at every time step as described below. The lateral boundary conditions (LBCs) for all forecasts are provided by the larger continental domain RDPS forecasts.

Fig. 2.
Fig. 2.

Schematic of setup for generating (a) the deterministic background states, (b) the forecasts initialized with analysis increments computed with NGD, LETKF, or EnVar DA approaches, and (c) the ensemble forecasts. The LHN approach is applied within the same 1-h time window, shown in (b), used for initializing forecasts with NGD, LETKF, and EnVar analysis increments.

Citation: Monthly Weather Review 148, 2; 10.1175/MWR-D-19-0199.1

The NGD, LETKF and EnVar DA experiments employ ensembles of short-term forecasts generated using the same 10 km model configuration as the deterministic forecasts. The ensemble forecasts are initialized by down-scaling the 256 analysis ensemble members from the global EnKF that currently employs a 39 km horizontal grid spacing (see section 2.1 of Houtekamer et al. 2019 for more detail on the EnKF configuration). The ensemble is centered on the RDPS analysis state by gradually adding to the model state through IAU the sum of two terms: 1) the ensemble perturbation (interpolated from the 39 km to the 10 km resolution grid) and 2) the RDPS analysis increment. For all ensemble members, the LBCs are taken from the same RDPS forecasts as used for generating the deterministic background states and, as a consequence, the ensemble has zero variance along the lateral boundaries. The ensemble forecasts are integrated 6 h beyond the analysis time to enable the evaluation of using either an ensemble of 6-h forecasts, 12-h forecasts (with the same valid time), or a combination of both, resulting in an ensemble of 512 members. Additive inflation is already applied to the analysis ensemble within the EnKF assimilation cycle to ensure the background ensemble has sufficient spread (see section 2.3 of Houtekamer et al. 2019).

b. Radar-derived precipitation assimilation

For this study, precipitation estimated from reflectivity measurements from weather radars over eastern North America (Fig. 1b) are the only assimilated observations when using the NGD, LHN, and LETKF approaches. Prior to assimilation, the polar reflectivity measurements are combined into 2D mosaics (pseudo-CAPPIs2 at an altitude of 1 km) and transformed into instantaneous precipitation rates (PR) following the procedure described by Jacques et al. (2018). The resulting composites and the accompanying quality index field, have a 10 km grid spacing and are available every 10 min. All assimilation approaches use the same measurements available in a 1-h window centered at the analysis time.

In the experiments that employ LHN, the observed instantaneous precipitation rates (available every 10 min) are used to adjust the model’s temperature tendencies at every time step between −0.5 and +0.5 h from the analysis time (Jacques et al. 2018). When using the NGD and LETKF approaches, the instantaneous precipitation rates are combined into 1-h precipitation accumulations. Only observations with a nonzero value for the quality index are included.

For both the NGD and LETKF approaches, the assimilated quantity is the 1-h accumulated precipitation (denoted ACC and in units of mm) transformed according to ln(ACC + 0.1). As discussed by Mahfouf et al. (2007), this log transformation is appropriate for precipitation because the observation error varies as a power law of the precipitation and therefore the observation error variance can be considered to be constant for the log-transformed variable [see Bishop (2019) for an alternative approach for addressing state-dependent observation error variance within an EnKF]. In addition, we also choose to reject observations where both the deterministic background state and either the observation or all ensemble members have zero precipitation (with the background state and ensemble members interpolated to the observation location). In such areas either the background state already perfectly agrees with the observations or the deterministic background state exactly equals all ensemble members. In either of these situations, if it occurs over a sufficiently large area, the deterministic background state should not be modified by the assimilation procedure (i.e., the analysis increment should be zero). It is noted, however, that when the deterministic background state and all ensemble members have zero precipitation, rejection of nonzero precipitation observations is only necessitated by the inability of the ensemble to provide useful information on the background probability distribution. If such a situation occurs frequently, it is likely a sign that the ensemble spread is too small and should be increased through an improved sampling of errors related to the model or other sources. Alternatively, ensemble inflation methods specifically designed to address this situation could also be considered (e.g., Yokota et al. 2018).

When applying the NGD and LETKF approaches, the spatial localization function is chosen to be the fifth-order polynomial of Gaspari and Cohn (1999) that becomes zero at a horizontal distance of 150 km (chosen based on a limited number of preliminary experiments with different distance values). No localization is applied to either the multivariate or vertical relationships. To limit the vertical spread of the information, the analysis increments are gradually tapered to zero toward the model top, starting above the tropopause. Consequently, at each horizontal grid point, the same weights are applied to all analysis variables and vertical levels throughout the troposphere and lower stratosphere.

A summary of the radar-derived precipitation assimilation experiments using the NGD, LETKF and LHN approaches with different ensemble sizes and values for the observation error std dev (for the log-transformed 1-h accumulated precipitation) is given in Table 1.

Table 1.

Configuration summaries for radar-derived precipitation DA experiments.

Table 1.

c. Assimilation of operational observations

EnVar is used to assimilate the full set of observations that are available within the model domain and that are currently assimilated in the operational deterministic prediction systems. This includes data from radiosondes, aircraft, surface stations, ships, buoys, satellite atmospheric motion vectors, satellite scatterometers, satellite microwave and infrared instruments, and global positioning system satellite radio occultation and ground-based instruments. The EnVar configuration is similar to that used by Bédard et al. (2019, manuscript submitted to Wea. Forecasting), except that a 3D, instead of 4D configuration of EnVar is used. Also, the hybrid covariances use an equal weighting between the static background-error covariance matrix and the ensemble-based covariance matrix. The ensemble covariances are estimated from the same 256 6-h short-term forecasts used for the NGD and LETKF experiments. The same spatial covariance localization is applied as in the operational RDPS, such that covariances are gradually reduced according to the fifth-order polynomial of Gaspari and Cohn (1999) and become zero at a horizontal distance of 2800 km and two scale heights in the vertical. Following the approach described by Bédard et al. (2019, manuscript submitted to Wea. Forecasting), the static covariance matrix is computed from a series of RDPS 48-h minus 24-h lagged forecast differences after horizontal interpolation onto the smaller model grid used in this study.

A summary of the assimilation experiments involving EnVar either alone or in combination with either the NGD or LHN approach for assimilating radar-derived precipitation is given in Table 2.

Table 2.

Configuration summaries for experiments assimilating all operational observations, both alone and in combination with radar-derived precipitation assimilation.

Table 2.

7. Results

Two types of results are presented to evaluate and compare the NGD approach with other approaches for assimilating radar-derived precipitation accumulations. First, for a single chosen representative date, a detailed examination of applying the NGD assimilation approach is conducted to gain insight into the approach itself and how it differs from the standard Gaussian LETKF approach. Then, results from one month of forecasts are presented that compare the ability of the NGD approach to improve short-term forecasts of precipitation and other variables in comparison with more conventional approaches.

a. Analyses at 1200 UTC 15 August 2016

The single case of assimilating radar-derived precipitation at 1200 UTC 15 August 2016 is examined in this section. Figure 3 shows the 1-h accumulated precipitation estimated from radar reflectivity measurements (Fig. 3a), from the deterministic model background state (Fig. 3b) and from the ensemble average of the 256-member ensemble of short-term 6-h forecasts (Fig. 3c). The ensemble spread std dev is also shown in terms of the log-transformed precipitation (Fig. 3d). While the overall spatial distribution of precipitation is similar in the observations and background state, many local details differ. In general, the background state appears to have too little area with precipitation in the southwest part of the domain, though this region is likely strongly affected by the nearby boundary of the inner model domain (indicated by the gray rectangle) beyond which the LBCs influence the model solution. Closer to the center of the model domain, precipitation in the background state appears to be more spread out and extends farther toward Lake Erie in the northeasterly direction than in the observations. As can be expected, the ensemble mean state is spatially smoother, contains a larger area with low precipitation values and has much lower peak values than the deterministic background state. In nearly all areas where the ensemble mean is larger than 0.1 mm h−1 (the lowest value used for color shading) the ensemble spread std dev in log-transformed units is greater than one. Taken together, the ensemble mean and spread give some indication that the observed precipitation values are reasonably well captured within the range of values in the ensemble, at least when considering the grid point values independently. However, it should be noted that stddev may not be the best measure for ensemble spread if the distribution deviates strongly from being Gaussian (as examined below).

Fig. 3.
Fig. 3.

(a) Observation, (b) background state, and background ensemble (c) mean and (d) spread std dev at 1200 UTC 15 Aug 2016. The observations, background state, and background ensemble mean are shown for the 1-h accumulated precipitation, whereas the background ensemble spread is shown for the log-transformed accumulated precipitation.

Citation: Monthly Weather Review 148, 2; 10.1175/MWR-D-19-0199.1

The normalized effective sample size (Neff/Ne) is shown in Fig. 4 from two NGD analyses that use 256 ensemble members. In areas far from observed precipitation, the normalized effective sample size equals one, indicating that the localized weight fields are all equal. When using a relatively small observation error std dev (Fig. 4a), the normalized effective sample size is reduced close to its minimum attainable value (1/Ne) over areas with nonzero observed precipitation, indicating that the localized weight field (7) is nonzero for very few and possibly only one member. When using a larger observation error std dev (Fig. 4c), the reduction in effective sample size is much less extreme, indicating that many more members receive a nonzero weight even within the area with precipitation. As described in section 2c, the final NGD analysis state smoothly reverts to the deterministic background state according to a linear function of the effective sample size. The weight given to the deterministic background state is shown in Figs. 4b and 4d. For both values of observation error std dev, this weight is essentially zero in all areas with a large number of nonzero precipitation observations, though when using a larger observation error std dev these areas are less extensive.

Fig. 4.
Fig. 4.

(a),(c) Normalized effective sample size Neff and (b),(d) the weight given to the deterministic background state in the NGD analysis when using an observation error std dev equal to either (top) 1 or (bottom) 5.

Citation: Monthly Weather Review 148, 2; 10.1175/MWR-D-19-0199.1

To better visualize the localized weight fields, the NGD approach is applied using only 16 ensemble members and an observation error std dev equal to 2. The resulting weight fields for all 16 members are shown in Fig. 5. Within the areas where the weighted ensemble average contributes more than half of the final NGD analysis (hatched areas), a single member generally dominates over distinct spatial patches with the transitions between these patches occurring quite rapidly. The horizontal line near the middle of the model domain indicates the locations used in Fig. 6 to show the weights for the 16 ensemble members when using various values of observation error std dev. Consistent with the results shown in Fig. 4, increases in observation error std dev correspond with increases in the number of members with nonzero weight and also a noticeably smoother spatial variation in the weight fields. While having weight fields with spatially smooth transitions is likely beneficial by providing a more continuous analysis state, it also corresponds with more members being involved in the weighted ensemble average and therefore, like the full ensemble average, an analysis state that is not itself a model solution and underestimates spatial gradients and peak values.

Fig. 5.
Fig. 5.

Individual localized weight fields computed with the NGD approach for each member of a 16-member ensemble when using an observation error std dev equal to 2. The hatched area denotes where the total weight on the ensemble members is greater than 0.5. The horizontal line near the center of the domain indicates the locations shown in the following two figures.

Citation: Monthly Weather Review 148, 2; 10.1175/MWR-D-19-0199.1

Fig. 6.
Fig. 6.

Detailed view of the localized weight fields at the locations corresponding to the horizontal line in the previous figure. Weights are computed with the NGD approach with a 16-member ensemble when using an observation error std dev equal to (a) 1, (b) 2, (c) 5, and (d) 10. The dashed gray curve is the total contribution of the weighted ensemble average to the final NGD analysis.

Citation: Monthly Weather Review 148, 2; 10.1175/MWR-D-19-0199.1

As shown in (11), LETKF also uses localized weight fields in the calculation of the analysis state. However, because these are computed under the assumption of Gaussian background errors and applied to the ensemble perturbations, the weights have both positive and negative values. The LETKF weights are shown in Fig. 7 for the same experimental configurations used for the NGD approach in Fig. 6. In addition to nearly half of the members having negative weights, the weight amplitudes are generally smaller for LETKF than for NGD when using the same observation error std dev. Also, nearly all members have nonzero weights in areas with precipitation observations when using the LETKF. This is in contrast with the NGD approach for which often only a small number of members have nonzero weights, except when using the largest value of observation error std dev. This ability of the NGD approach to completely ignore ensemble members locally that do not agree with the observations is potentially an advantage relative to the LETKF approach.

Fig. 7.
Fig. 7.

As in Fig. 6, but showing the localized weight fields computed with the LETKF approach.

Citation: Monthly Weather Review 148, 2; 10.1175/MWR-D-19-0199.1

The analysis and analysis increment of 1-h accumulated precipitation and the increments of water vapor mixing ratio at the model level near 850 hPa are shown in Fig. 8 when using either the NGD or LETKF approach. These results are obtained from using σo = 1 and 256 ensemble members. Similarly, Fig. 9 shows the same results when using σo = 5. With the larger value of observation error std dev, the NGD precipitation analysis becomes smoother and tends to revert to the ensemble mean state. In contrast, the LETKF precipitation analysis does not appear to become smoother, but instead reverts to background state in a more direct way (cf. Figs. 8b and 9b with Fig. 3b). The analysis increments for log-transformed precipitation and water vapor are generally similar when using the NGD or LETKF approach, but with larger amplitudes for NGD, especially for water vapor. Closer examination of the analysis-minus-observation difference (not shown) reveals that the LETKF analysis is able to fit the observations more closely than the NGD analysis when σo = 1, but the opposite is true when σo = 5. It is interesting to note that a consequence of rejecting observations where both the observation and deterministic background state have no precipitation is that the NGD analysis state can contain spuriously large precipitation values in these areas. This is seen in northwestern Georgia when using σo = 1 where high precipitation amounts (Fig. 8a) appear at locations where both the observation (Fig. 3a) and background state (Fig. 3b) have zero precipitation.

Fig. 8.
Fig. 8.

Analysis of the (a),(b) 1-h accumulated precipitation rate (ACC) and increments of (c),(d) log-transformed precipitation and (e),(f) water vapor mixing ratio (WV) at the model level near 850 hPa obtained with the (left) NGD and (right) LETKF approaches that employ a 256-member ensemble and an observation error std dev equal to 1.

Citation: Monthly Weather Review 148, 2; 10.1175/MWR-D-19-0199.1

Fig. 9.
Fig. 9.

As in Fig. 8, but using an observation error std dev equal to 5.

Citation: Monthly Weather Review 148, 2; 10.1175/MWR-D-19-0199.1

The main motivation for exploring a non-Gaussian approach for assimilating radar-derived precipitation measurements originates from the understanding that precipitation and other related variables can have strongly non-Gaussian distributions. To assess the Gaussianity of the distributions being considered, the SciPy library function “normaltest” (Jones et al. 2001; D’Agostino and Pearson 1973) is applied to the 256-member ensemble for several variables at each grid point (Fig. 10). This function provides a nonnegative test statistic, with values increasing from zero as the kurtosis and skewness diverge from that of a Gaussian distribution. Based on this test statistic, the function also provides the probability that the supplied ensemble of values is consistent with a Gaussian distribution.

Fig. 10.
Fig. 10.

(left) The test statistic indicating the degree to which the distribution of the chosen variable differs from Gaussian and (right) the corresponding probability that the distribution is Gaussian. These are shown for the (a),(b) log-transformed precipitation, (c),(d) water vapor mixing ratio, and (e),(f) temperature. Water vapor and temperature are from the model level near 850 hPa.

Citation: Monthly Weather Review 148, 2; 10.1175/MWR-D-19-0199.1

For the log-transformed accumulated precipitation, the resulting test statistic has very high values (Fig. 10a) nearly everywhere where the ensemble has nonzero spread. Corresponding to this, the probability is near zero for testing the null hypothesis that the distribution is Gaussian (Fig. 10b). This indicates that the precipitation, even though it has been log transformed, is strongly non-Gaussian. For the water vapor mixing ratio (Figs. 10c,d) and temperature (Figs. 10e,f), both at the model level near 850 hPa, the test also indicates non-Gaussian distributions, but to a lesser degree. For water vapor, the distribution is consistently non-Gaussian in the areas with precipitation, but more consistent with being Gaussian in other areas. For temperature, the test statistic only has elevated values in a few small regions that may be located near the boundary between areas with and without precipitation.

An example of combining the NGD approach for assimilating radar-derived precipitation accumulations and EnVar for assimilating all other observations by using the NGD analysis as the background state for EnVar is shown in Fig. 11. Results are included from using the two approaches described in section 5: NGD + EnVar and NGD + EnVarBlend. When used independently to correct the same background state, the NGD (Fig. 11a) and EnVar (Fig. 11b) approaches produce water vapor increments (at the model level near 850 hPa) that have some noticeable similarities, especially near the northwest edge of the main area with precipitation (indicated by the gray dashed line). The simple approach for combining the two approaches (NGD + EnVar, Fig. 11c) produces increments that are larger in some areas where precipitation is occurring than when using either of the approaches independently. The reduction in the background error std dev in the improved strategy (NGD + EnVarBlend, Fig. 11d) reduces the increments in these areas so that they resemble more the increment obtained when using NGD alone.

Fig. 11.
Fig. 11.

Water vapor mixing ratio increments at the model level near 850 hPa from (a) assimilating precipitation data with the NGD_512_2 experiment; (b) assimilating all conventional and satellite data with EnVar; (c) using the NGD analysis state as the background state for EnVar (NGD + EnVar); and (d) also using the NGD analysis state as the background state for EnVar, but with the EnVar background error std dev reduced to zero in areas with many nonzero precipitation observations (NGD + EnVarBlend). The gray dashed line indicates the region referred to in the main text.

Citation: Monthly Weather Review 148, 2; 10.1175/MWR-D-19-0199.1

b. Forecast impacts from assimilating radar-derived precipitation accumulations

This section presents results comparing a series of forecasts obtained from using the NGD approach with those obtained from using other DA methods during the month of August 2016. Several conventional scores based on the contingency table, as defined by Schaefer (1990), are used to evaluate the impact on precipitation forecasts of the different approaches for assimilating radar-derived precipitation. Based on the instantaneous precipitation rates estimated from radar reflectivity composites, the following (partially interrelated) scores were computed from 113 forecasts during August 2016: the frequency bias index (FBI), critical success index (CSI), false alarm ratio (FAR), and probability of detection (POD). The scores are shown from 0.5 h before to 9 h after the analysis time. Additional scores, including the minimum radius for a skilled forecast (Dixon et al. 2009), were also examined, but are not presented as they showed a similar overall signal as at least one of the scores listed above.

Figure 12 shows these scores for the NGD and LETKF experiments with σo = 2 and using either ensembles with 256 (6-h forecast) or 512 (6- and 12-h forecast) members. The NoDA experiment is also shown for comparison in this and all subsequent figures of this type. As indicated by the FBI, both DA approaches result in a similar overall reduction of up to approximately 20% in the areal extent of precipitation (with respect to 1 mm h−1 threshold) that develops during the 1-h period while the IAU procedure is active. The other scores all show rapid improvement that develops during the IAU period and then gradually decays until a forecast lead time of approximately 6 or 7 h, when they become similar to the NoDA experiment. The CSI, FAR, and POD scores are all generally improved by a small, but statistically significant,3 amount when using the 512 versus 256 ensemble members. Therefore, all subsequent comparisons only include experiments with 512 members. Perhaps not surprisingly, additional experiments (not shown) using just ensembles of 256 12-h forecast members resulted in degraded precipitation forecasts, relative to those shown.

Fig. 12.
Fig. 12.

Precipitation forecast scores for the NoDA (dotted gray), NGD_256_2 (orange dashed), NGD_512_2 (orange solid), LETKF_256_2 (purple dashed), and LETKF_512_2 (purple solid) experiments. The following scores are based on the contingency table values for the threshold of 1 mm h−1: frequency bias index (FBI), critical success index (CSI), false alarm ratio (FAR), and the probability of detection (POD). The gray shaded area indicates the times during which the IAU procedure is applied using analysis increments computed with either the NGD or LETKF approach.

Citation: Monthly Weather Review 148, 2; 10.1175/MWR-D-19-0199.1

In Fig. 13 the NGD and LETKF approaches are each compared when using three different values for σo (1, 2, and 5 for NGD and 0.5, 1, and 2 for LETKF) with respect to only the FBI and CSI scores. For the NGD approach (Fig. 13a), the FBI has a strong dependence on σo such that larger error std dev corresponds with a larger overall reduction in precipitation. This is likely due to the increased spatial smoothing that results from including more ensemble members in the weighted ensemble average in (10) as the error std dev is increased. The highest CSI is obtained when using σo = 2. For the LETKF approach (Fig. 13b), the relationship between the FBI and σo is weaker than for the NGD approach, though early in the forecast the smallest bias (i.e., FBI closest to 1) is also obtained by using the smallest value of error std dev. At the very beginning of the forecast, the highest CSI is obtained when using σo = 0.5, however the CSI from using σo = 1 is very similar after about 1 h and at longer forecast lead times has a very small, but statistically significant, improvement.

Fig. 13.
Fig. 13.

As in Fig. 12, but only showing the FBI and CSI for (a) NGD_512_1 (purple), NGD_512_2 (green), and NGD_512_5 (orange) experiments and (b) LETKF _512_0.5 (purple), LETKF_512_1 (green), and LETKF_512_2 (orange) experiments.

Citation: Monthly Weather Review 148, 2; 10.1175/MWR-D-19-0199.1

The configurations of the NGD and LETKF approaches with the best overall CSI (when ignoring forecast lead times less than 1 h) in Fig. 13 are compared with each other and with LHN in Fig. 14. Overall, the scores with the NGD and LETKF approaches are similar, with a slight benefit for FAR with NGD and a slight benefit for POD with LETKF. In contrast with the NGD and LETKF approaches, LHN produces a large increase in the FBI during the 1-h period while it is applied. During this 1-h period the CSI and POD also both increase rapidly and attain values higher than with the NGD and LETKF approaches. However, these scores decay more rapidly with LHN, such that beyond a forecast lead time of about 1.5 h and up to about 5 (for POD) or 6 h (for CSI) they are statistically significantly lower than for the other approaches. With respect to FAR, forecasts produced with LHN result in similar minimum values as the other approaches, but with a more rapid convergence toward the values of the NoDA experiment.

Fig. 14.
Fig. 14.

As in Fig. 12, but for the LHN (black), NGD_512_2 (orange), and LETKF_512_1 (purple) experiments. Note that LHN is applied only for the times indicated by the gray shaded area.

Citation: Monthly Weather Review 148, 2; 10.1175/MWR-D-19-0199.1

In addition to examining precipitation forecasts, the assimilation of radar-derived precipitation measurements also has an impact on other important atmospheric variables. Figure 15 shows the impact on temperature and the zonal wind component from assimilating only radar-derived precipitation with the LHN (Figs. 15a,b), NGD (Figs. 15c,d), or LETKF (Figs. 15e,f) approach. The relative change in RMSE with respect to the NoDA experiment for aircraft observations is shown as a function of pressure and forecast lead time. In general, all three approaches result in statistically significant reductions (with the confidence level shown when greater than 90%) of temperature and wind forecast errors for most pressure levels and forecast lead times, except for temperature forecasts during the first few forecast hours when using LHN. The largest improvements for both variables were obtained with the NGD approach. Though not shown here, a comparison between NGD and LETKF also demonstrates a small, but statistically significant benefit with NGD relative to LETKF.

Fig. 15.
Fig. 15.

Forecast scores against aircraft (left) temperature and (right) zonal wind for (a),(b) LHN, (c),(d) NGD_512_2, and (e),(f) LETKF_512_1. In all cases the quantity shown is the relative change (in percentage) in the RMSE compared with the NoDA experiment such that negative values (green) correspond with improved forecasts relative to the NoDA experiment. The numbers in each box, when present, indicate the level of statistical significance of the difference.

Citation: Monthly Weather Review 148, 2; 10.1175/MWR-D-19-0199.1

c. Forecast impacts from assimilating all operational observations in combination with precipitation data

Precipitation forecast scores that compare two strategies for combining the NGD approach with EnVar are shown in Fig. 16. Using EnVar alone to assimilate all conventional and satellite observations has almost no impact on the precipitation scores, except for a decrease in the FBI. Compared with using the NGD approach alone, the simple strategy of using the NGD analysis state as the background state for EnVar (NGD + EnVar) has little impact on the FBI, but has a noticeable negative impact on the other forecasts scores. When the background error std dev of EnVar is reduced in areas where many radar measurements are assimilated, the combination of NGD and EnVar (NGD + EnVarBlend) results in precipitation forecasts with similar skill as when using the NGD approach alone. This suggests that the corrections made by EnVar when assimilating all other observations are not consistent with the corrections made by assimilating radar measurements with NGD.

Fig. 16.
Fig. 16.

As in Fig. 12, but for the EnVar (black dashed), NGD_512_2 (purple), NGD + EnVar (green), and EnVar + EnVarBlend (orange) experiments.

Citation: Monthly Weather Review 148, 2; 10.1175/MWR-D-19-0199.1

The improved strategy for combining the NGD approach with EnVar is compared with using LHN, both alone and in combination with EnVar (Fig. 17). By combining LHN with EnVar, both the CSI and FAR are improved by a statistically significant amount, while the POD is slightly degraded. This is in contrast with the impact of combining NGD with EnVar which did not produce any improvement relative to using NGD alone. Comparing the use of LHN and NGD when both are combined with EnVar still shows some benefits from using the NGD approach that are statistically significant for forecast lead times beyond 1.5 h, especially for the CSI and FAR scores.

Fig. 17.
Fig. 17.

As in Fig. 12, but for the LHN (black), LHN + EnVar (black dashed), and NGD + EnVarBlend (orange) experiments.

Citation: Monthly Weather Review 148, 2; 10.1175/MWR-D-19-0199.1

Similar to Fig. 15, the impact on forecasts of temperature and the zonal wind component as measured with aircraft observations is shown in Fig. 18. The relative change in RMSE with respect to using EnVar alone is shown for the three experiments that combine precipitation assimilation with EnVar. The combination of LHN with EnVar has some negative impacts (relative to EnVar alone) at very short forecast lead times, but becomes mostly positive after about 3 h, especially for zonal wind. The simple strategy for combining the NGD approach with EnVar (NGD + EnVar) results in only small impacts that are somewhat negative near the surface and positive in the upper troposphere as compared with using EnVar alone. The improved strategy for combining the NGD approach with EnVar (NGD + EnVarBlend), while improving the precipitation forecasts (as seen in Fig. 16), results in statistically significant degradations to the temperature and zonal wind forecasts for lead times less than 6 h. This degradation is caused by the reduction of the EnVar background error std dev in areas with precipitation that essentially causes all observations other than precipitation in such areas to be ignored.

Fig. 18.
Fig. 18.

As in Fig. 15, but for (a),(b) LHN + EnVar, (c),(d) NGD + EnVar, and (e),(f) NGD + EnVarBlend experiments. In all cases the quantity shown is the relative change (in percentage) in the RMSE compared with the EnVar experiment such that negative values (green) correspond with improved forecasts relative to the EnVar experiment.

Citation: Monthly Weather Review 148, 2; 10.1175/MWR-D-19-0199.1

8. Conclusions

In this study, a new non-Gaussian approach for deterministic DA is presented and evaluated in the context of radar-derived precipitation assimilation. The new NGD approach produces precipitation forecasts of comparable quality as when applying the LETKF ensemble mean update equation to the deterministic DA problem. This is somewhat surprising since the distribution of the background ensemble can be strongly non-Gaussian in areas with observed precipitation and therefore the LETKF should have difficulty producing accurate analysis increments for the important (and unobserved) model variables. An interesting result of this study is the finding that the NGD experiment that produced the best precipitation forecasts has an effective sample size close to one wherever precipitation is observed. This means the NGD analysis state is locally equal to a weighted average of only a few (and at times only one) ensemble members for all model variables and over all vertical levels within the troposphere. The precipitation forecasts with the NGD approach improve upon those produced with LHN after a lead time of 1.5 h, both when assimilating radar-derived precipitation accumulations alone or in combination with using EnVar for assimilating all other observations. The forecasts of temperature and zonal wind are also improved with the NGD approach relative to LHN, when used in isolation. However, in combination with EnVar, the forecasts of temperature and zonal wind are degraded with the NGD approach as compared with using LHN. This points to the need for an improved strategy for combining the NGD approach with EnVar.

Several new strategies for combining the NGD approach with EnVar may be examined in a future study. One possibility is to use the localized weight fields from the NGD assimilation to modify the ensemble covariances used in EnVar, similar to the procedure described by Slivinski et al. (2015) for combining the particle filter and EnKF. However, it is not clear that such an approach would be successful since the best results with the NGD approach were obtained with a configuration that produced very few members with nonzero weights. Alternatively, the full LETKF algorithm could be used to assimilate radar-derived precipitation and then combined with EnVar by using the updated analysis ensemble produced by the LETKF to specify the EnVar background error covariances. The use of the two-step procedure to combine NGD with EnVar should also be compared against assimilating all observations simultaneously, including radar-derived precipitation, within EnVar. There are likely some benefits from assimilating all observations simultaneously, but it is not clear if these outweigh the potential benefits of using a non-Gaussian approach for assimilating radar measurements. The assimilation of radar-derived precipitation in EnVar could potentially benefit from applying scale-dependent covariance localization (Caron and Buehner 2018), since the amount of localization needed for the related analysis variables (e.g., precipitation, water vapor, cloud condensate) is much higher than for other analysis variables.

As the next step in this research, it is planned to evaluate approaches for assimilating radar-derived precipitation in the more realistic context of 1-hourly cycled experiments, instead of the simplified setting of the present study. It is expected that the improvements from assimilating radar measurements should accumulate to some extent through cycling, though use of the NGD approach in this context will first require establishing an effective strategy for combining it with the assimilation of all other observations. Since the model currently used at ECCC for convective-scale prediction has a horizontal grid spacing of 2.5 km (Milbrandt et al. 2016), future research must also consider how the NGD and other possible approaches for assimilating radar-derived precipitation can be applied to initialize deterministic forecasts with a higher spatial resolution than the ensemble, which will likely remain at 10 km grid spacing for the foreseeable future. While the quality of forecasts from using NGD and Gaussian DA approaches were found to be comparable in the present study, this would need to be reevaluated in the context of a higher-resolution forecast model that can more explicitly resolve convective processes.

Acknowledgments

The initial idea of the NGD approach and motivation for this study benefited from discussions with Jonathan Poterjoy, Chris Snyder, Isztar Zawadzki, and Andrés Pérez Hortal. Stéphane Laroche, Craig Bishop, and two anonymous reviewers provided helpful comments that improved an earlier version of the manuscript. Barbara Casati provided helpful advice on computing statistical significance for the precipitation forecast scores.

REFERENCES

  • Ban, J., Z. Liu, X. Zhang, X.-Y. Huang, and H. Wang, 2017: Precipitation data assimilation in WRFDA 4D-Var: Implementation and application to convection-permitting forecasts over United States. Tellus, 69A, 1368310, https://doi.org/10.1080/16000870.2017.1368310.

    • Search Google Scholar
    • Export Citation

  • Bédard, J., and M. Buehner, 2019: A practical assimilation approach to extract smaller-scale information from observations with spatially correlated errors: An idealized study. Quart. J. Roy. Meteor. Soc., https://doi.org/10.1002/qj.3687, in press.

    • Crossref
    • Export Citation

  • Bick, T., and Coauthors, 2016: Assimilation of 3D radar reflectivities with an ensemble Kalman filter on the convective scale. Quart. J. Roy. Meteor. Soc., 142, 1490–1504, https://doi.org/10.1002/qj.2751.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Bishop, C. H., 2019: Data assimilation strategies for state-dependent observation error variances. Quart. J. Roy. Meteor. Soc., 145, 217–227, https://doi.org/10.1002/qj.3424.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Bishop, C. H., B. J. Etherton, and S. J. Majumdar, 2001: Adaptive sampling with the ensemble transform Kalman filter. Part I: Theoretical aspects. Mon. Wea. Rev., 129, 420–436, https://doi.org/10.1175/1520-0493(2001)129<0420:ASWTET>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Bloom, S. C., L. L. Takacs, A. M. da Silva, and D. Ledvina, 1996: Data assimilation using incremental analysis updates. Mon. Wea. Rev., 124, 1256–1271, https://doi.org/10.1175/1520-0493(1996)124<1256:DAUIAU>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Buehner, M., 2005: Ensemble-derived stationary and flow-dependent background-error covariances: Evaluation in a quasi-operational NWP setting. Quart. J. Roy. Meteor. Soc., 131, 1013–1043, https://doi.org/10.1256/qj.04.15.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Buehner, M., J. Morneau, and C. Charette, 2013: Four-dimensional ensemble-variational data assimilation for global deterministic weather prediction. Nonlinear Processes Geophys., 20, 669–682, https://doi.org/10.5194/npg-20-669-2013.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Buehner, M., and Coauthors, 2015: Implementation of deterministic weather forecasting systems based on ensemble-variational data assimilation at Environment Canada. Part I: The global system. Mon. Wea. Rev., 143, 2532–2559, https://doi.org/10.1175/MWR-D-14-00354.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Caron, J., and M. Buehner, 2018: Scale-dependent background error covariance localization: Evaluation in a global deterministic weather forecasting system. Mon. Wea. Rev., 146, 1367–1381, https://doi.org/10.1175/MWR-D-17-0369.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Caron, J., T. Milewski, M. Buehner, L. Fillion, M. Reszka, S. Macpherson, and J. St-James, 2015: Implementation of deterministic weather forecasting systems based on ensemble–variational data assimilation at Environment Canada. Part II: The regional system. Mon. Wea. Rev., 143, 2560–2580, https://doi.org/10.1175/MWR-D-14-00353.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Caumont, O., V. Ducrocq, É. Wattrelot, G. Jaubert, and S. Pradier-Vabre, 2010: 1D+3DVar assimilation of radar reflectivity data: A proof of concept. Tellus, 62A, 173–187, https://doi.org/10.1111/j.1600-0870.2009.00430.x.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • D’Agostino, R., and E. S. Pearson, 1973: Tests for departure from normality. Biometrika, 60, 613–622, https://doi.org/10.1093/biomet/60.3.613.

    • Search Google Scholar
    • Export Citation

  • Dixon, M., Z. Li, H. Lean, N. Roberts, and S. Ballard, 2009: Impact of data assimilation on forecasting convection over the United Kingdom using a high-resolution version of the Met Office Unified Model. Mon. Wea. Rev., 137, 1562–1584, https://doi.org/10.1175/2008MWR2561.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Duda, J. D., X. Wang, Y. Wang, and J. R. Carley, 2019: Comparing the assimilation of radar reflectivity using the direct GSI-based ensemble–variational (EnVar) and indirect cloud analysis methods in convection-allowing forecasts over the continental United States. Mon. Wea. Rev., 147, 1655–1678, https://doi.org/10.1175/MWR-D-18-0171.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Gao, J., and D. J. Stensrud, 2012: Assimilation of reflectivity data in a convective-scale, cycled 3DVAR framework with hydrometeor classification. J. Atmos. Sci., 69, 1054–1065, https://doi.org/10.1175/JAS-D-11-0162.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Gao, J., M. Xue, and D. J. Stensrud, 2013: The development of a hybrid EnKF-3DVAR algorithm for storm-scale data assimilation. Adv. Meteor., 2013, 512656, https://doi.org/10.1155/2013/512656.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Gaspari, G., and S. E. Cohn, 1999: Construction of correlation functions in two and three dimensions. Quart. J. Roy. Meteor. Soc., 125, 723–757, https://doi.org/10.1002/qj.49712555417.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Girard, C., and Coauthors, 2014: Staggered vertical discretization of the Canadian Environmental Multiscale (GEM) model using a coordinate of the log-hydrostatic-pressure type. Mon. Wea. Rev., 142, 1183–1196, https://doi.org/10.1175/MWR-D-13-00255.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Gustafsson, N., and Coauthors, 2018: Survey of data assimilation methods for convective-scale numerical weather prediction at operational centres. Quart. J. Roy. Meteor. Soc., 144, 1218–1256, https://doi.org/10.1002/qj.3179.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Hamill, T. H., and J. S. Whitaker, 2001: Distance-dependent filtering of background error covariance estimates in an ensemble Kalman filter. Mon. Wea. Rev., 129, 2776–2790, https://doi.org/10.1175/1520-0493(2001)129<2776:DDFOBE>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Houtekamer, P. L., and H. L. Mitchell, 2001: A sequential ensemble Kalman filter for atmospheric data assimilation. Mon. Wea. Rev., 129, 123–137, https://doi.org/10.1175/1520-0493(2001)129<0123:ASEKFF>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Houtekamer, P. L., M. Buehner, and M. De La Chevrotière, 2019: Using the hybrid gain algorithm to sample data assimilation uncertainty. Quart. J. Roy. Meteor. Soc., 145, 35–56, https://doi.org/10.1002/qj.3426.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Hunt, B. R., E. Kostelich, and I. Szunyogh, 2007: Efficient data assimilation for spatiotemporal chaos: A local ensemble transform Kalman filter. Physica D, 230, 112–126, https://doi.org/10.1016/j.physd.2006.11.008.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Jacques, D., D. Michelson, J.-F. Caron, and L. Fillion, 2018: Latent heat nudging in the Canadian regional deterministic prediction system. Mon. Wea. Rev., 146, 3995–4014, https://doi.org/10.1175/MWR-D-18-0118.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Jones, C. D., and B. Macpherson, 1997: A latent heat nudging scheme for assimilation of precipitation data into an operation mesoscale model. Meteor. Appl., 4, 269–277, https://doi.org/10.1017/S1350482797000522.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Jones, E., and Coauthors, 2001: SciPy: Open source scientific tools for Python. SciPy, accessed 21 May 2019, http://www.scipy.org/.

  • Kawabata, T., T. Kuroda, H. Seko, and K. Saito, 2011: A cloud-resolving 4DVAR assimilation experiment for a local heavy rainfall event in the Tokyo metropolitan area. Mon. Wea. Rev., 139, 1911–1931, https://doi.org/10.1175/2011MWR3428.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Kong, A., J. Liu, and W. Wong, 1994: Sequential imputations and Bayesian missing data problems. J. Amer. Stat. Assoc., 89, 278–288, https://doi.org/10.2307/2291224.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Kretschmer, M., B. R. Hunt, and E. Ott, 2015: Data assimilation using a climatologically augmented local ensemble transform Kalman filter. Tellus, 67A, 26617, https://doi.org/10.3402/tellusa.v67.26617.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Lopez, P., 2011: Direct 4D-Var assimilation of NCEP stage IV radar and gauge precipitation data at ECMWF. Mon. Wea. Rev., 139, 2098–2116, https://doi.org/10.1175/2010MWR3565.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Lorenc, A. C., 2003: The potential of the ensemble Kalman filter for NWP—A comparison with 4D-Var. Quart. J. Roy. Meteor. Soc., 129, 3183–3203, https://doi.org/10.1256/qj.02.132.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Lorenc, A. C., 2017: Improving ensemble covariances in hybrid variational data assimilation without increasing ensemble size. Quart. J. Roy. Meteor. Soc., 143, 1062–1072, https://doi.org/10.1002/qj.2990.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Mahfouf, J.-F., B. Brasnett, and S. Gagnon, 2007: A Canadian precipitation analysis (CaPA) project: Description and preliminary results. Atmos.–Ocean, 45, 1–17, https://doi.org/10.3137/ao.v450101.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Milbrandt, J. A., S. Bélair, M. Faucher, M. Vallée, M. L. Carrera, and A. Glazer, 2016: The Pan-Canadian high resolution (2.5 km) deterministic prediction system. Wea. Forecasting, 31, 1791–1816, https://doi.org/10.1175/WAF-D-16-0035.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Posselt, D. J., and C. H. Bishop, 2018: Nonlinear data assimilation for clouds and precipitation using a gamma inverse-gamma ensemble filter. Quart. J. Roy. Meteor. Soc., 144, 2331–2349, https://doi.org/10.1002/qj.3374.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Poterjoy, J., and J. L. Anderson, 2016: Efficient assimilation of simulated observations in a high-dimensional geophysical system using a localized particle filter. Mon. Wea. Rev., 144, 2007–2020, https://doi.org/10.1175/MWR-D-15-0322.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Poterjoy, J., R. A. Sobash, and J. L. Anderson, 2017: Convective-scale data assimilation for the Weather Research and Forecasting Model using the local particle filter. Mon. Wea. Rev., 145, 1897–1918, https://doi.org/10.1175/MWR-D-16-0298.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Poterjoy, J., L. Wicker, and M. Buehner, 2019: Progress toward the application of a localized particle filter for numerical weather prediction. Mon. Wea. Rev., 147, 1107–1126, https://doi.org/10.1175/MWR-D-17-0344.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Potthast, R., A. Walter, and A. Rhodin, 2019: A localized adaptive particle filter within an operational NWP framework. Mon. Wea. Rev., 147, 345–362, https://doi.org/10.1175/MWR-D-18-0028.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Robert, S., D. Leuenberger, and H. Künsch, 2018: A local ensemble transform Kalman particle filter for convective-scale data assimilation. Quart. J. Roy. Meteor. Soc., 144, 1279–1296, https://doi.org/10.1002/qj.3116.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Schaefer, J. T., 1990: The critical success index as an indicator of warning skill. Wea. Forecasting, 5, 570–575, https://doi.org/10.1175/1520-0434(1990)005<0570:TCSIAA>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Schraff, C., H. Reich, A. Rhodin, A. Schomburg, K. Stephan, A. Periáñez, and R. Potthast, 2016: Kilometre-scale ensemble data assimilation for the COSMO model (KENDA). Quart. J. Roy. Meteor. Soc., 142, 1453–1472, https://doi.org/10.1002/qj.2748.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Slivinski, L., E. Spiller, A. Apte, and B. Sandstede, 2015: A hybrid particle–ensemble Kalman filter for Lagrangian data assimilation. Mon. Wea. Rev., 143, 195–211, https://doi.org/10.1175/MWR-D-14-00051.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Sun, J., and N. A. Crook, 1997: Dynamical and microphysical retrieval from Doppler radar observations using a cloud model and its adjoint. Part I: Model development and simulated data experiments. J. Atmos. Sci., 54, 1642–1661, https://doi.org/10.1175/1520-0469(1997)054<1642:DAMRFD>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Sun, J., and Coauthors, 2014: Use of NWP for nowcasting convective precipitation: Recent progress and challenges. Bull. Amer. Meteor. Soc., 95, 409–426, https://doi.org/10.1175/BAMS-D-11-00263.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Surcel, M., I. Zawadzki, and M. K. Yau, 2014: On the filtering properties of ensemble averaging for storm-scale precipitation forecasts. Mon. Wea. Rev., 142, 1093–1105, https://doi.org/10.1175/MWR-D-13-00134.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Tong, M., and M. Xue, 2005: Ensemble Kalman filter assimilation of Doppler radar data with a compressible nonhydrostatic model: OSS experiments. Mon. Wea. Rev., 133, 1789–1807, https://doi.org/10.1175/MWR2898.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Wang, Y., and X. Wang, 2017: Direct assimilation of radar reflectivity without tangent linear and adjoint of the nonlinear observation operator in the GSI-based EnVar system: Methodology and experiment with the 8 May 2003 Oklahoma City tornadic supercell. Mon. Wea. Rev., 145, 1447–1471, https://doi.org/10.1175/MWR-D-16-0231.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Wattrelot, E., O. Caumont, and J. Mahfouf, 2014: Operational implementation of the 1D+3D-Var assimilation method of radar reflectivity data in the AROME model. Mon. Wea. Rev., 142, 1852–1873, https://doi.org/10.1175/MWR-D-13-00230.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Yokota, S., H. Seko, M. Kunii, H. Yamauchi, and E. Sato, 2018: Improving short-term rainfall forecasts by assimilating weather radar reflectivity using additive ensemble perturbations. J. Geophys. Res. Atmos., 123, 9047–9062, https://doi.org/10.1029/2018JD028723.

    • Crossref
    • Search Google Scholar
    • Export Citation
1

However, as shown by Bédard and Buehner (2019), some types of observation error correlations can be accounted for implicitly by additionally assimilating observations after applying spatial differences, without any modification to the DA algorithm.

2

Constant-altitude plan position indicators.

3

Statistical significance was measured for all precipitation verification scores using a Bootstrap approach by resampling 1000 times the differences in the scores between selected pairs of experiments and applying a 90% confidence interval.

Save
Cover Monthly Weather Review
Monthly Weather Review

  • Ban, J., Z. Liu, X. Zhang, X.-Y. Huang, and H. Wang, 2017: Precipitation data assimilation in WRFDA 4D-Var: Implementation and application to convection-permitting forecasts over United States. Tellus, 69A, 1368310, https://doi.org/10.1080/16000870.2017.1368310.

    • Search Google Scholar
    • Export Citation

  • Bédard, J., and M. Buehner, 2019: A practical assimilation approach to extract smaller-scale information from observations with spatially correlated errors: An idealized study. Quart. J. Roy. Meteor. Soc., https://doi.org/10.1002/qj.3687, in press.

    • Crossref
    • Export Citation

  • Bick, T., and Coauthors, 2016: Assimilation of 3D radar reflectivities with an ensemble Kalman filter on the convective scale. Quart. J. Roy. Meteor. Soc., 142, 1490–1504, https://doi.org/10.1002/qj.2751.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Bishop, C. H., 2019: Data assimilation strategies for state-dependent observation error variances. Quart. J. Roy. Meteor. Soc., 145, 217–227, https://doi.org/10.1002/qj.3424.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Bishop, C. H., B. J. Etherton, and S. J. Majumdar, 2001: Adaptive sampling with the ensemble transform Kalman filter. Part I: Theoretical aspects. Mon. Wea. Rev., 129, 420–436, https://doi.org/10.1175/1520-0493(2001)129<0420:ASWTET>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Bloom, S. C., L. L. Takacs, A. M. da Silva, and D. Ledvina, 1996: Data assimilation using incremental analysis updates. Mon. Wea. Rev., 124, 1256–1271, https://doi.org/10.1175/1520-0493(1996)124<1256:DAUIAU>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Buehner, M., 2005: Ensemble-derived stationary and flow-dependent background-error covariances: Evaluation in a quasi-operational NWP setting. Quart. J. Roy. Meteor. Soc., 131, 1013–1043, https://doi.org/10.1256/qj.04.15.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Buehner, M., J. Morneau, and C. Charette, 2013: Four-dimensional ensemble-variational data assimilation for global deterministic weather prediction. Nonlinear Processes Geophys., 20, 669–682, https://doi.org/10.5194/npg-20-669-2013.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Buehner, M., and Coauthors, 2015: Implementation of deterministic weather forecasting systems based on ensemble-variational data assimilation at Environment Canada. Part I: The global system. Mon. Wea. Rev., 143, 2532–2559, https://doi.org/10.1175/MWR-D-14-00354.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Caron, J., and M. Buehner, 2018: Scale-dependent background error covariance localization: Evaluation in a global deterministic weather forecasting system. Mon. Wea. Rev., 146, 1367–1381, https://doi.org/10.1175/MWR-D-17-0369.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Caron, J., T. Milewski, M. Buehner, L. Fillion, M. Reszka, S. Macpherson, and J. St-James, 2015: Implementation of deterministic weather forecasting systems based on ensemble–variational data assimilation at Environment Canada. Part II: The regional system. Mon. Wea. Rev., 143, 2560–2580, https://doi.org/10.1175/MWR-D-14-00353.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Caumont, O., V. Ducrocq, É. Wattrelot, G. Jaubert, and S. Pradier-Vabre, 2010: 1D+3DVar assimilation of radar reflectivity data: A proof of concept. Tellus, 62A, 173–187, https://doi.org/10.1111/j.1600-0870.2009.00430.x.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • D’Agostino, R., and E. S. Pearson, 1973: Tests for departure from normality. Biometrika, 60, 613–622, https://doi.org/10.1093/biomet/60.3.613.

    • Search Google Scholar
    • Export Citation

  • Dixon, M., Z. Li, H. Lean, N. Roberts, and S. Ballard, 2009: Impact of data assimilation on forecasting convection over the United Kingdom using a high-resolution version of the Met Office Unified Model. Mon. Wea. Rev., 137, 1562–1584, https://doi.org/10.1175/2008MWR2561.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Duda, J. D., X. Wang, Y. Wang, and J. R. Carley, 2019: Comparing the assimilation of radar reflectivity using the direct GSI-based ensemble–variational (EnVar) and indirect cloud analysis methods in convection-allowing forecasts over the continental United States. Mon. Wea. Rev., 147, 1655–1678, https://doi.org/10.1175/MWR-D-18-0171.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Gao, J., and D. J. Stensrud, 2012: Assimilation of reflectivity data in a convective-scale, cycled 3DVAR framework with hydrometeor classification. J. Atmos. Sci., 69, 1054–1065, https://doi.org/10.1175/JAS-D-11-0162.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Gao, J., M. Xue, and D. J. Stensrud, 2013: The development of a hybrid EnKF-3DVAR algorithm for storm-scale data assimilation. Adv. Meteor., 2013, 512656, https://doi.org/10.1155/2013/512656.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Gaspari, G., and S. E. Cohn, 1999: Construction of correlation functions in two and three dimensions. Quart. J. Roy. Meteor. Soc., 125, 723–757, https://doi.org/10.1002/qj.49712555417.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Girard, C., and Coauthors, 2014: Staggered vertical discretization of the Canadian Environmental Multiscale (GEM) model using a coordinate of the log-hydrostatic-pressure type. Mon. Wea. Rev., 142, 1183–1196, https://doi.org/10.1175/MWR-D-13-00255.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Gustafsson, N., and Coauthors, 2018: Survey of data assimilation methods for convective-scale numerical weather prediction at operational centres. Quart. J. Roy. Meteor. Soc., 144, 1218–1256, https://doi.org/10.1002/qj.3179.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Hamill, T. H., and J. S. Whitaker, 2001: Distance-dependent filtering of background error covariance estimates in an ensemble Kalman filter. Mon. Wea. Rev., 129, 2776–2790, https://doi.org/10.1175/1520-0493(2001)129<2776:DDFOBE>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Houtekamer, P. L., and H. L. Mitchell, 2001: A sequential ensemble Kalman filter for atmospheric data assimilation. Mon. Wea. Rev., 129, 123–137, https://doi.org/10.1175/1520-0493(2001)129<0123:ASEKFF>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Houtekamer, P. L., M. Buehner, and M. De La Chevrotière, 2019: Using the hybrid gain algorithm to sample data assimilation uncertainty. Quart. J. Roy. Meteor. Soc., 145, 35–56, https://doi.org/10.1002/qj.3426.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Hunt, B. R., E. Kostelich, and I. Szunyogh, 2007: Efficient data assimilation for spatiotemporal chaos: A local ensemble transform Kalman filter. Physica D, 230, 112–126, https://doi.org/10.1016/j.physd.2006.11.008.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Jacques, D., D. Michelson, J.-F. Caron, and L. Fillion, 2018: Latent heat nudging in the Canadian regional deterministic prediction system. Mon. Wea. Rev., 146, 3995–4014, https://doi.org/10.1175/MWR-D-18-0118.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Jones, C. D., and B. Macpherson, 1997: A latent heat nudging scheme for assimilation of precipitation data into an operation mesoscale model. Meteor. Appl., 4, 269–277, https://doi.org/10.1017/S1350482797000522.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Jones, E., and Coauthors, 2001: SciPy: Open source scientific tools for Python. SciPy, accessed 21 May 2019, http://www.scipy.org/.

  • Kawabata, T., T. Kuroda, H. Seko, and K. Saito, 2011: A cloud-resolving 4DVAR assimilation experiment for a local heavy rainfall event in the Tokyo metropolitan area. Mon. Wea. Rev., 139, 1911–1931, https://doi.org/10.1175/2011MWR3428.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Kong, A., J. Liu, and W. Wong, 1994: Sequential imputations and Bayesian missing data problems. J. Amer. Stat. Assoc., 89, 278–288, https://doi.org/10.2307/2291224.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Kretschmer, M., B. R. Hunt, and E. Ott, 2015: Data assimilation using a climatologically augmented local ensemble transform Kalman filter. Tellus, 67A, 26617, https://doi.org/10.3402/tellusa.v67.26617.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Lopez, P., 2011: Direct 4D-Var assimilation of NCEP stage IV radar and gauge precipitation data at ECMWF. Mon. Wea. Rev., 139, 2098–2116, https://doi.org/10.1175/2010MWR3565.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Lorenc, A. C., 2003: The potential of the ensemble Kalman filter for NWP—A comparison with 4D-Var. Quart. J. Roy. Meteor. Soc., 129, 3183–3203, https://doi.org/10.1256/qj.02.132.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Lorenc, A. C., 2017: Improving ensemble covariances in hybrid variational data assimilation without increasing ensemble size. Quart. J. Roy. Meteor. Soc., 143, 1062–1072, https://doi.org/10.1002/qj.2990.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Mahfouf, J.-F., B. Brasnett, and S. Gagnon, 2007: A Canadian precipitation analysis (CaPA) project: Description and preliminary results. Atmos.–Ocean, 45, 1–17, https://doi.org/10.3137/ao.v450101.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Milbrandt, J. A., S. Bélair, M. Faucher, M. Vallée, M. L. Carrera, and A. Glazer, 2016: The Pan-Canadian high resolution (2.5 km) deterministic prediction system. Wea. Forecasting, 31, 1791–1816, https://doi.org/10.1175/WAF-D-16-0035.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Posselt, D. J., and C. H. Bishop, 2018: Nonlinear data assimilation for clouds and precipitation using a gamma inverse-gamma ensemble filter. Quart. J. Roy. Meteor. Soc., 144, 2331–2349, https://doi.org/10.1002/qj.3374.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Poterjoy, J., and J. L. Anderson, 2016: Efficient assimilation of simulated observations in a high-dimensional geophysical system using a localized particle filter. Mon. Wea. Rev., 144, 2007–2020, https://doi.org/10.1175/MWR-D-15-0322.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Poterjoy, J., R. A. Sobash, and J. L. Anderson, 2017: Convective-scale data assimilation for the Weather Research and Forecasting Model using the local particle filter. Mon. Wea. Rev., 145, 1897–1918, https://doi.org/10.1175/MWR-D-16-0298.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Poterjoy, J., L. Wicker, and M. Buehner, 2019: Progress toward the application of a localized particle filter for numerical weather prediction. Mon. Wea. Rev., 147, 1107–1126, https://doi.org/10.1175/MWR-D-17-0344.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Potthast, R., A. Walter, and A. Rhodin, 2019: A localized adaptive particle filter within an operational NWP framework. Mon. Wea. Rev., 147, 345–362, https://doi.org/10.1175/MWR-D-18-0028.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Robert, S., D. Leuenberger, and H. Künsch, 2018: A local ensemble transform Kalman particle filter for convective-scale data assimilation. Quart. J. Roy. Meteor. Soc., 144, 1279–1296, https://doi.org/10.1002/qj.3116.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Schaefer, J. T., 1990: The critical success index as an indicator of warning skill. Wea. Forecasting, 5, 570–575, https://doi.org/10.1175/1520-0434(1990)005<0570:TCSIAA>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Schraff, C., H. Reich, A. Rhodin, A. Schomburg, K. Stephan, A. Periáñez, and R. Potthast, 2016: Kilometre-scale ensemble data assimilation for the COSMO model (KENDA). Quart. J. Roy. Meteor. Soc., 142, 1453–1472, https://doi.org/10.1002/qj.2748.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Slivinski, L., E. Spiller, A. Apte, and B. Sandstede, 2015: A hybrid particle–ensemble Kalman filter for Lagrangian data assimilation. Mon. Wea. Rev., 143, 195–211, https://doi.org/10.1175/MWR-D-14-00051.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Sun, J., and N. A. Crook, 1997: Dynamical and microphysical retrieval from Doppler radar observations using a cloud model and its adjoint. Part I: Model development and simulated data experiments. J. Atmos. Sci., 54, 1642–1661, https://doi.org/10.1175/1520-0469(1997)054<1642:DAMRFD>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Sun, J., and Coauthors, 2014: Use of NWP for nowcasting convective precipitation: Recent progress and challenges. Bull. Amer. Meteor. Soc., 95, 409–426, https://doi.org/10.1175/BAMS-D-11-00263.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Surcel, M., I. Zawadzki, and M. K. Yau, 2014: On the filtering properties of ensemble averaging for storm-scale precipitation forecasts. Mon. Wea. Rev., 142, 1093–1105, https://doi.org/10.1175/MWR-D-13-00134.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Tong, M., and M. Xue, 2005: Ensemble Kalman filter assimilation of Doppler radar data with a compressible nonhydrostatic model: OSS experiments. Mon. Wea. Rev., 133, 1789–1807, https://doi.org/10.1175/MWR2898.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Wang, Y., and X. Wang, 2017: Direct assimilation of radar reflectivity without tangent linear and adjoint of the nonlinear observation operator in the GSI-based EnVar system: Methodology and experiment with the 8 May 2003 Oklahoma City tornadic supercell. Mon. Wea. Rev., 145, 1447–1471, https://doi.org/10.1175/MWR-D-16-0231.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Wattrelot, E., O. Caumont, and J. Mahfouf, 2014: Operational implementation of the 1D+3D-Var assimilation method of radar reflectivity data in the AROME model. Mon. Wea. Rev., 142, 1852–1873, https://doi.org/10.1175/MWR-D-13-00230.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Yokota, S., H. Seko, M. Kunii, H. Yamauchi, and E. Sato, 2018: Improving short-term rainfall forecasts by assimilating weather radar reflectivity using additive ensemble perturbations. J. Geophys. Res. Atmos., 123, 9047–9062, https://doi.org/10.1029/2018JD028723.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • <sc>Fig</sc>. 1.
    View in gallery
    Fig. 1.

    (a) The model surface height field over the limited-area model domain used in this study. The gray box in this and similar later figures indicates the boundary of the inner domain where the model evolves free of direct influence from the LBCs. (b) The spatial coverage of radar data within the model domain shown in terms of the quality index for the precipitation rate estimated from radar reflectivity at 1200 UTC 15 Aug 2016.

  • <sc>Fig</sc>. 2.
    View in gallery
    Fig. 2.

    Schematic of setup for generating (a) the deterministic background states, (b) the forecasts initialized with analysis increments computed with NGD, LETKF, or EnVar DA approaches, and (c) the ensemble forecasts. The LHN approach is applied within the same 1-h time window, shown in (b), used for initializing forecasts with NGD, LETKF, and EnVar analysis increments.

  • <sc>Fig</sc>. 3.
    View in gallery
    Fig. 3.

    (a) Observation, (b) background state, and background ensemble (c) mean and (d) spread std dev at 1200 UTC 15 Aug 2016. The observations, background state, and background ensemble mean are shown for the 1-h accumulated precipitation, whereas the background ensemble spread is shown for the log-transformed accumulated precipitation.

  • <sc>Fig</sc>. 4.
    View in gallery
    Fig. 4.

    (a),(c) Normalized effective sample size Neff and (b),(d) the weight given to the deterministic background state in the NGD analysis when using an observation error std dev equal to either (top) 1 or (bottom) 5.

  • <sc>Fig</sc>. 5.
    View in gallery
    Fig. 5.

    Individual localized weight fields computed with the NGD approach for each member of a 16-member ensemble when using an observation error std dev equal to 2. The hatched area denotes where the total weight on the ensemble members is greater than 0.5. The horizontal line near the center of the domain indicates the locations shown in the following two figures.

  • <sc>Fig</sc>. 6.
    View in gallery
    Fig. 6.

    Detailed view of the localized weight fields at the locations corresponding to the horizontal line in the previous figure. Weights are computed with the NGD approach with a 16-member ensemble when using an observation error std dev equal to (a) 1, (b) 2, (c) 5, and (d) 10. The dashed gray curve is the total contribution of the weighted ensemble average to the final NGD analysis.

  • <sc>Fig</sc>. 7.
    View in gallery
    Fig. 7.

    As in Fig. 6, but showing the localized weight fields computed with the LETKF approach.

  • <sc>Fig</sc>. 8.
    View in gallery
    Fig. 8.

    Analysis of the (a),(b) 1-h accumulated precipitation rate (ACC) and increments of (c),(d) log-transformed precipitation and (e),(f) water vapor mixing ratio (WV) at the model level near 850 hPa obtained with the (left) NGD and (right) LETKF approaches that employ a 256-member ensemble and an observation error std dev equal to 1.

  • <sc>Fig</sc>. 9.
    View in gallery
    Fig. 9.

    As in Fig. 8, but using an observation error std dev equal to 5.

  • <sc>Fig</sc>. 10.
    View in gallery
    Fig. 10.

    (left) The test statistic indicating the degree to which the distribution of the chosen variable differs from Gaussian and (right) the corresponding probability that the distribution is Gaussian. These are shown for the (a),(b) log-transformed precipitation, (c),(d) water vapor mixing ratio, and (e),(f) temperature. Water vapor and temperature are from the model level near 850 hPa.

  • <sc>Fig</sc>. 11.
    View in gallery
    Fig. 11.

    Water vapor mixing ratio increments at the model level near 850 hPa from (a) assimilating precipitation data with the NGD_512_2 experiment; (b) assimilating all conventional and satellite data with EnVar; (c) using the NGD analysis state as the background state for EnVar (NGD + EnVar); and (d) also using the NGD analysis state as the background state for EnVar, but with the EnVar background error std dev reduced to zero in areas with many nonzero precipitation observations (NGD + EnVarBlend). The gray dashed line indicates the region referred to in the main text.

  • <sc>Fig</sc>. 12.
    View in gallery
    Fig. 12.

    Precipitation forecast scores for the NoDA (dotted gray), NGD_256_2 (orange dashed), NGD_512_2 (orange solid), LETKF_256_2 (purple dashed), and LETKF_512_2 (purple solid) experiments. The following scores are based on the contingency table values for the threshold of 1 mm h−1: frequency bias index (FBI), critical success index (CSI), false alarm ratio (FAR), and the probability of detection (POD). The gray shaded area indicates the times during which the IAU procedure is applied using analysis increments computed with either the NGD or LETKF approach.

  • <sc>Fig</sc>. 13.
    View in gallery
    Fig. 13.

    As in Fig. 12, but only showing the FBI and CSI for (a) NGD_512_1 (purple), NGD_512_2 (green), and NGD_512_5 (orange) experiments and (b) LETKF _512_0.5 (purple), LETKF_512_1 (green), and LETKF_512_2 (orange) experiments.

  • <sc>Fig</sc>. 14.
    View in gallery
    Fig. 14.

    As in Fig. 12, but for the LHN (black), NGD_512_2 (orange), and LETKF_512_1 (purple) experiments. Note that LHN is applied only for the times indicated by the gray shaded area.

  • <sc>Fig</sc>. 15.
    View in gallery
    Fig. 15.

    Forecast scores against aircraft (left) temperature and (right) zonal wind for (a),(b) LHN, (c),(d) NGD_512_2, and (e),(f) LETKF_512_1. In all cases the quantity shown is the relative change (in percentage) in the RMSE compared with the NoDA experiment such that negative values (green) correspond with improved forecasts relative to the NoDA experiment. The numbers in each box, when present, indicate the level of statistical significance of the difference.

  • <sc>Fig</sc>. 16.
    View in gallery
    Fig. 16.

    As in Fig. 12, but for the EnVar (black dashed), NGD_512_2 (purple), NGD + EnVar (green), and EnVar + EnVarBlend (orange) experiments.

  • <sc>Fig</sc>. 17.
    View in gallery
    Fig. 17.

    As in Fig. 12, but for the LHN (black), LHN + EnVar (black dashed), and NGD + EnVarBlend (orange) experiments.

  • <sc>Fig</sc>. 18.
    View in gallery
    Fig. 18.

    As in Fig. 15, but for (a),(b) LHN + EnVar, (c),(d) NGD + EnVar, and (e),(f) NGD + EnVarBlend experiments. In all cases the quantity shown is the relative change (in percentage) in the RMSE compared with the EnVar experiment such that negative values (green) correspond with improved forecasts relative to the EnVar experiment.

Fig. 1.

(a) The model surface height field over the limited-area model domain used in this study. The gray box in this and similar later figures indicates the boundary of the inner domain where the model evolves free of direct influence from the LBCs. (b) The spatial coverage of radar data within the model domain shown in terms of the quality index for the precipitation rate estimated from radar reflectivity at 1200 UTC 15 Aug 2016.
Fig. 2.

Schematic of setup for generating (a) the deterministic background states, (b) the forecasts initialized with analysis increments computed with NGD, LETKF, or EnVar DA approaches, and (c) the ensemble forecasts. The LHN approach is applied within the same 1-h time window, shown in (b), used for initializing forecasts with NGD, LETKF, and EnVar analysis increments.
Fig. 3.

(a) Observation, (b) background state, and background ensemble (c) mean and (d) spread std dev at 1200 UTC 15 Aug 2016. The observations, background state, and background ensemble mean are shown for the 1-h accumulated precipitation, whereas the background ensemble spread is shown for the log-transformed accumulated precipitation.
Fig. 4.

(a),(c) Normalized effective sample size Neff and (b),(d) the weight given to the deterministic background state in the NGD analysis when using an observation error std dev equal to either (top) 1 or (bottom) 5.
Fig. 5.

Individual localized weight fields computed with the NGD approach for each member of a 16-member ensemble when using an observation error std dev equal to 2. The hatched area denotes where the total weight on the ensemble members is greater than 0.5. The horizontal line near the center of the domain indicates the locations shown in the following two figures.
Fig. 6.

Detailed view of the localized weight fields at the locations corresponding to the horizontal line in the previous figure. Weights are computed with the NGD approach with a 16-member ensemble when using an observation error std dev equal to (a) 1, (b) 2, (c) 5, and (d) 10. The dashed gray curve is the total contribution of the weighted ensemble average to the final NGD analysis.
Fig. 7.

As in Fig. 6, but showing the localized weight fields computed with the LETKF approach.
Fig. 8.

Analysis of the (a),(b) 1-h accumulated precipitation rate (ACC) and increments of (c),(d) log-transformed precipitation and (e),(f) water vapor mixing ratio (WV) at the model level near 850 hPa obtained with the (left) NGD and (right) LETKF approaches that employ a 256-member ensemble and an observation error std dev equal to 1.
Fig. 9.

As in Fig. 8, but using an observation error std dev equal to 5.
Fig. 10.

(left) The test statistic indicating the degree to which the distribution of the chosen variable differs from Gaussian and (right) the corresponding probability that the distribution is Gaussian. These are shown for the (a),(b) log-transformed precipitation, (c),(d) water vapor mixing ratio, and (e),(f) temperature. Water vapor and temperature are from the model level near 850 hPa.
Fig. 11.

Water vapor mixing ratio increments at the model level near 850 hPa from (a) assimilating precipitation data with the NGD_512_2 experiment; (b) assimilating all conventional and satellite data with EnVar; (c) using the NGD analysis state as the background state for EnVar (NGD + EnVar); and (d) also using the NGD analysis state as the background state for EnVar, but with the EnVar background error std dev reduced to zero in areas with many nonzero precipitation observations (NGD + EnVarBlend). The gray dashed line indicates the region referred to in the main text.
Fig. 12.

Precipitation forecast scores for the NoDA (dotted gray), NGD_256_2 (orange dashed), NGD_512_2 (orange solid), LETKF_256_2 (purple dashed), and LETKF_512_2 (purple solid) experiments. The following scores are based on the contingency table values for the threshold of 1 mm h−1: frequency bias index (FBI), critical success index (CSI), false alarm ratio (FAR), and the probability of detection (POD). The gray shaded area indicates the times during which the IAU procedure is applied using analysis increments computed with either the NGD or LETKF approach.
Fig. 13.

As in Fig. 12, but only showing the FBI and CSI for (a) NGD_512_1 (purple), NGD_512_2 (green), and NGD_512_5 (orange) experiments and (b) LETKF _512_0.5 (purple), LETKF_512_1 (green), and LETKF_512_2 (orange) experiments.
Fig. 14.

As in Fig. 12, but for the LHN (black), NGD_512_2 (orange), and LETKF_512_1 (purple) experiments. Note that LHN is applied only for the times indicated by the gray shaded area.
Fig. 15.

Forecast scores against aircraft (left) temperature and (right) zonal wind for (a),(b) LHN, (c),(d) NGD_512_2, and (e),(f) LETKF_512_1. In all cases the quantity shown is the relative change (in percentage) in the RMSE compared with the NoDA experiment such that negative values (green) correspond with improved forecasts relative to the NoDA experiment. The numbers in each box, when present, indicate the level of statistical significance of the difference.
Fig. 16.

As in Fig. 12, but for the EnVar (black dashed), NGD_512_2 (purple), NGD + EnVar (green), and EnVar + EnVarBlend (orange) experiments.
Fig. 17.

As in Fig. 12, but for the LHN (black), LHN + EnVar (black dashed), and NGD + EnVarBlend (orange) experiments.
Fig. 18.

As in Fig. 15, but for (a),(b) LHN + EnVar, (c),(d) NGD + EnVar, and (e),(f) NGD + EnVarBlend experiments. In all cases the quantity shown is the relative change (in percentage) in the RMSE compared with the EnVar experiment such that negative values (green) correspond with improved forecasts relative to the EnVar experiment.
All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 2113 1236 351
PDF Downloads 872 131 5