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  • View in gallery

    Scheme for the evaluation of the properties along a slant path. The slant profile is built as a departure from a column profile, given gradients of each relevant quantity in the neighborhood, and the geometry of the slant path. This path departs from the vertical by a zenith angle θ, along azimuth φ, measured from the footprint toward the satellite.

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    Observed brightness temperature in the 60°–90°N latitude region, for ATMS channels (a) 9 and (c) 15 for a 6-h period centered at 0000 UTC 1 Jan 2017. The difference in the simulation of these observations, when using vertical and slant profiles is shown for channels (b) 9 and (d) 15.

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    Standard deviation of OMB brightness temperatures evaluated from slant profiles, normalized by the standard deviation of OMB using vertical profiles, for (a) ATMS channels and (c) AMSU-A NOAA-18 channels. The variation with zenith angle of the standard deviation normalized to the nadir view is shown for (b) channel 9 of ATMS and (d) channel 9 of AMSU-A. The statistics were calculated over different latitudinal bands: global (black), high latitude (red), midlatitude (blue), and tropics (green). The validation is from 1 to 11 Jan 2017, representing 0.5 million observations for the global data.

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    Evolution of the standard deviation of OMB (K) for channel 9 of AMSU-A on board MetOp-1 satellite. The horizontal axis is the analysis time. The observations are used after bias correction. The control and modified experiments are labeled as G252 (blue) and FB (red). The overall standard deviation is marked with a dot on the right of the figure for each experiment.

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    Standard deviation of ATMS channel 9 OMB (K), for the control G252 (blue) and modified FB (red) experiments, as a function of scan positions. The statistics are calculated based on bias-corrected global observations for the period 1–31 Jan 2017.

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    (a) Ratio of operational OMB standard deviation (, historical source to generate the original a priori errors in Table 3) against the standard deviation of OMB of the control 1 (G252, blue curve) and the slant 1 experiment (FB, red curve), for AMSU-A on board NOAA-18. The OMB standard deviations over control 1 and slant 1 were calculated over the 1–31 Jan 2017 period. (b) As in (a), but for ATMS instrument.

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    Comparison of the experiments slant 1 (FB) and slant 2 (FBSA) against control 1 (G252). Plots show the mean of the normalized difference, in percent, of the standard deviation of the geopotential height field against its own analysis, as a function of the pressure level and forecast range. (a) Global mean, boreal winter, slant 1 (FB) vs control 1 (G252), . (b) As in (a), but for the North Pole (60°–90°N). (c) Global mean, boreal winter, slant 2 (FBSA) vs control 1 (G252). (d) As in (c), but for the North Pole. See the description of each experiment in Table 2. Red (blue) is the percentage decrease (increase) in standard deviation of experiment compared to the control. Red thus indicates a better performance of the gradient experiments. Black dots mark statistical significance above the 95% confidence level, based on a Fisher F test.

  • View in gallery

    Comparison of the experiments slant 2 (FBSA) against control 2 (S2), for boreal (top) winter and (bottom) summer. Plots show the mean of the normalized difference, in percent of the standard deviation of the geopotential height field, against its own analysis (as in Fig. 7). (a) Global mean, boreal winter. (b) As in (a), but for the North Pole (60°–90°N). (c) Global mean, boreal summer. (d) As in (c), but for the South Pole region (austral winter). See the description of each experiment in Table 2. Red (blue) is the percentage decrease (increase) in standard deviation of experiment compared to the control. Red thus indicates a better performance of the gradient experiments. Black dots mark statistical significance above the 95% confidence level, based on a Fisher F test.

  • View in gallery

    (a) Zonal mean of the normalized difference in the standard deviation (against its own analysis) of the geopotential height field in boreal summer (15 Jun–31 Aug 2016), between the experiment slant 2 (FBSA) and control 2 (S2), for the 12-h forecast: . These runs use updated error statistics, see Table 2. Red (blue) is the percentage decrease (increase) in standard deviation of experiment compared to the control. Red thus indicates a better performance of the gradient experiment. Black dots mark statistical significance above the 95% confidence level, based on a Fisher F test. (b)–(d) As in (a), but at 24-, 36-, and 48-h forecast.

  • View in gallery

    As in Fig. 9, but for the boreal winter (15 Dec 2016–28 Feb 2017).

  • View in gallery

    Normalized standard deviation of differences between the slant 2 (FBSA) experiment against the independent SABER observations for temperature at different forecast hours: . Red indicates a better performance of the gradient experiment. The statistics cover the period between 1 Jan and 28 Feb 2017, for global data. Around 3 × 106 SABER observations are used. Black dots mark statistical significance above the 95% confidence level, based on a Fisher F test.

  • View in gallery

    Scores of experiment slant 3 (FBSA2) against its control 2 (S2) for boreal winter. (a),(b) As in Figs. 8a and 8b, but evaluated over the global domain and North Pole region, respectively, but local horizontal gradients are obtained over domains of 50-km radius (instead of 100 km), thus they are less filtered against higher spatial frequencies.

  • View in gallery

    Bias (dashed curves) and standard deviation (solid curves) of the geopotential height from slant 3 (FBSA2, red) experiment, relative to slant 2 (FBSA, blue), when compared against radiosondes. (top) Global domain (about 55 000 radiosondes) and (bottom) for the northern circumpolar region (60°–90°N, about 6500 radiosondes). (a),(c) 144-h forecasts and (b),(d) 192-h forecasts. Numbers within the shaded boxes on the left (right) are the level of confidence, in percent, that the biases (standard deviation) from the two experiments are different. The color shading indicates which experiment has a better score: red for better performance of slant 3 (FBSA2: gradients evaluated over a smaller domain).

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Impact of Slant-Path Radiative Transfer in the Simulation and Assimilation of Satellite Radiances in Environment Canada’s Weather Forecast System

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  • 1 Environment and Climate Change Canada, Dorval, Quebec, Canada
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Abstract

Slant satellite-viewing geometry is investigated for the simulation and assimilation of radiances in Environment Canada’s weather forecast system. The standard approach is to extract from a short-term forecast (trial field) a one-dimensional vertical profile, located at the ground footprint of the observation location, to compute the model equivalent of the observation. Since in general, the lines of sight are not vertical, the observation operator can be improved by interpolating the trial field to the slant path. The interpolation relies on the estimation of horizontal gradients of atmospheric variables, extracted at each altitude layer, and representative of a neighborhood of approximately 100 km. Using these slant profiles of the trial fields yields significant improvements in the simulation and assimilation of radiance observations. The impact of this modification in assimilation and forecasts is evaluated over two periods of two months. Both short-term and long-term effects are found, obtained not only from the more accurate interpolation of the trial fields, but also from a revision of the observation error suggested by the lower observation minus trial field difference statistics. The error standard deviation of the geopotential height shows, in the upper troposphere and lower stratosphere, in 12–48-h forecasts, up to 10% reduction in the zonal value at higher latitudes, and 1%–2% reduction in the global value. Given these improved statistics, a revision of the radiance error statistics was explored. Applying these revised statistics suggests that the impact of a slant operator extends into longer range.

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: Josep M. Aparicio, josep.aparicio@canada.ca

Abstract

Slant satellite-viewing geometry is investigated for the simulation and assimilation of radiances in Environment Canada’s weather forecast system. The standard approach is to extract from a short-term forecast (trial field) a one-dimensional vertical profile, located at the ground footprint of the observation location, to compute the model equivalent of the observation. Since in general, the lines of sight are not vertical, the observation operator can be improved by interpolating the trial field to the slant path. The interpolation relies on the estimation of horizontal gradients of atmospheric variables, extracted at each altitude layer, and representative of a neighborhood of approximately 100 km. Using these slant profiles of the trial fields yields significant improvements in the simulation and assimilation of radiance observations. The impact of this modification in assimilation and forecasts is evaluated over two periods of two months. Both short-term and long-term effects are found, obtained not only from the more accurate interpolation of the trial fields, but also from a revision of the observation error suggested by the lower observation minus trial field difference statistics. The error standard deviation of the geopotential height shows, in the upper troposphere and lower stratosphere, in 12–48-h forecasts, up to 10% reduction in the zonal value at higher latitudes, and 1%–2% reduction in the global value. Given these improved statistics, a revision of the radiance error statistics was explored. Applying these revised statistics suggests that the impact of a slant operator extends into longer range.

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: Josep M. Aparicio, josep.aparicio@canada.ca

1. Introduction

Radiance observations are one of the most important sources of data for numerical weather prediction (NWP) (Cardinali 2009; Buehner et al. 2018). It is therefore important to optimize the use of these observations in operational data assimilation systems. Improvement to the observation operator used in satellite data assimilation is the subject of this paper. The standard approach in NWP is to perform radiative transfer calculations on interpolated one-dimensional (1D) vertical columns of the background model fields, at the location of the observation, and to produce from them simulated estimations of these radiance observations. The difference between the observation and its model equivalent, referred to as innovation, is then assimilated, resulting in a modified atmospheric field more compatible with the observations. Under this standard approach, slant observations are assumed to depend on the viewing angle, but only through a geometric dependence of the optical depth; the background field is assumed to be locally horizontally uniform and spherically symmetric. For channels with mean sensitivity in the stratosphere, this translates into significant horizontal displacements between the vertical and slant line of sight. For example, for a channel with maximum sensitivity around 15 km and a satellite observation at a 60° zenith angle, the horizontal displacement is 26 km. This corresponds to one grid point at the current resolution of the Canadian global model.

The impact of atmospheric horizontal gradients on slant-path radiative transfer simulations of radiances has been the subject of previous studies. Poli et al. (2005) and Joiner and Poli (2005) analyzed the case of Atmospheric Infrared Sounder (AIRS) and Advanced Microwave Sounding Unit-A (AMSU-A). They discuss that the effect of the presence of atmospheric horizontal gradients on these sounders’ data is generally small, and below the instrument noise level for most of the channels. The impact would be limited to high-peaking (sensitivity at high levels) channels at large zenith angles. Felder et al. (2007) shows that ozone horizontal gradients can cause up to 0.5% RMS error in the calculated radiance from the Solar Backscatter Ultraviolet/2 instrument. Since these studies, the accuracy and resolution of NWP fields has improved, and as a result, horizontal gradients from model fields are better depicted. In addition, recent sounders have lower noise levels, for instance, the Cross-Track Infrared Sounder (CrIS; Likun et al. 2013), and scan wider swaths, such as the Advanced Technology Microwave Sounder (ATMS; Kim et al. 2014).

Benefits have been found of constructing a forward model that retains more extensive information of the local background field, not only the vertical structure, but also along the horizontal azimuth, in simulations (Bormann et al. 2007) of limb observations of radiances from the Michelson Interferometer for Passive Atmospheric Sounding (MIPAS), and of GPS radio occultation bending angles (Healy et al. 2007). In those studies, a two-dimensional (2D) interpolator was used, within the observation operator, which constructs a series of vertical profiles along the observation azimuthal plane, to provide a richer representation of the local state of the atmosphere. A similar 2D interpolator was used in Bormann (2017), to evaluate the impact of slant-path radiative transfer on radiance assimilation. Improved simulations of radiances were reported for high-peaking temperature-sounding channels at mid- and high latitudes, and for high-peaking humidity-sounding channels at midlatitudes. Those results also indicate improvements lasting up to 3 days in forecasts in the upper troposphere and stratosphere, especially at high latitudes. That study also suggests that whereas model fields can capture horizontal structures representing various spatial scales, only the larger scales may be relevant for the purpose of describing a slant path compatible with the horizontal resolution of the observations, and their vertical sensitivity.

Similarly, the local variability of the atmosphere is also noticeable from radiosonde observations (Laroche and Sarrazin 2013). Besides variations in time, since the radiosonde’s ascent is slow, the ensemble of observations in a balloon launch are also geographically nonlocal, because of horizontal drift with winds, and proper consideration of these effects was found to lead to a noticeable positive impact in assimilation accuracy and forecast performance.

In this paper, the impact of local horizontal variability of temperature T, pressure P, and specific humidity q on the slant-path radiative transfer, in the simulation and assimilation of satellite radiances, is explored using the Environment and Climate Change Canada (ECCC) weather forecast system (Charron et al. 2012; Buehner et al. 2015). It is proposed to focus on information that is additional to that contained in a standard background 1D profile, namely to add a 1D profile of horizontal gradients, allowing interpolation on the slant path. The atmosphere contains variability at many scales, from unresolved subgrid scales, to planetary. Since the horizontal region involved in an observation by a satellite microwave sounder covers a few tens of kilometers, we consider here the impact of horizontal variability at scales of the order of 100 km. These intermediate and larger scales are forecasted more accurately than the finer scales. As well, the large scales appear smooth within the horizontal vicinity of the observation, and could be described locally with a relatively simple linear function. It is thus proposed that the additional information added to the standard 1D profile, besides the values of relevant thermodynamic variables (P, T, q), are the linear horizontal gradients of these variables, averaged over this neighborhood, around the observation ground footprint. The sensitivity of the results to the size of the neighborhood considered will also be examined.

This paper is organized as follows. The methodology is presented in section 2. The impact of slant-path calculations on innovation statistics is shown in section 3. These statistics indicate which channels are more impacted. Results from assimilation experiments are analyzed in section 4. Since statistics indicate reduction of the standard deviation of observation minus background-simulated radiances (OMB), the observation errors are reduced accordingly in an assimilation cycle. The impact of the scale of the horizontal gradients, as well as that of reduced observation error, is evaluated. Section 5 summarizes the findings of this study and presents an outlook of future work in view of operational implementation.

2. Methodology

The simulation of satellite radiance observations is performed using the Radiative Transfer for the Television and Infrared Observation Satellite (TIROS) Operational Vertical Sounder (TOVS) (RTTOV), version 10, model (Matricardi et al. 2004) using, as input, model information on the state of the atmosphere and surface, as well as the viewing zenith angle. Let us call the entire description of an atmospheric state, as produced by a weather forecast system, for a given instant. The relevant information to estimate the expected value of an observation is only a small fraction of . The standard approach for NWP applications is to reduce the full general atmospheric state to only a local description , in the vicinity of the observation, where the dimension has been greatly reduced:
e1
This local information is later provided to an observation operator H to estimate the expected value E of an observation: . A classical choice of localization has been the reduction of to a 1D object: all vertical layers are retained, but only at a single (latitude, longitude) point. Let us call this localization choice a column, . As a 1D object, and since we are considering radiance observations, it describes a profile of, for instance, pressure P, temperature T, and moisture q, or some equivalent set, discretized as a function of altitude. This choice of localization assumes that most of the properties of the atmosphere that are relevant for a given observation are already expressed in this vertical profile, and neglects the impact of any difference present in the neighborhood (any horizontal gradient). This has proven to be very useful, but is also known to be imperfect, as indicated in section 1.
To better describe the actual state of the atmosphere, the local state vector must retain more information from the background state than is contained in a column . More extensive local states have been explored in Bormann et al. (2007) and Bormann (2017), which retain an entire 2D slice of the model atmosphere, along the azimuth of the line of sight. We will refer to these extended state vectors as “slices,” or . Their dimension is substantially larger than that of , although still much smaller than that of :
e2

For example, Bormann (2017) used 6 profiles to sample 160 km of horizontal extent of the azimuthal plane. In this study, the possibility of increasing minimally the size of the local state vector beyond a column is explored. During the construction of the state vector, besides the local column located at the ground footprint, the neighborhood around it is also considered. At each altitude level, and for each relevant variable (P, T, q), the average east–west (E–W) and north–south (N–S) gradients over this neighborhood are evaluated and retained. Since the satellite-viewing angle varies from nadir to about 50° for microwave sounders on board polar-orbiting satellites, the horizontal region of interest spans a few tens of kilometers off the footprint. The gradients are computed at a comparable scale, which is defined here by a 100-km radius around the ground location. Of all the spectrum of horizontal variability of the relevant variables, this gradient thus retains the larger scales, and neglects the smaller scales.

In Table 1, a summary is presented of the dimensionality of the different state vectors mentioned. The construction of a state vector of finite size involves a vertical discretization into n levels. For state-of-the-art NWP systems, , and in the specific case of ECCC’s global system, . A column state vector that expresses the thermodynamic variables can be defined by the following collection, or a similar set:
e3
where h is the height from mean sea level, and , are the discretized vertical layers. The gradient-aware extension proposed here consists of
e4
where represents the horizontal gradient of a field, at constant altitude.
Table 1.

Dimensionality of several state vectors: the entire description of ECCC’s global system that is presently operational, as an example of full state vector, a local column vector, an ensemble of column vectors (a slice), using 6 columns as in Bormann (2017), or 31 as in Bormann et al. (2007), and the gradient description explored in this study, which is a 3D description. The dimensionality of columns and slices is based on the number of vertical layers of ECCC’s system.

Table 1.

For this study, the horizontal gradients on east (x) and north (y) directions are evaluated through least squares fits to the background fields within the averaging region. In each of the 81-level discretization of the model, from the ground to the model lid at 0.1 hPa, and for each required variable, the values in the neighboring horizontal points within the averaging region (of approximately 100-km radius), are fitted to a linear function. In ECCC’s global system, the horizontal grid is a Yin–Yang array (Kageyama and Sato 2004; Qaddouri and Lee 2011), with a horizontal discretization step of approximately 25 km.

In these grids, the density of horizontal grid points is very uniform, although not completely constant. At this resolution, the neighborhood of a given observation spans about 8 × 8 horizontal grid points . A field X at constant altitude (e.g., temperature), will present a set of values through this neighborhood. Let us name , the eastward and northward distance of any given point with respect to the ground footprint of the observation. We then express the field as a central value , and two gradients , obtained from a fit to a linear function within the neighborhood:
e5
The fit itself acts as a low-pass filter, and retains only the properties of the field X that are of larger scale. The cutoff scale is determined by the size of the neighborhood . Besides this baseline radius of 100 km, the use of a cutoff at smaller scale (retaining a larger fraction of the local structure) has also been explored, as detailed in section 4.
Once the local state vector [Eq. (4)] is built, and given the azimuth and zenith angle of an observation, either a standard column profile, or a slant profile, can be produced (see Fig. 1). Given an observation, the vertical profile is extracted as usual from the appropriate model fields, interpolating linearly from the four adjacent grid points to the ground footprint of the observation location, and ignoring local horizontal gradients, a collection , which is identical to [Eq. (3)]. The evaluation of the slant thermodynamic profile is determined using the local gradients obtained through the fit [see Eq. (5)]:
eq1
e6
eq2
where is the vector that expresses the horizontal gradient of a field. Expressed in the local eastward and northward coordinate system, . The vector expresses the horizontal component of the unit vector along of the slant propagation path, from the footprint toward the satellite. In this local coordinate system, , where θ is the zenith angle with respect to vertical direction, and φ its azimuth with respect to the north (see Fig. 1). Both θ and φ are extracted from the observation data received at ECCC. In the set of Eq. (6), a linear expression is used for the temperature field profile, whereas a log-linear expression is used for pressure and moisture fields. These gradients effectively supplement the control vector. The resulting vertical and slant thermodynamic profiles have the same size and shape, and both can be fed to the RTTOV observation operator to obtain simulated observations. By construction, these slant and vertical profiles are identical for an observation at nadir, since in this case .
Fig. 1.
Fig. 1.

Scheme for the evaluation of the properties along a slant path. The slant profile is built as a departure from a column profile, given gradients of each relevant quantity in the neighborhood, and the geometry of the slant path. This path departs from the vertical by a zenith angle θ, along azimuth φ, measured from the footprint toward the satellite.

Citation: Monthly Weather Review 146, 12; 10.1175/MWR-D-18-0126.1

In an assimilation experiment, the cost function J and its gradient may be written as (for simplicity, these are here expressed with a 3DVar approach):
e7
where is an atmospheric state, is the background state, and is the ensemble of observations. Both and are the background and observation error covariance matrix, respectively. We will focus on the observation terms . In the standard column approach, the operator acts upon column state vectors: . Although the operator acts upon a column, it takes into account the satellite-viewing angle. The atmosphere represented by this column state vector is a three-dimensional object whose horizontal gradients have been neglected.
To compute the cost function in an atmosphere that is not completely spherical, we may choose instead , a gradient-aware operator that acts upon gradient state vectors , where horizontal gradients are retained, as in Eq. (4). In that case, should also use , the Jacobian of that operator, which is a moderately complex computation, as the gradients involve many more grid points around the observation. In this work, instead, the difference between the output of vertical and slant forward model operators is rearranged as an offset to the observation:
e8
This expression defines the quantity . If assimilated with a column operator, this offset observation leads to the same contribution to the cost function, as the original observation using a slant operator:
e9
The derivatives of these, however, are not identical, although is very close to . We will assume, for the purpose of the minimization, that . The Jacobian of the column operator already expresses the dependence with respect to the vertical stratification of the atmosphere, which is the largest dependence. It does not contain, however, the dependences with respect to the horizontal gradients. Also supporting this approximation, for the purpose of data assimilation, analysis increments are forced to be spatially smooth at horizontal length scales below some cutoff, as a result of background error correlations (Buehner et al. 2005). It is worth mentioning that ECCC’s background model resolution for the global system is of 25 km, whereas the analysis increment is evaluated only at a resolution of 50 km. As for the observations, the footprint resolution is approximately 45 km for the AMSU-A instrument and 32 km for the ATMS instrument. As a consequence, it is here assumed that the use of slant-path calculations in the cost function accounts for the bigger portion of the impact through the innovations . It is, on the other hand, assumed that the impact through a difference between the Jacobian derivatives of the vertical and slant profiles is negligible.

In a similar study, Bormann (2017) used a 2D interpolator in the observation’s azimuthal plane to resolve the slant-path profile from model fields. The dependence of the observation operator with respect to the horizontal structure was fully implemented in that study, as the slant profile is built within the observation operator. Also, the Jacobian derivatives and the analysis increments are calculated over this slant profile.

In the approach taken in this study, the slant offset is evaluated, and the offset observations are used directly within the standard variational data assimilation system, with a standard column observation operator, which as shown in Eq. (9), allows the evaluation of the slant cost function. Consequently, the additional cost is limited to the evaluation of this observation offset. The value of the observation offset was kept constant during the cost function minimization. In the tests below, it was implemented as a calculation outside the variational data assimilation. The computing cost can be reduced further, once it is identified that only some instruments and channels are practically impacted: a range of temperature channels sensitive to the upper troposphere and stratosphere, as shown below.

The gradient-dependent offsets of the radiance observations were extracted for two key instruments on board several polar-orbiting platforms. These are the AMSU-A, on board platforms NOAA-15/18/19 and MetOp-1/2; and ATMS, on board the Joint Polar Satellite System (JPSS). The choice was made to focus on channels that are mostly sensitive to temperature in the upper stratosphere and stratosphere. These upper-level fields present horizontal variability at larger scales. The coarse-resolution horizontal gradients that are evaluated to calculate the slant profiles may not be optimal to capture the small-scale variability that may develop in the moisture field. Hence, attention was focused on temperature-sensitive channels. The results of section 3 provide additional support for this choice of channels.

3. Simulation of observations

In this section, the innovation statistics of the simulated radiance observations, with and without the estimation of a slant-path offset, are compared. Background fields are obtained from short-range forecasts from ECCC’s global weather forecast system; they represent a 6-h window, at lead times between 3 and 9 h from analysis time, discretized in time increments of 15 min. For each observation, the vertical profile is extracted from these model fields, interpolating to the observation location, a collection . The slant profile is also obtained, using also the local gradients, as in Eq. (6). Both profiles may be provided to RTTOV, obtaining the corresponding estimations of brightness temperature.

The differences between the brightness temperatures derived from the vertical and slant profiles, for channels 9 and 15 of the ATMS instrument, during the 6-h period centered at 0000 UTC 1 January 2017, are shown in Fig. 2. For channel 9, which is a temperature channel whose sensitivity peaks near the tropopause, the difference in brightness temperature can reach 0.5 K. Channel 15 is a temperature channel whose sensitivity peaks near 2 hPa, and in that case the brightness temperature difference can be as large as 1.2 K. As expected, the largest differences take place for large zenith angles, in regions where large horizontal gradients are present.

Fig. 2.
Fig. 2.

Observed brightness temperature in the 60°–90°N latitude region, for ATMS channels (a) 9 and (c) 15 for a 6-h period centered at 0000 UTC 1 Jan 2017. The difference in the simulation of these observations, when using vertical and slant profiles is shown for channels (b) 9 and (d) 15.

Citation: Monthly Weather Review 146, 12; 10.1175/MWR-D-18-0126.1

OMB statistics of brightness temperatures, using both vertical and slant profiles were evaluated and compared for the various microwave channels. The trial fields were obtained from a reference cycle, which corresponds to the operational system at ECCC in 2017. The radiance observations were those from this cycle, bias corrected with standard procedures (Garand et al. 2005; Buehner et al. 2015), and thinned at the scale of 150 km.

The standard deviation of the innovations, (), referring to the vertical profiles, and (), for the slant profiles, can be obtained. The statistics were evaluated over the 10-day period of 1–11 January 2017. Similar results were obtained for 10-day period of 1–11 July 2016 (not shown). The overall relative change of the innovation standard deviation, comparing the slant versus the vertical estimations, can be defined as . This ratio should be less than 1 if the use of a slant profile improves the estimation of the brightness temperature. The results for the different channels are shown in Fig. 3, for ATMS and AMSU-A (on board NOAA-18), averaged over global and broad latitudinal bands. The impact in all cases is positive, but for several channels, it remains modest. With this sample size, a normalized ratio smaller than 99.7% is statistically significant with a 95% confidence level. The temperature channels that peak in the upper troposphere and lower stratosphere are the most impacted, especially over mid- and high-latitude regions. There are 2% and 1.7% reductions in normalized for channel 9 of ATMS and AMSU-A, respectively, calculated over the global domain. The reduction can reach 3.5% and 5.5% when calculated over high-latitude regions. These statistics are in agreement with results presented in Bormann (2017) using a 2D operator.

Fig. 3.
Fig. 3.

Standard deviation of OMB brightness temperatures evaluated from slant profiles, normalized by the standard deviation of OMB using vertical profiles, for (a) ATMS channels and (c) AMSU-A NOAA-18 channels. The variation with zenith angle of the standard deviation normalized to the nadir view is shown for (b) channel 9 of ATMS and (d) channel 9 of AMSU-A. The statistics were calculated over different latitudinal bands: global (black), high latitude (red), midlatitude (blue), and tropics (green). The validation is from 1 to 11 Jan 2017, representing 0.5 million observations for the global data.

Citation: Monthly Weather Review 146, 12; 10.1175/MWR-D-18-0126.1

The impact of gradients can be expected a priori to increase with higher zenith angle. This is also shown in Fig. 3, in terms of reduction of the ratio of standard deviation of innovations, between nadir and slant path, at different zenith angles. Whereas there is almost no reduction near nadir, at large viewing angles, the error reduction for channel 9 exceeds 6% and 4% of normalized , for ATMS and AMSU-A instruments, respectively. The impact at higher zenith is more pronounced over higher-latitude regions, as expected. The impact on the moisture channels of the ATMS instrument is modest.

4. Assimilation experiments

Given that the slant-path radiative transfer leads to OMB reduction for some channels, assimilation experiments were launched to test the impact in NWP framework. Experiments were conducted for two periods of two and a half months, in boreal winter (15 December 2016–28 February 2017), and boreal summer (15 June–31 August 2016). The global assimilation system is a 4D ensemble variational (4D-EnVar) system (Buehner et al. 2015). The background error covariances are obtained from the combination of flow-dependent estimates from an ensemble Kalman filter (EnKF) and a static climatological estimate that varies with latitude and month. Between the surface and 40 hPa, the background error covariances are the mean of dynamic and static covariances. Above 10 hPa, only the static covariances are used. Between 40 and 10 hPa, there is a linear transition. As mentioned earlier, the model contains 81 hybrid levels, extending from the surface to 0.1 hPa. The assimilation window spans 6 h, and the background state is discretized in time increments of 15 min.

As will be shown further, summer 2016 results show larger impact of using gradients, compared to winter 2017. For brevity, some of the summer 2016 results are presented but only the results for the boreal winter experiments are discussed in detail. Table 2 summarizes the main experiments that were carried out. The control 1 (G252), follows current standard operational procedures at ECCC. Radiances are simulated using vertical profiles of the background field (i.e., column local state vectors). In a first test toward slant background profiles, slant 1 (FB), the slant thermodynamic profiles [Eq. (6)] are evaluated, and simulated radiances are evaluated from them. The radiance observations are modified using the offset, following Eq. (8). Given the results shown in Fig. 3, where the impact in certain channels is much larger, only the radiances from AMSU-A, on board NOAA-15/18/19 and MetOp-1/2; and ATMS, on board JPSS are modified; radiances from other instruments were kept unchanged. In this first test, as in the control, the a priori error statistics attributed to radiance data are taken unmodified from the control experiment, using current operational values. We evaluate these a priori statistics from historical statistics, here obtained from operational data evaluated in January 2015. Radiance observations are quality controlled, bias corrected, and thinned within each experiment. Following standard operational procedures at ECCC, radiance observations are dynamically debiased based on a methodology described by Buehner et al. (2015) and cloud-affected radiances are not assimilated. Since bias corrections parameters are dynamically evaluated, they evolve with time within each experiment. Comparison of this time evolution between different experiments shows only minor differences on the global bias of the various channels.

Table 2.

Definition of the experiments described in this paper. The initial control run is , which uses a classic vertical radiance operator. An initial test for a slant operator is , which uses the same a priori error statistics. Error statistics were reevaluated from each of these. New control and slant runs were relaunched, resulting in a second control , and a second slant experiment . A final run tested a different averaging scale to evaluate horizontal gradients.

Table 2.

Within each 6-h analysis window, was calculated for each radiance channel of each instrument. For the purpose of this evaluation, only the data that are common to the control and test experiments are selected, after quality control. An example of OMB statistics is shown in Fig. 4, for bias-corrected observations of channel 9 of AMSU-A/MetOp-1. Other instruments show similar results. As was shown in section 3, channel 9 is particularly sensitive to the thermodynamic gradients applied in Eq. (6). Throughout the cycle, the slant 1 (FB) experiment maintains smaller , compared to the control 1 (G252) . As a reference, the overall value of of each experiment is marked as a dot on the right of Fig. 4. A reduction of 2.8% is noted for the slant-path experiment.

Fig. 4.
Fig. 4.

Evolution of the standard deviation of OMB (K) for channel 9 of AMSU-A on board MetOp-1 satellite. The horizontal axis is the analysis time. The observations are used after bias correction. The control and modified experiments are labeled as G252 (blue) and FB (red). The overall standard deviation is marked with a dot on the right of the figure for each experiment.

Citation: Monthly Weather Review 146, 12; 10.1175/MWR-D-18-0126.1

The dependence of with respect to the scan position is shown in Fig. 5, for the control and slant experiments. These statistics are calculated for the period of 1–31 January 2017, again for channel 9 of the ATMS instrument. An overall tendency of the outer scan positions (larger zenith angle) to present larger is clear. In agreement with the OMB statistics shown previously (Fig. 3), the use of slant paths, evaluated with linear horizontal gradients, leads to a reduction in , which is more pronounced at the outer scan positions. As expected, the nadir scan positions are insensitive to the presence of gradients, as their line of sight is vertical, matching the column operators. With the introduction of these gradients, the curve of versus scan position flattens, although not completely, indicating that the gradients have provided information that was relevant for the simulation of the side-looking observations, but this improvement may still be incomplete to account for all effects of the slant path. Despite this, a large fraction of the potential improvement of a nonvertical state vector has been extracted: the statistical difference between central and side observations is much smaller in the slant 1 (FB) experiment than in the control.

Fig. 5.
Fig. 5.

Standard deviation of ATMS channel 9 OMB (K), for the control G252 (blue) and modified FB (red) experiments, as a function of scan positions. The statistics are calculated based on bias-corrected global observations for the period 1–31 Jan 2017.

Citation: Monthly Weather Review 146, 12; 10.1175/MWR-D-18-0126.1

In ECCC’s system, the a priori observation error statistics for each channel and instrument are based on some , a historical global estimate of , which is then scaled by a heuristically determined factor referred to as inflation factor . The a priori observation errors and inflation factors are shown in Table 3 for assimilated channels (5–15) of the ATMS instrument, for the various experiments mentioned. The inflation factor is expected to account for, among other things, neglected error correlations, arising, for instance, from radiative transfer, or representativeness error [refer to Garand et al. (2013) for more details]. A reduction in the a priori observation error could increase the number of rejections since observations whose OMB difference exceeds 3 are discarded. Given that the a priori estimation of the observation error statistics are determined from innovation statistics, Figs. 3, 4, and 5 also suggest the standard deviation of observation error could be reduced. In both the control 1 (G252) and the slant 1 (FB) experiments, the a priori observation error statistics were the same, and they do not depend on the viewing angle. As shown in Fig. 5, the resulting OMB statistics do in fact have some dependence on the scan position, although reduced for the slant-path experiments.

Table 3.

Observation error statistics for assimilated channels (5–15) of the ATMS instrument. Original () a priori observation error statistics, evaluated in a previous run (the operational system during Jan 2015), and applied both to the control 1 () and slant 1 () tests. Also shown are reevaluated error statistics , using the vertical operator, estimated from , and applied to produce the control 2 () test. Reevaluated error statistics were obtained with the slant operator, estimated from the slant 1 () test, and applied to produce the slant 2 () test. Last column (IF) shows the inflation factor that has been attributed to each channel, and that relates , with a historical estimate over some reference period of a previous run.

Table 3.

The historic statistics (dating January 2015 and used to derive ), were compared against , obtained from each experiment. This is shown for different channels of AMSUA on board NOAA-18, in Fig. 6a, and of ATMS channels, in Fig. 6b. Results indicate that for some channels the assigned observation error should be updated. For channels 9–11 of AMSU-A and 7–13 of ATMS, the appears to be slightly overestimated in the modified experiment. The reduction in the observation operator error with the introduction of the gradients implies that the a priori estimation of the standard deviation of the observation error could also be reduced further for the affected channels of AMSU-A and ATMS instruments.

Fig. 6.
Fig. 6.

(a) Ratio of operational OMB standard deviation (, historical source to generate the original a priori errors in Table 3) against the standard deviation of OMB of the control 1 (G252, blue curve) and the slant 1 experiment (FB, red curve), for AMSU-A on board NOAA-18. The OMB standard deviations over control 1 and slant 1 were calculated over the 1–31 Jan 2017 period. (b) As in (a), but for ATMS instrument.

Citation: Monthly Weather Review 146, 12; 10.1175/MWR-D-18-0126.1

To test the effect of a modification in the a priori observation error, the list of observation error statistics, for all ATOVS and ATMS channels and instruments, was updated, using statistics of the newly obtained global for each channel of each instrument of the modified experiment (i.e., using the atmospheric gradients, cycle slant 1, FB), and then running a new experiment for the same period (slant 2, FBSA), also simulating the radiances with gradients. To be able to compare fairly with the control, the same operation of reevaluation of the observation error statistics was performed over the same period of time, and using the same data, over the control experiment, and then also rerun. This results in a new control 2 (S2). The original observation error statistics, and the modified statistics that are used in the second control and second experiment cycles, are presented in Table 3.

Comparing against the control 1 experiment (G252), results indicate a reduction in the standard deviation for the geopotential height in slant1 (FB), at short term. This is shown in Figs. 7a (global) and 7b (Arctic). Bormann (2017) shows similar order of magnitude reduction in forecast errors, up to day 3 at high latitudes. Further, the slant2 experiment (FBSA), with reevaluated error statistics, is also compared against the same control 1, and results are shown in Figs. 7c (global) and 7d (Arctic). Besides a similar short-term impact as slant 1, additional impact at longer term is obtained.

Fig. 7.
Fig. 7.

Comparison of the experiments slant 1 (FB) and slant 2 (FBSA) against control 1 (G252). Plots show the mean of the normalized difference, in percent, of the standard deviation of the geopotential height field against its own analysis, as a function of the pressure level and forecast range. (a) Global mean, boreal winter, slant 1 (FB) vs control 1 (G252), . (b) As in (a), but for the North Pole (60°–90°N). (c) Global mean, boreal winter, slant 2 (FBSA) vs control 1 (G252). (d) As in (c), but for the North Pole. See the description of each experiment in Table 2. Red (blue) is the percentage decrease (increase) in standard deviation of experiment compared to the control. Red thus indicates a better performance of the gradient experiments. Black dots mark statistical significance above the 95% confidence level, based on a Fisher F test.

Citation: Monthly Weather Review 146, 12; 10.1175/MWR-D-18-0126.1

To consider a control where the same reevaluation of error statistics has taken place, Fig. 8 shows a comparison between slant 2 (FBSA) and control 2 (S2). The error statistics had been reevaluated in both, independently. The boreal winter season is presented in Figs. 8a (global) and 8b (North Pole, local winter), and the boreal summer season in Figs. 8c (global) and 8d (South Pole, local winter). The longer-term impact obtained in the comparison against control 1 is found again.

Fig. 8.
Fig. 8.

Comparison of the experiments slant 2 (FBSA) against control 2 (S2), for boreal (top) winter and (bottom) summer. Plots show the mean of the normalized difference, in percent of the standard deviation of the geopotential height field, against its own analysis (as in Fig. 7). (a) Global mean, boreal winter. (b) As in (a), but for the North Pole (60°–90°N). (c) Global mean, boreal summer. (d) As in (c), but for the South Pole region (austral winter). See the description of each experiment in Table 2. Red (blue) is the percentage decrease (increase) in standard deviation of experiment compared to the control. Red thus indicates a better performance of the gradient experiments. Black dots mark statistical significance above the 95% confidence level, based on a Fisher F test.

Citation: Monthly Weather Review 146, 12; 10.1175/MWR-D-18-0126.1

The geographical distribution of the immediate impact, at short term, is presented in Figs. 9 and 10, which show, by latitude, the reduction of forecast error in slant 2 (FBSA), compared to the control 2 (S2) experiment, respectively for boreal summer and boreal winter. In both seasons, the reduction is statistically significant at high latitudes in the upper troposphere and lower stratosphere. There is up to 10% reduction in the zonal mean of normalized differences in standard deviation of geopotential height, for 12-h forecasts in the Arctic during the boreal winter experiment (Fig. 10a). The magnitude of the reduction decreases with forecast lead time. The variation of other variables (e.g., temperature) presents a similar pattern. The short-range reduction in the increments is due to the implementation of slant-path radiative transfer for observation simulation, and it is noticeable up to 36 h when the experiments are compared against their own analysis.

Fig. 9.
Fig. 9.

(a) Zonal mean of the normalized difference in the standard deviation (against its own analysis) of the geopotential height field in boreal summer (15 Jun–31 Aug 2016), between the experiment slant 2 (FBSA) and control 2 (S2), for the 12-h forecast: . These runs use updated error statistics, see Table 2. Red (blue) is the percentage decrease (increase) in standard deviation of experiment compared to the control. Red thus indicates a better performance of the gradient experiment. Black dots mark statistical significance above the 95% confidence level, based on a Fisher F test. (b)–(d) As in (a), but at 24-, 36-, and 48-h forecast.

Citation: Monthly Weather Review 146, 12; 10.1175/MWR-D-18-0126.1

Fig. 10.
Fig. 10.

As in Fig. 9, but for the boreal winter (15 Dec 2016–28 Feb 2017).

Citation: Monthly Weather Review 146, 12; 10.1175/MWR-D-18-0126.1

The comparisons indicate the following:

  • Between the control 1 (G252), and the slant 1 (FB) initial slant experiment, a short-term impact (below 2 days) is obtained, notably in the Arctic region (local winter), peaking in the mid- to lower stratosphere (20–100 hPa). This is attributed to the use of the local gradients.
  • Between the control 1 (G252) and the slant 2 (FBSA) experiment, with revised observation error statistics, this positive impact at short term is similar or slightly improved. Impact at longer term also appears.
  • Revising also the observation error statistics of the control, the comparison between the control 2 (S2) and the slant 2 (FBSA) experiments, the long-term impact between them (up to 10 days) is obtained as well. This feature is missing during the experiments with no observation error revision, and hence it is attributed to the lower a priori observation errors in slant 2.

Using slant-path radiative transfer for assimilation of radiances, Bormann (2017) found statistically significant benefits in short-range forecast up to day 4 in the stratosphere. Observation errors were not modified in their experiment. In this study, the comparison of the radiance data rejection rates indicates that the slant 1 (FB) experiment, against the control 1 (G252), has an increased number of assimilated observations, by a small amount: 0.1% of the data volume. The modified observation error statistics decreases the number of assimilated observations in the slant 2 (FBSA) experiment, compared against slant 1 (FB), also by about 0.1%. This was expected as the smaller a priori observation error imposes stricter selection criteria from the 3 quality check.

The comparison of slant 2 (FBSA) versus control 2 (S2) experiments was also evaluated against independent observations. For this purpose, the temperature retrievals from the Sounding of the Atmosphere using Broadband Emission Radiometry (SABER) instrument (Brown et al. 2006) were used. This is a microwave limb sounder. Although its observations barely reach below the tropopause, it has a good ability to evaluate the entire stratosphere globally. Only the standard deviation was used as a statistical measure. The standard deviations between experiments and SABER temperature retrievals were calculated for the period 1 January–28 February 2017.

Figure 11 illustrates that there is a reduction in the normalized difference in the standard deviation of slant 2 (FBSA) and SABER observations at short-range forecasts in the upper stratosphere and stratopause (1–0.3 hPa). This reduction is maintained at longer lead times, and extends into a reduction at longer range and lower altitude, in the midstratosphere (7–10 days and 100–5 hPa), a feature that also appears in the forecast versus analysis evaluation (Fig. 7).

Fig. 11.
Fig. 11.

Normalized standard deviation of differences between the slant 2 (FBSA) experiment against the independent SABER observations for temperature at different forecast hours: . Red indicates a better performance of the gradient experiment. The statistics cover the period between 1 Jan and 28 Feb 2017, for global data. Around 3 × 106 SABER observations are used. Black dots mark statistical significance above the 95% confidence level, based on a Fisher F test.

Citation: Monthly Weather Review 146, 12; 10.1175/MWR-D-18-0126.1

Since the cutoff scale for the evaluation of the gradient may be a relevant parameter, the sensitivity of the results to the size of the averaging domain for the evaluation of the gradients was evaluated. A 50-km-averaging radius around the ground location was chosen, running a new slant 3 (FBSA2) experiment . The forecast is compared against its own analysis, and the results are shown in Fig. 12. When compared to the slant 2 (FBSA) experiment (which uses gradients evaluated over a 100-km domain, Fig. 7), the results indicate similar performance for short-range forecasts. However, at medium to long range, the performance appears degraded when the smaller domain is used to calculate the gradients. Radiosonde observations were also used to validate this conclusion. Figure 13 shows the comparison of geopotential height bias (dashed curves) and standard deviation (solid curve) for slant 2 (FBSA) (blue) and slant 3 (FBSA2) (red) experiments for 144- and 192-h forecasts. A deterioration of the standard deviation of geopotential height is significant (at the 90% level) above 150 hPa. This confirms that the initial choice of the domain size (~200 km) was reasonable, given the current properties of the assimilation system. This choice could be reviewed if the horizontal resolution of the model and/or that of the observations were significantly modified.

Fig. 12.
Fig. 12.

Scores of experiment slant 3 (FBSA2) against its control 2 (S2) for boreal winter. (a),(b) As in Figs. 8a and 8b, but evaluated over the global domain and North Pole region, respectively, but local horizontal gradients are obtained over domains of 50-km radius (instead of 100 km), thus they are less filtered against higher spatial frequencies.

Citation: Monthly Weather Review 146, 12; 10.1175/MWR-D-18-0126.1

Fig. 13.
Fig. 13.

Bias (dashed curves) and standard deviation (solid curves) of the geopotential height from slant 3 (FBSA2, red) experiment, relative to slant 2 (FBSA, blue), when compared against radiosondes. (top) Global domain (about 55 000 radiosondes) and (bottom) for the northern circumpolar region (60°–90°N, about 6500 radiosondes). (a),(c) 144-h forecasts and (b),(d) 192-h forecasts. Numbers within the shaded boxes on the left (right) are the level of confidence, in percent, that the biases (standard deviation) from the two experiments are different. The color shading indicates which experiment has a better score: red for better performance of slant 3 (FBSA2: gradients evaluated over a smaller domain).

Citation: Monthly Weather Review 146, 12; 10.1175/MWR-D-18-0126.1

5. Discussion and conclusions

In this study, a coarse-resolution estimate of the model fields’ horizontal gradients was used to approximate the model’s local variability, and to build slant profiles for the simulation and assimilation of radiance observations. This background profile contains the vertical information of the required variables, and also their respective horizontal local gradients, which can be considered in the formulation of a forward operator. The resulting description of the horizontal structure is effectively a low-pass-filtered field, with a cutoff scale of the order of 100 km.

The gradients associated to structures larger than the cutoff are used to estimate the difference between a vertical and a slant-path observation. These observation-space offsets were used to map the slant observations to a virtual observation on an atmosphere with no horizontal gradient. This mapping leads to the same contribution to the cost function as a slant observation. It was assumed that the Jacobian of the slant profile is well approximated by the Jacobian of the vertical profile. This approximation is similar to that used in practice, where Jacobians are kept constant in the iteration process leading to the analysis. The slant-path mapping was performed for two key instruments on board several polar-orbiting platforms: AMSUA, on board NOAA-15/18/19 and MetOp-1/2; and ATMS, on board JPSS.

The impact is shown to be significant (e.g., up to 6% reduction in standard deviation of error for ATMS) for the simulation of radiances at high zenith angles, for channels sensitive to the upper troposphere and lower stratosphere, and in mid and higher latitudes. A set of four boreal winter and summer assimilation experiments (vertical/slant background and original/updated a priori error) were performed to examine the impact of 1) slant-path radiative transfer in the assimilation cycle and 2) the effect of reduced observation error in experiments where slant-path effects are considered. The variation of observation error as a function of scan position was not considered. Results indicate that in the stratosphere, forecasts up to three days are improved linked to slant-path modifications of the forward operator. The modified observation minus background statistics suggests that the a priori observation error statistics may be updated. Assimilation tests suggest impacts at longer range as a result of this adjustment. The forecasts were also compared against independent SABER observations, to corroborate the short- and long-range improvements in the slant-path experiments. The improvement of the use of slant-path radiative transfer is limited to radiances sensitive to the upper atmosphere, at large zenith angles. The results show a qualitatively similar improvement to those from previous studies. Finally, using a smaller cutoff scale for the low-pass estimation of gradients (reducing the horizontal fitting domain, from 100-km radius to 50 km), did not improve the results.

This proposed approach has been tested within the context of ECCC’s global assimilation and forecast system, in the configuration of the present operational runs, and is conceptually ready for operational implementation. Although it resorts to a theoretical control vector that, above the classical column, is supplemented with horizontal gradients of column variables, it does not modify the size of the state vector that is required during minimization, which is still a 1D column. The impact of horizontal gradients can instead be applied as a gradient-dependent offset in observation space, which can be evaluated outside the variational minimization program. It, therefore, impacts minimally on computing resources. As an alternative to the approach used in this study, we have started developing a slant 1D interpolator to be used within ECCC’s global assimilation system, as opposed to offline. In that approach, the interpolation would consider the four grid points surrounding the point of intersection of line of sight at a given level. This new operator is being built with its adjoint and tangent linear counterparts, and overcomes the limitations of the current approach (i.e., using the adjoint and tangent linear of the vertical operator). The computational impact of such an approach on the system has yet to be evaluated. Further work will be dedicated to test the new 1D interpolator against the current approach. We intend to test as well the impact of slant-path radiative transfer for moisture-sensitive channels.

Acknowledgments

The authors wish to thank Dr. Sylvain Heilliette for his technical support. We also thank Dr. Mark Buehner, Dr. Stéphane Laroche, and the three anonymous reviewers for their constructive comments, which helped us to enhance the quality of this manuscript.

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