1. Introduction
Subseasonal-to-seasonal forecasting at 2-week to 12-month lead times has become an increasingly relevant research area for the environmental modeling community due in large part to a stronger societal demand for reliable information about high-impact climate events (e.g., floods, droughts) further in advance of their onset. Several features in the Earth system have been investigated to determine to what extent they may be exploited as sources of predictability. One such source is the land surface, which directly modulates atmospheric flows, as indicated by the significant moisture and energy fluxes at the land–air boundary. Soil moisture variability can be regarded as a proxy for these interactions (Koster and Suarez 2003; Seneviratne et al. 2010). Earlier studies have shown that satellite and in situ soil moisture observations, when assimilated into a land surface model, improved the representation of soil moisture and other land surface states for short time scales (a few days) (Draper and Reichle 2015); within a coupled land–atmosphere modeling framework, improved land surface estimates resulted in more realistic evapotranspiration and precipitation (Koster et al. 2002). Earlier work has also demonstrated that enhanced soil moisture initialization has similar implications for the reproduction of large-scale atmospheric modes with anomalies persisting on intradecadal and multidecadal time scales (e.g., Kuenzer et al. 2009; Ford et al. 2016). Given a fixed set of soil moisture observations, we wish to examine how much useful information may be extracted by data assimilation systems having different levels of complexity.
Modern land surface assimilation has roots stemming from the Kalman filter (Kalman 1960), a state estimation method that takes into account the input uncertainties. The Kalman filter analysis is optimal if 1) the observation error and forecast error are mutually uncorrelated, unbiased, and white (temporally uncorrelated); 2) the model state is linearly related to the observations; and 3) the model dynamics are linear. Land surface models are driven by nonlinear processes, which suggests that an assimilation system would be more effective at producing improved state estimates if it did not rely so heavily on linearity assumptions.
The extended Kalman filter (EKF; Evensen 1992) explicitly allows for the first-order treatment of weakly nonlinear model processes. The EKF requires the state-dependent linearization [or tangent linear model (TLM)] of the model operator to propagate error estimates forward in time. The code in operational land models is not differentiable analytically (due to, for example, the presence of conditional statements), so the linearization must be determined numerically. This additional computational expense increases with the number of model variables that are included in the analysis state vector (Reichle et al. 2002a). Deciding which variables to include and which to disregard when performing this computation requires a substantial amount of application-specific expertise to ensure that the resultant TLM approximation is useful. The EKF’s use of closure approximations and its assumption of linear error growth may still lead to filter failure if nonlinearities are strong enough. One could simply increase the frequency at which the state update (or analysis) is performed or revise the TLM to include higher-order terms (e.g., Miller et al. 1994). Such engineering has been shown to mitigate the threat of filter instabilities; however, the enhanced filter performance may not be enough to justify the extra computational burden.
The ensemble Kalman filter (EnKF; Burgers et al. 1998) entirely avoids the TLM and allows for the fully nonlinear propagation of the forecast errors. This is achievable because the complete probability distribution of the errors is approximated with a finite ensemble of model states at each update step. An added benefit of the EnKF is that it can more readily account for dynamic horizontal forecast error correlations (difficult for the EKF due to computational reasons) and permits the more careful treatment of model error. The EnKF has been shown to be capable in a broad range of applications (Houtekamer and Zhang 2016). In the context of soil moisture state estimation, it has been shown that the EnKF performs at least as well as the EKF when controlling for computational effort (Reichle et al. 2002a). But even though the ensemble of the EnKF contains information about higher-order statistical moments that is advanced between analyses, the actual update equation explicitly considers only its mean and covariance, making the EnKF suboptimal for problems dominated by nonlinearity and/or non-Gaussianity.
Many other advanced data assimilation algorithms have been introduced that may be beneficial for approaching the nonlinear problems highlighted thus far. Such methods include reduced-rank filters (Verlaan and Heemink 1997), particle filters (Gordon et al. 1993; Xiong et al. 2006; Nakano et al. 2007), and the increasingly popular and broad suite of hybrid methods that consider input from both sequential and variational components when performing the analysis. Of particular interest here is the rank histogram filter (RHF) of Anderson (2010), which is similar to the EnKF but does not assume that the ensemble prior and observation distributions are normal.
It has been demonstrated that the RHF can be a competitive alternative to the EnKF even for global numerical weather prediction problems (Anderson 2010). The models used in these earlier examples have typically had dynamics meant to mimic the chaotic nature of a geophysical fluid such as the atmosphere or the ocean (Zhang et al. 2009). Land surface models, in contrast, are dissipative—any ensemble of initial perturbations will collapse toward a single common trajectory with time if something is not done to forcibly maintain its spread (Reichle et al. 2002b). The dynamical regime of the model undoubtedly affects the performance characteristics of any ensemble data assimilation method. Therefore, a more comprehensive assessment of the RHF is needed that explores how the RHF behaves in a land surface application. The main objective of the present paper is to examine the performance differences between the EnKF and the RHF for soil moisture estimation within the context of a synthetic experiment using the NASA Catchment land surface model. A secondary objective is to assess the impact of adaptive error covariance inflation (section 2c) on the performance of the EnKF and RHF. We begin in section 2 with a review of the two ensemble filter methods. Section 3 discusses the details of the land surface model and outlines the experimental design. Section 4 presents results, and section 5 provides conclusions.
2. Review of ensemble filtering methods
a. Ensemble Kalman filter
b. Rank histogram filter
The RHF is an ensemble filtering method that can accommodate any distribution for the ensemble prior or observation. Figures 6 and 7 in Anderson (2010) provide useful schematics that show how the RHF works. Here, we will briefly describe its update procedure.
To construct the ensemble prior, the RHF applies the forward operator
Available observations are assimilated serially, one at a time, which assumes that the observation errors are independent. Given the use of Gaussian tails, the range of possible state values is not constrained in the regions of the distributions outside of the ensemble. This is obviously not appropriate for bounded quantities. It is possible to force boundedness on the posterior ensemble as was mentioned by Anderson (2010), but the subsequent regression may still result in physically impossible values for the unobserved quantities. [This problem is solved by the marginal adjustment RHF of Anderson (2020), a recent modification of the RHF.] As is discussed in section 3c, we impose extra-analysis physical consistency checks to reduce the harmful effects of unbounded posterior distributions.
In summary, perhaps the most important feature of the RHF is that, unlike the EnKF, the RHF does not restrict the ensemble distribution of model states (prior or posterior) to be Gaussian. This novel aspect of the RHF is the focus of the present study.
c. Addressing systematic error with inflation
By design, the EnKF updates the magnitudes of the ensemble covariances during each analysis such that the norm of the error covariance matrix after the analysis is typically smaller and necessarily no larger than it was before the analysis; the user-defined observation error covariance matrix is usually kept static, meaning that, with successive analysis cycles, more and more of the information provided by the observations will tend to be ignored and the data assimilation system will eventually stop altering the state estimate, a phenomenon known as filter divergence.
Filter divergence can often be avoided via appropriate use of a noise application scheme like the one employed during all of the land model integrations in this study (see section 3b). Such a strategy is an inflation method because it helps ensure that the ensemble spread does not get too small. Several different inflation formulations have been proposed in the literature, each being shown to generally improve ensemble representativeness and performance (e.g., Anderson and Anderson 1999; Whitaker and Hamill 2012).
As stated previously, Anderson (2009) models the inflation factor using a normal distribution, where the forecasted inflation distribution is the most recent (damped) posterior inflation distribution. This study uses the adaptive inflation algorithm of El Gharamti (2018), which improves on that of Anderson (2009) by instead modeling the inflation factor using an inverse-gamma distribution, yielding a more stable implementation that completely avoids negative values, and better combats overdispersion.
3. Land model and experiment setup
a. Land surface model
The land surface model used here is the Catchment model of the Goddard Earth Observing System (GEOS) modeling and assimilation framework (Koster et al. 2000). The catchment-based approach considers the effects of subgrid heterogeneity on the horizontal structure of hydrological land surface processes. The model has been shown to be viable when used stand-alone (Reichle et al. 2019) as well as when coupled to a general circulation model (Gelaro et al. 2017).
The model prognostic variables include soil temperature and heat content, snow water equivalent, and soil water. The latter are water excess and deficit variables that measure the total amount of water stored in the catchment and the departure of a layer’s water content from the equilibrium vertical profile. The excess and deficit variables include the surface excess, root-zone excess, and catchment deficit, which correspond to three nested layers of 0–5, 0–100, and 0 cm to the (spatially varying) bedrock depth, respectively. These soil water prognostic variables are used to diagnose the corresponding volumetric soil moisture values–surface moisture content (sfmc), root-zone moisture content (rzmc), and profile moisture content (prmc). The distribution of rzmc is used to spatially partition a catchment fractionally into three different hydrological regimes: 1) saturated: ground surface completely saturated, 2) transpiration-sufficient: surface not saturated but transpiration proceeds without stress or severe water limitation (moisture-stress evapotranspiration is not supported), and 3) wilting: soil moisture is below the wilting point and therefore too dry for any transpiration to occur. Different parameterization schemes for the surface energy balance and soil water transfer are used in the different hydrological regimes, a distinction that is ignored in more conventional, layer-based land surface models.
b. Ensemble spread and stochastic perturbations
c. Experiment design
The domain for the synthetic experiment consists of 18 globally distributed locations mainly clustered in North America and Europe (Reichle et al. 2019). The locations were selected because each is also a validation site for the NASA SMAP soil moisture products. As a group, the sites have a wide array of soil and vegetation properties that provide challenges to real-world land–assimilation efforts. Site selection was made in anticipation of future work that will rely on the in situ measurements for validation of real data assimilation experiments (please see appendix A for validation site details). The synthetic experiment is carried out at each site independently from the others. MERRA-2 data (Gelaro et al. 2017) are used for the surface meteorological forcing inputs. The Catchment model parameters and boundary conditions are those used in Version 4 of the SMAP L4_SM system (Reichle et al. 2019).
Conducting the experiment in a synthetic environment allows for the fine-grained control of every aspect of the experiment. The potential usefulness of the results obtained in a real-world context is made more apparent when the parameters of the experiment are set in a sensible way. An important consideration is how to simulate model error, which can be difficult to characterize given that it is unknown in the real world. Studies focusing on land data assimilation have tried to account for model error in a synthetic framework by using distinct meteorological forcing datasets and/or model parameters for the truth and data assimilation runs (e.g., Reichle and Koster 2003). Here, we follow a similar rationale and prescribe model error by requiring that the truth be generated using meteorological forcing for the years 2000–09 while the data assimilation experiments are run with forcing corresponding to the years 2010–19; additionally, the data assimilation ensembles were initialized with the 1 January 2010 model states taken from the end of the experiment period; stated differently, we are trying to estimate the truth given intentionally incorrect (or error-prone) meteorological forcing and initial conditions. A schematic of the approach is provided in Fig. 1 and is explained in more detail below.
Schematic of the synthetic experiment showing the 10-yr offset of the meteorological forcing used to drive the truth run (single-member) and that used to drive the assimilation run (100-member ensemble).
Citation: Journal of Hydrometeorology 24, 6; 10.1175/JHM-D-22-0046.1
The Catchment model initial conditions were obtained as follows. First, the model state was spun up from a cold start by running the model four times through the 20-yr period from 1 January 2000 through 31 December 2019. After this spinup, the model was run from 1 January 2000 through 31 December 2009. This 10-yr period serves as the truth for the assimilation experiments (Fig. 1). Daily sfmc observations of the truth were generated by perturbing the true value with a random sample of normally distributed noise with mean zero and standard deviation equal to 0.03 m3 m−3. The standard deviation was chosen to be roughly equal to typical errors in SMAP soil moisture retrievals (Colliander et al. 2022).
Once the observations and truth states had been recorded, the model was reset to 1 July 2009 at which point a 100-member ensemble was initialized, each with an identical copy of the truth state at that time (Fig. 1). To spin up the ensemble, each member was run for another 6 months through 31 December 2009 with a unique set of AR(1) perturbations applied to a subset of the meteorological forcing and prognostic variables (see section 3b). The perturbations were samples from either lognormal or normal distributions with mean 1 and 0, respectively. The time scale of the AR(1) correlation for each perturbation time series was chosen to provide a physically meaningful error time scale; there was no prescription of cross-correlated errors (see Table B1 in appendix B). The ensemble was then run from 1 January 2000 through 31 December 2009, using the (intentionally wrong) 2010–19 meteorological forcing data and 1 January 2010 ensemble of initial conditions. The ensemble trajectory during this 10-yr period was recorded as the open loop simulation. The open-loop establishes the base-level performance of the modeling system without the benefits of soil moisture data assimilation.
Finally, the data assimilation experiments consist of ensemble simulations that are identical to the open loop but with the assimilation of the synthetic daily observations of surface soil moisture. To mimic the quality controls in place for real-world satellite soil moisture retrieval algorithms, soil moisture observations were not assimilated if any one of the ensemble members indicated that snow was on the ground or that the soil was frozen. After an analysis was performed, the updated quantities were passed through a physical consistency check procedure, and, if necessary, were adjusted to remain within the physical limitations encoded in the model.
There were four assimilation experiments performed, each one highlighting a different filter/inflation combination: 1) EnKF without adaptive inflation, 2) RHF without adaptive inflation, 3) EnKF with adaptive inflation, and 4) RHF with adaptive inflation. [Note that from here onward, “inflation” is used to refer to the adaptive inflation algorithm of El Gharamti (2018) described in section 2c, and not to the general application of model prognostics and forcing perturbations outlined in section 3b]. Each case used the exact same Catchment model, meteorological forcing, initial ensemble, and member-specific temporal sequence of AR1 perturbations and had access to the same set of sfmc observations.
The update of the state and of the inflation parameters is handled wholly by NCARs Data Assimilation Research Testbed (DART; Anderson et al. 2009), which allows for any discretized computer model to be interfaced with a suite of sequential ensemble data assimilation methods and common inflation techniques. In these experiments, the prior state vector provided to DART is not the entire model state vector but rather a subset consisting of only the surface excess and root-zone excess model prognostic variables, which matches the EnKF state vector of the SMAP L4_SM system (Reichle et al. 2019, their section 2.2.2). The adaptive inflation algorithm was set to inflate only the ensemble prior; the mean and standard deviation of the inflation distribution were initialized to 1.0 and 0.6, respectively, and were not allowed to be negative; before the inflation factor was updated and applied to the ensemble, the difference between the inflation factor and 1 was multiplied by a damping constant (equal to 0.9), the result of which was then added to 1 [see Eq. (8)].
d. Validation
1) Performance metrics and normalized information contribution
The impact on model skill of assimilating soil moisture observations was evaluated using several performance metrics, namely, the anomaly correlation coefficient (ACC), the mean absolute bias (MAB), and the unbiased root-mean-square error (ubRMSE). Each metric was applied to the three diagnostic soil moisture content variables output by the model (sfmc, rzmc, prmc).
The computation for the ACC is identical to that of the regular Pearson correlation coefficient but with the extra step of first removing the mean seasonal cycle from each of the two time series. The mean seasonal cycle was determined using the mean trajectory of the open loop (no meaningful difference in performance was seen when the climatologies of the respective experiments were used instead of the open loop mean). On a given day, the average value across all available years was taken from the mean open-loop run; the resulting time series (of 365 daily values) was then smoothed with an equal-weighted, 15-day average centered on the day of interest; leap-day quantities (29 February) were discarded in all input time series prior to the calculation of the anomaly correlation coefficient. The MAB was determined by taking the absolute value of the difference between two time series, then taking the average value of the result. The ubRMSE computes the root-mean-square error of two time series after having first subtracted the respective mean values from each.
Confidence intervals for the ACC were computed using a Fisher Z transform of the anomaly correlation values. As in Draper et al. (2012), the effective sample size of the input time series was reduced to account for inherent autocorrelations in the data. This consideration leads to progressively larger uncertainties for moisture variables with more temporal memory. The confidence intervals for the MAB and ubRMSE were computed using a two-tailed t test with 10 000 replicates.
2) Normalized innovations
Innovation statistics will also be used to help characterize filter performance. The EnKF assumes that the observations and model estimates are unbiased and uncorrelated, meaning that a histogram of the innovations, after having been scaled with the inverse of the expected error standard deviation (
4. Results and discussion
a. Soil moisture
Improved accuracy of the soil moisture state has the most potential to improve near-surface atmospheric flows in regions with strong land–atmosphere coupling (Koster and Suarez 2003). One such location is the Great Plains of the United States, where the Little Washita test site is positioned. For portions of the analysis that follow, we focus our attention on the Little Washita site, given that it is representative of where we would expect to see maximal impact of assimilating soil moisture observations. In some cases, we also illustrate results for the sites at Carman, St. Josephs, and Tonzi Ranch to highlight interesting aspects. We did not detect any meaningful general dependence of the assimilation performance on site-specific environmental features (soil properties, land cover type, climate) or the ensemble itself (e.g., innovations statistics). If such a signal is to be detected, a much larger number of sites is likely needed, which is beyond the scope of the present study.
Figure 2 shows the sfmc and rzmc time series at the Little Washita validation site during June–August (JJA) 2009, the final year of the experiment period. The same plot was created for different years (not pictured) to see if there was a trend in the difference between the experimental configurations that were tested (e.g., EnKF versus the RHF), but no such trend was found; hence the year 2009 was chosen arbitrarily for presentation purposes. Pictured are the truth trajectory and the ensemble mean values of the open loop and data assimilation runs; also shown are the 100 individual members of the open loop ensemble (gray lines). The peaks in the soil moisture trajectories coincide with precipitation events and are typically followed by relatively slower dry-down periods, which are marked by little to no precipitation. Note that the open loop ensemble spread is small during precipitation events and increases during the dry-down periods. For both sfmc and rzmc, the mean of the open loop at this site is generally drier than the truth at the Little Washita site, meaning that the assimilation of sfmc observations should add moisture to correct the systematic dry bias coming from the model error specification (the incorrect meteorological forcing). Qualitatively, this was achieved for each of the four data assimilation runs as indicated by the smaller differences between their ensemble mean values and the truth (relative to the difference between the open loop and the truth). Also noticeable are the modest improvements in correlation with the truth. None of the filters responded particularly well to surface soil moisture observations following relatively strong precipitation events during this time period. There seems to be rapid peaking at the onset of the wetting episodes but not of a high enough magnitude. This last point can be explained by recalling that the analysis is effectively a weighted average of the model simulation (ensemble mean) and the assimilated observation, the weights being inversely proportional to the variance of the respective errors. The observation error was intentionally held fixed for the entirety of the experiment period whereas the ensemble estimate of the model forecast error (the ensemble spread) was changing over time. During heavy precipitation events, the model ensemble was increasingly confident that the soil was nearing saturation as evidenced by relatively small values of ensemble spread. As a result, the observation information is not given as much relative weight in the averaging process, sometimes leading to insufficient soil wetting. The severity of this effect is mitigated as the precipitation ceases and the soil dries, allowing the ensemble spread to increase to more climatologically normal values (boosting responsiveness of the ensemble to information introduced by the assimilated observations).
Time series of volumetric soil moisture content (m3 m−3) for (a) sfmc and (b) rzmc at Little Washita during JJA 2009 (OL: open ploop, EnKF: ensemble Kalman filter, RHF: rank histogram filter, EI: EnKF with adaptive inflation enabled, RI: RHF with adaptive inflation enabled).
Citation: Journal of Hydrometeorology 24, 6; 10.1175/JHM-D-22-0046.1
Figure 3 shows the average NICACC (Fig. 3a), NICMAB (Fig. 3b), and NICubRMSE (Fig. 3c) metrics for sfmc, rzmc, and prmc, along with their corresponding 95% confidence intervals. The EnKF had a positive NICCC for each of the three soil moisture variables, indicating a systematic improvement over the open loop. The NICACC scores are ∼0.51 for sfmc, ∼0.41 for rzmc, and ∼0.42 for prmc. The NICMAB for the EnKF was positive for sfmc and rzmc with respective values of ∼0.16 and 0.03 but was near zero for prmc (∼0). The assimilation of soil moisture observations with the EnKF improved model skill in terms of the bias for sfmc but seems to have led to no real performance benefit for the deeper soil moisture reservoirs of the catchment (rzmc and prmc) relative to the open loop. The EnKF NICubRMSE scores are all positive with a value of ∼0.32 for sfmc and a more modest ∼0.25 for both rzmc and prmc. Generally, the EnKF provides a clear net benefit over the open loop for model estimates of soil moisture.
Normalized information contribution metrics for (a) anomaly correlation coefficients, (b) mean absolute bias, and (c) unbiased root-mean-square error (ubRMSE). Metrics are averaged across all 18 validation sites. The entire 10-yr experiment period was considered. Confidence intervals pictured are 95%.
Citation: Journal of Hydrometeorology 24, 6; 10.1175/JHM-D-22-0046.1
The RHF had a NICACC of ∼0.52 for sfmc, ∼0.41 for both rzmc and prmc, performing significantly better than the EnKF for sfmc and just as well for rzmc and prmc. For NICMAB, the RHF showed scores of ∼0.23, ∼0.12, and ∼0.08 for sfmc, rzmc, and prmc, respectively, performing significantly better than the EnKF in each case. Each of the NICMAB values for the RHF were positive, meaning that the RHF had a positive effect on model performance in terms of model bias relative to the open loop. The RHF had NICubRMSE values of ∼0.33 for sfmc and ∼0.25 for both rzmc and prmc, effectively identical to those for the EnKF. In all cases, NICυ values for the RHF were positive and never significantly worse than those for the EnKF. The most notable difference between the two filters is seen for the bias, where the RHF shows itself to be the significantly better filter, increasing the NICMAB for each of the three soil moisture variables by ∼0.05 over the EnKF.
With the addition of adaptive inflation, the RHF achieved NICυ values at least as good as its noninflated variant, but in only one case was that improvement statistically significant (NICACC for sfmc). Similarly, the use of adaptive inflation with the EnKF always improved performance but significantly so in only two cases (NICACC for sfmc and rzmc). In terms of NICACC, the inflated RHF performed significantly better than all of the other filter experiments for sfmc, but such a difference did not translate to the unobserved soil moisture variables (rzmc and prmc), for which no one filter configuration showed itself to be best. The NICMAB and NICMAB subpanels illustrate that inflation generally does improve the MAB and ubRMSE values but never in a statistically significant sense.
The NICυ value magnitudes across all experiments and metrics indicate that the impact of soil moisture assimilation on skill was always positive and never negative. The largest improvements over the open loop were observed for the anomaly correlations (NICACC ranging from ∼0.40 to ∼0.53) and the smallest skill improvements were for the bias (NICMAB ranging from ∼0 to ∼0.24). The NICυ metrics show that the RHF performed somewhat better than the EnKF and that adaptive inflation mostly results in minor skill improvements for both filters compared to the respective noninflated filter variants.
b. Ensemble representativeness
The rank histograms for the open loop and for each of the four assimilation experiments are shown in Fig. 4 for a representative set of locations, including St. Josephs, Carman, Tonzi Ranch, and Little Washita. The rank histograms were created by considering every prior ensemble used over the experiment period, then ordering (or ranking) all the members of this augmented set from low to high according to the surface soil moisture state value; these bins are then populated using all the available observations (over the 10 years). The bins corresponding to each experiment are not identical, but the observations used to populate the bins are the same. To reduce noise in the illustrations, the counts of adjacent bins were averaged (the smoothing has no impact on the major features of the histograms). It is important to note that a rank histogram says nothing about whether the ensemble or the observations are representative of the actual system variable being estimated (soil moisture in this case). It simply provides a way to gauge if an ensemble is representative of the space spanned by the observations. One-to-one correspondence between the two would be evidenced by a flat line for a given experiment. None of the ensembles in Fig. 4 has a histogram with such a horizontal profile. At St. Josephs (Fig. 4a), the rank histogram for the open loop shows some underdispersion (as evidenced by the gentle “U” shape) and a forecast ensemble that is biased wet compared to the observations. Data assimilation does not seem to correct for either issue in any meaningful way as evidenced by the close tracking of the filter and open loop histogram profiles; in fact, data assimilation seems to exaggerate the bias (note the higher left edge of the histograms of the assimilation experiments compared to that of the open loop).
Rank histograms of SFMC observations for each assimilation experiment at (a) St. Josephs, (b) Carman, (c) Tonzi Ranch, and (d) Little Washita. All 10 years of available data were considered.
Citation: Journal of Hydrometeorology 24, 6; 10.1175/JHM-D-22-0046.1
All the rank histograms at Carman (Fig. 4b) also show biased and underdispersive ensembles but instead with forecasts that tend to be too dry with respect to the observations; there is no clear distinction between the open loop and the assimilation experiments at this location, except, again, at the left-most edge where the assimilation experiments show slightly wetter ensembles than for the open loop.
The open loop rank histogram at Tonzi Ranch (Fig. 4c) does not show an obvious global dispersion problem, but it does exhibit a forecast ensemble that is too dry compared to the observations. The assimilation experiments reduce the severity of the bias (given by the relatively flatter outer left edge).
At Little Washita (Fig. 4d), the rank histogram profile for the open loop indicates a forecast ensemble biased wet compared to the observations; but the filter experiments show more general underdispersion with little to no bias (as evidenced by their roughly symmetrical U shapes). The results imply that data assimilation leads to small changes in the ensemble’s representativeness of the observation space; most of these changes are concentrated on the left side of the histogram figures, where assimilating the soil moisture observations clearly leads to generally wetter ensembles. The main aspects of the rank histograms do not vary coherently with filter type or whether inflation was used, nor do they seem to depend on climatology (dry versus wet).
Figure 5 shows time series of the open loop and ensemble prior standard deviations of surface and root-zone soil moisture at the same four locations during JJA 2009. At St. Josephs (Fig. 5a), the standard deviation for the open loop ensemble had a JJA 2009 mean value of 0.026 m3 m−3, ranging from 0.01 to 0.042 m3 m−3 over the 3-month time period. The mean standard deviations were somewhat lower for the RHF (0.024 m3 m−3) and higher for the EnKF (0.29 m3 m−3), both having roughly similar ranges from ∼0.01 to ∼0.045 m3 m−3. The time series for the inflated cases were identical to those of their respective noninflated counterparts. The variability of the time series does not show a clear dependence on experimental configuration. At Carman (Fig. 5b), the time series for the open loop standard deviations shows a mean value of approximately 0.031 m3 m−3 and a 3-month range of 0.012–0.045 m3 m−3. The RHF standard deviation mean is 0.024 m3 m−3 while that for the EnKF is 0.026 m3 m−3. Inflation does not impact the mean for the EnKF but does lead to a small reduction for the RHF to a value of 0.023 m3 m−3. Here, the time series corresponding to the data assimilation experiments do not track the open loop case as closely (e.g., in early August) as was seen for St. Josephs. The open loop at Tonzi Ranch (Fig. 5c) had a mean standard deviation of 0.062 m3 m−3, with a range of 0.037–0.078 m3 m−3. The spreads in the ensemble priors are significantly lower than that for the open loop, with mean values of 0.046 m3 m−3 for the EnKF and 0.042 m3 m−3 for the RHF (the values were the same for the respective inflated variants of each filter). At Little Washita (Fig. 5d), the ensemble standard deviations for the open loop showed a mean value of 0.026 m3 m−3 (ranging from 0.010 to 0.038 m3 m−3). The EnKF variants are relatively less dispersed, having mean values of 0.032 m3 m−3 and 0.033 m3 m−3 for the noninflated and inflated forms, respectively. The RHF variants track each other closely, with means of 0.027 m3 m−3 and a common range of 0.011–0.046 m3 m−3.
Time series examples of volumetric soil moisture content (m3 m−3) standard deviation of data assimilation experiment ensembles (priors) and of the open loop ensemble for JJA 2009 at (a) St. Josephs, (b) Carman, (c) Tonzi Ranch, and (d) Little Washita.
Citation: Journal of Hydrometeorology 24, 6; 10.1175/JHM-D-22-0046.1
Generally, the RHF tends to have smaller spreads than the EnKF (the same is true for the relative differences between the inflated variants); the EnKF variants, as we have previously shown, exhibit positive performance gains over the open loop, but the large standard deviations relative to the RHF can be interpreted as evidence that the EnKF successfully compensates for the slight inappropriateness of its assumptions by boosting the ensemble spread.
The ensemble spread in the inflated EnKF closely tracks that of the noninflated EnKF, and the same is true for the RHF variants. In fact, at St. Josephs and Tonzi Ranch there is no discernible difference in ensemble spread between the inflated and noninflated filter versions across the entire JJA 2009 period (Figs. 5a,c), and only very small differences are seen at Carman and Little Washita (Figs. 5b,d). It remains unclear why the inflation does not lead to larger differences in ensemble spread during JJA 2009.
Of note is that the standard deviations of the open loop are not necessarily always greater than or equal to those of the noninflated filter configurations, which seems nonsensical if one neglects the boundedness of the sfmc problem being considered here; when an ensemble is especially wet or dry, its spread tends to collapse as the mean approaches the extremes. One example of this behavior can be seen clearly at Little Washita by comparing Figs. 1a and 5d, where the open loop shows significantly smaller standard deviations than the filters during time periods for which the filter ensemble sfmc values are moderate and the open loop ensemble has a relatively much wetter soil column. Across sites, it was also observed that inflation values tended to be closer to one during periods when the mean of the open loop was closer to the truth trajectory.
c. Measuring non-Gaussianity
Figure 6 illustrates the normalized innovation distributions of the assimilation experiments at the same representative locations as in Fig. 4 and over the entire 10-yr experiment period. Distributions for the EnKF and RHF are shown in the left column, and distributions for the filter variants with adaptive inflation are shown in the right column. For reference, the dotted line pictures the unit normal distribution (mean = skewness = 0, variance = 1), which is the distribution of the normalized innovations of the EnKF when all its assumptions are satisfied (see section 1). Each of the distributions in Fig. 6 is biased and skewed, clearly showing the systematic departure of the soil moisture assimilation problem considered here from those assumptions.
Representative set of normalized innovations distributions at (a),(b) St. Josephs, (c),(d) Carman, (e),(f) Tonzi Ranch, and (g),(h) Little Washita for the (left) noninflated and (right) inflated filter variants. Histograms were computed over the entire assimilation period (1 Jan 2000–31 Dec 2009). Quantities listed in key show the mean, standard deviation (sdev), and skewness of each distribution.
Citation: Journal of Hydrometeorology 24, 6; 10.1175/JHM-D-22-0046.1
At St. Josephs (Figs. 6a,b), the dry biases for the noninflated filters are the same at −0.17; the standard deviations are greater than 1 (1.40 and 1.46 for the EnKF and RHF, respectively), suggesting that the ensembles are underestimating the expected background error covariance; the skewness values for the EnKF and the RHF are nearly identical at −0.41 and −0.40, respectively. The negative skewness values imply that the probability density tends to be somewhat more concentrated at the higher end of the distribution when compared to a Gaussian (the relationship is reversed for positive skewness). The use of inflation (Fig. 6b) did not lead to any meaningful change in the mean, standard deviation, or skewness for either filter; whatever difference in statistics was observed seems to depend more on filter type (RHF versus EnKF) than it does on use of inflation.
Carman (Figs. 6c,d) instead shows wet biases (0.15 and 0.18 for the EnKF and RHF, respectively); the standard deviations are 1.35 for the EnKF and 1.39 for the RHF; also shown are negative skewness values with −0.34 and −0.27 for the EnKF and RHF. As at St. Josephs, inflation leads to negligible change in the distribution statistics considered here.
At Tonzi Ranch (Figs. 6e,f), the innovations indicate a systematic wet bias relative to expectation with values of 0.13 and 0.15 for the EnKF and RHF; the respective standard deviations are 1.12 and 1.16, with skewness values of −0.38 for the EnKF and −0.34 for the RHF. The normalized innovation statistics for the corresponding inflated cases are nearly identical. As at Carman and St. Josephs, the differences in skewness are more dependent on filter type than on whether inflation was used.
The bias of the normalized innovations at Little Washita was 0.04 for the EnKF and 0.08 for the RHF (observations tended to be wetter than the forecasts); the EnKF had a standard deviation and skewness of 1.67 and −0.74, respectively, with the RHF having similar values (standard deviation of 1.76 and skewness of −0.67). Consideration of inflation (Fig. 6h) shows little to no change for the RHF, which had a reduction of 0.01 in the mean and one of 0.02 in both the standard deviation and skewness; somewhat larger differences are present between the EnKF variants where now the inflated case shows an increase of 0.03 in the bias and a 0.02 reduction in the standard deviation of the normalized innovations distribution.
When looking at similar plots for all 18 locations (not pictured), there is no apparent relationship between the mean, bias, and skewness. Also of note is that, for each filter, inflation always led to a skewness magnitude that was less than or equal to that for the corresponding noninflated case. The RHF, inflated or not, tended to lead to skewness values closer to zero than the EnKF. These results imply that adaptive inflation was more likely to generate ensembles with relatively dry outlier members; these drier outliers were also more common for the EnKF than for the RHF. The results also show that the standard deviations of the normalized innovations distributions were generally larger for the RHF than for the EnKF, irrespective of inflation usage, a finding that is in line with the tendency of the RHF to have smaller ensemble spreads than the EnKF (as previously discussed).
To better quantify and understand the effects of non-Gaussianity on filter performance, we use the Kolmogorov–Smirnoff (KS) statistic to test if a sample comes from a normal distribution. This is achieved by comparing the cumulative distribution functions of the sample and of a continuous normal distribution having the same mean and variance. The KS statistic is defined as the maximum distance between the two functions across the range of the random variable. Assuming that the sample is sufficiently large, higher KS statistics generally communicate more confidence that the sample distribution does not sample from the reference distribution. Here, we test every ensemble prior that was used as input for an analysis at each location and use a significance level of 5%. We take the null hypothesis to be that the ensemble prior is indeed Gaussian. This analysis was carried out for each of the 18 locations for every ensemble prior during JJA 2009, which is representative of the 10-yr experiment period.
The KS values across all locations ranged from ∼0.5 to 0.65, which are all quite large. Every corresponding p value was orders of magnitude smaller than the target significance level. This suggests that the evidence allows us to reject the null hypothesis at each of the analysis times in JJA 2009; in other words, we can be confident that none of the prior distributions were Gaussian. When comparing the average KS value for each filter, the RHF had slightly higher values than the EnKF ensemble at all 18 sites. The difference does not depend on whether inflation was used (this distinction is present when making the same comparisons for the posterior ensembles). Also noted was that the mean KS values for the open loop were usually significantly higher than those of the data assimilation experiments. This result highlights how the update procedures used by the two filters lead to the destruction of non-Gaussian information in the ensemble priors, with the RHF maintaining more non-Gaussianity in the ensemble than the EnKF. In Fig. 7, Little Washita is used as a representative example of the relative relationships of the KS time series for the open loop and for the ensemble priors corresponding to the four different data assimilation experiments. Here, the KS values for the RHF tend to be somewhat higher than those for the EnKF. The inflated EnKF shows the lowest KS readings. Note the relatively higher KS values for the open loop compared to all the assimilation cases.
Time series of Kolmogorov–Smirnov values for the ensemble prior at the Little Washita site during JJA 2009.
Citation: Journal of Hydrometeorology 24, 6; 10.1175/JHM-D-22-0046.1
5. Summary and conclusions
In this paper, we presented results highlighting the differences between the EnKF and RHF when surface soil moisture observations were assimilated in a synthetic experiment. There were four cases: each filter was used with and without an adaptive inflation technique. The synthetic framework allows for more control of the experimental details, namely, the better contextualization of model error, and the complete avoidance of representativity issues associated with using complex forward operators. In all the metrics observed, the RHF typically performs on par with the EnKF, but often slightly better. This consistent improvement suggests that the non-Gaussianity present in the problem was significant enough to be exploited by the RHF. Since the adaptive inflation was shown to be beneficial for both filters, the deficiencies in ensemble spread were indeed relevant to performance and therefore requiring treatment. Further testing in a real-observation context would aid in understanding how the quality of the system error specifications affects the performance gap between the two filters.
The RHF allows for the non-Gaussian representation of the forecast and observation distributions. In our experiments, the observation error distributions were deliberately kept Gaussian to isolate the effects of the forecast-error parameterization. The RHF performed better than the open loop for all the NICυ metrics examined here. This was not the case for the EnKF, which showed near-zero NICMAB values for prmc. Evaluation of the KS statistics suggests that the RHF, inflated or not, allows for improved conservation of the ensemble’s non-Gaussian information. Given the results here, we propose that it is worthwhile to further examine the RHF as a potential alternative to the EnKF for more complex land assimilation problems (including using bias corrected filters, modifying a filter for spatially distributed application, or assimilating multiple types of observations).
Acknowledgments.
Emmanuel Dibia was supported by the NASA Fellowship Activity NNH18ZHA003N, Rolf Reichle was supported by the NASA SMAP mission, and Xin-Zhong Liang by NOAA Center for Atmospheric Science and Meteorology (NCAS-M) under Grant NA16SEC481006. Computational resources were provided by the NASA High-End Computing program through the NASA Center for Climate Simulation. The GEOS source code is available under the NASA Open‐Source Agreement (https://github.com/GEOS-ESM).
Data availability statement.
MERRA-2 data can be obtained from the Goddard Earth Sciences Data and Information Services Center at https://earthdata.nasa.gov/eosdis/daacs/gesdisc.
APPENDIX A
SMAP L4_SM Validation Sites
Figure A1 and Table A1 describe the locations of the 18 SMAP L4_SM validation sites that were tested in the study.
Map of SMAP L4_SM validation sites.
Citation: Journal of Hydrometeorology 24, 6; 10.1175/JHM-D-22-0046.1
Latitude and longitude of SMAP L4_SM validation sites, along with corresponding climate regime and MODIS IGBP land cover type.
APPENDIX B
Ensemble Perturbations
Table B1 describes the types of perturbations used to maintain the ensemble during the forecast period.
Meteorological forcing variables that are perturbed to maintain ensemble spread along with their corresponding perturbation standard deviations (S values) and time scales.
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