1. Introduction
A large body of evidence supports the hypothesis that land data assimilation systems (LDAS) can enhance the precision (i.e., correlation with truth) of multilayer soil moisture (SM) estimates derived from land surface models (LSMs) via the assimilation of satellite-based surface SM (SSM) information (Bolten and Crow 2012; Blankenship et al. 2016; Carrera et al. 2019; Muñoz-Sabater et al. 2019; Mladenova et al. 2019; Dong et al. 2019; Reichle et al. 2007, 2019, 2021). However, considerably less evidence exists that LDAS can leverage such improvements in land surface water states to simultaneously improve the accuracy (i.e., mean absolute error) of land surface water fluxes.
For example, despite continued evidence suggesting the vital role of prestorm SM conditions in determining storm runoff efficiency, recent multibasin studies have identified only relatively small streamflow improvements associated with the assimilation of SSM into a hydrologic model (De Santis et al. 2021) and stressed the complexity of factors impacting the relative success of SSM data assimilation with regard to streamflow (Massari et al. 2015). Reflecting the nonrobust nature of existing LDAS streamflow results, previous studies have suggested that, at least for high-flow events, hydrologic forecasting systems would be better served by using SSM time series to correct precipitation inputs—rather than directly assimilating them (Chen et al. 2014; Massari et al. 2018). Likewise, improvements in LSM surface energy fluxes realized upon the assimilation of SSM observations have been mixed and/or modest in nature (Peters-Lidard et al. 2011; Lu et al. 2019; Crow et al. 2020). Therefore, even after two decades of LDAS research and implementation, inconsistent improvements in water flux accuracy remain a persistent shortcoming of SSM-based LDAS and constitute a serious impediment to the effective use of such systems for flux-focused applications like hydrological forecasting (requiring surface infiltration versus runoff flux partitioning) and numerical weather prediction (NWP) (requiring surface latent- versus sensible-heat flux partitioning).
Given that LDAS rely on physical representations imbedded in LSMs to correct water fluxes using land surface state observations, an important candidate source for this limitation is the internal LSM representation of state–flux coupling (i.e., their prescribed relationship between water states and water fluxes). Indeed, recent work suggests that LSMs commonly misrepresent the strength of the relationship between SM and both surface runoff (Qs) and evapotranspiration (ET) (Dirmeyer et al. 2018; Lei et al. 2018; Crow et al. 2018, 2019; Dong et al. 2020a). Such systematic errors can degrade the ability of an LDAS to accurately link surface water states and fluxes and, therefore, appropriately leverage SSM observations to constrain Qs or ET. Despite this intuitive connection, the downstream impacts of LSM SM versus ET and SM versus Qs coupling errors have not been closely examined in LDAS. For example, Crow et al. (2020) hypothesized that the overcoupling of SM and ET in the LSM component of an NWP system can, in certain circumstances, cause SSM data assimilation to degrade ET and screen-level air temperature forecasts while simultaneously improving SM estimates. However, their results are based on a simplified/linear water balance model that may not be fully reflective of state-of-the-art, nonlinear LSMs. Likewise, while past work has identified a general tendency for LSMs to under couple SM and Qs (see, e.g., Crow et al. 2018), the impact of this under coupling has not yet been extended to include LDAS applied to hydrologic forecasting.
Here, we apply a series of synthetic fraternal twin data assimilation (SFTDA) experiments to examine the sensitivity of SM, Qs, and ET analyses from an LDAS to the presence of bias in the strength of assumed SM versus Qs and SM versus ET relationships contained within an assimilation LSM. In this way, we seek to motivate and develop new best practices for LDAS applied to flux-based forecasting applications.
2. Approach
SFTDA experiments are based on specifying a “nature” LSM to generate true observations that serve both as a source of synthetic observations and as a benchmark to evaluate final analysis results and an “assimilation” LSM that synthetic observations are assimilated into, and statistical models for independent errors in both the assimilation LSM as well as the assimilated synthetic observations. Consequently, a SFTDA experiment consists of four basic steps: (i) a single integration of the nature LSM with applied model error perturbations to generate a nature run representing synthetic truth; (ii) the degradation of preselected state estimates acquired from the nature run via random perturbations that are statistically consistent with an assumed observation error model; (iii) the assimilation of these synthetic observations back into the (unperturbed) assimilation LSM; and (iv) the evaluation of subsequent data assimilation state and flux results (i.e., the “analysis”) versus the nature run. For appropriate context, data assimilation results should be compared with open loop (OL) results reflecting assimilation LSM estimates in the absence of data assimilation. Here, we define the OL baseline as the ensemble mean of perturbed, but nonupdated, assimilation LSM integrations. This ensures that SFTDA results will converge exactly to the OL in the case of infinite observation error (where no weight is applied to assimilated observations).
A defining characteristic of a SFTDA experiment is the controlled introduction of systematic differences between the nature and assimilation LSM. In this way, SFTDA experiments capture the common case where an assimilation LSM contains systematic errors relative to the true natural system. Here, we examine systematic errors in the assimilation LSM regarding the strength of the relationship between SM states and both Qs and ET fluxes. In this way, we can study how systematic error in an LSM’s representation of water state–water flux coupling impacts their ability to enhance the estimation of land surface water fluxes via the assimilation of land surface state observations. While a range of land observations can be assimilated into LSMs, we conform here to the common LDAS case of assimilating remotely sensed SSM retrievals. In addition, since the preprocessing of remotely sensed SSM retrievals is a necessary strategy for addressing systematic differences between assimilated observations and the assimilation LSM, we also apply well-established rescaling strategies for SSM observations.
In all cases, the LSM is version 3.6 of the Noah land surface model with multiparameterization options (NoahMP v3.6; Niu et al. 2011; Xia et al. 2017). By varying either the physical approach that NoahMP uses to estimate Qs or the parameterization of surface soil evaporation resistance (see section 2c below), we generate four separate NoahMP configurations that exhibit a range of SSM versus total runoff (Q; surface runoff plus baseflow) coupling strengths as well as four (additional) configurations that capture a comparable range of SSM versus ET coupling strengths. By utilizing different pairs of these NoahMP configurations as either the nature or assimilation LSM in our SFTDA experiments, we systematically examine the impact of incorrect state–flux coupling physics on the performance of an LDAS. Additional methodological details are provided below.
a. Spatial domains
To examine the impact of varying landcover and climate conditions on results, SFTDA experiments are conducted for two separate regional domains within the southern United States. The first domain, hereinafter the southeast or “SE” domain, corresponds to the geographic box: 31.25°–36.75°N, 93.00°–86.00°W. This domain has a humid/subtropical climate characterized by temperate winters and hot/humid summers with an even seasonal distribution of rainfall. Landcover is approximately 50% forest by area with smaller fractions of agricultural fields (∼20%) and wetlands (∼10%).
The second domain, hereinafter the southern Great Plains or “SGP” domain, corresponds to the geographic box: 34.25°–39.75°N, 103.00°–96.00°W. This domain has a continental climate with long, cold winters and warm summers and is characterized by a relative precipitation minimum during the late summer. Land cover is a mixture of fragmented forests, range land, and agricultural fields—particularly winter wheat. In summary, relative to the SE domain, the SGP domain is more arid, produces less Q, contains lower levels of biomass, and has a more pronounced seasonal SM cycle due to warm/dry conditions found in the summer and cold/wet conditions in the winter.
b. NoahMP modeling
All NoahMP results are based on 15-min/0.125°-resolution NoahMP v3.6 simulations between 1 April 2015 and 30 June 2017 generated within version 7.4.2 of the NASA Land Information System (NASA LIS; Kumar et al. 2006) and spun up from a cold start on 1 January 2010. The baseline LIS configuration used is available for download (https://github.com/NASA-LIS/LISF/releases/tag/v7.4.2-public).
Meteorological forcing data are taken from the North American Land Data Assimilation System phase 2 (NLDAS-2; Xia et al. 2012) dataset. This product is, in turn, based on the North American Regional Reanalysis for all fields except precipitation. Instead, NLDAS-2 hourly precipitation accumulations are acquired by downscaling daily accumulation totals obtained from a rain gauge analysis using ground-based weather radar data (Xia et al. 2012). NoahMP is run with four different SM layers: 0–10, 10–40, 40–100, and 100–200 cm. We assume that the 0–10-cm layer provides SSM estimates and that a depth-based weighted average of the top three layers (i.e., 0–10, 10–40, and 40–100 cm) corresponds to root-zone SM (RZSM). Soil texture and vegetation lookup tables are based on State Soil Geographic and United States Geological Survey classifications, respectively.
c. NoahMP configurations and coupling metrics
To generate multiple sets of SFTDA experiments, we define NoahMP configurations that exhibit various levels of water state–water flux coupling strength. Among LSMs, NoahMP is notable in that it is designed to run with multiple parameterization packages for key land surface processes. For example, it contains four different runoff configurations: the simplified groundwater (SIM GW) case, the simplified TOPMODEL (SIM TOP) case, the free-drainage (FD) lower boundary assumption case, and the Biosphere Atmosphere Transfer Scheme (BATS) case. See Niu et al. (2011) for a complete discussion of each configuration. As described in Crow et al. (2019), these configurations make notably different predictions regarding the impact of prestorm SSM levels on storm-scale runoff coefficients (RC; the fraction of storm-scale precipitation converted into storm-scale Q). To quantify these differences, Crow et al. (2018, 2019) define the unitless SSM runoff coupling strength (SRCS) as the temporal Spearman rank correlation coefficient between prestorm SSM levels and subsequent storm-scale values of RC—as sampled across many nonwinter storm events. In these studies, the temporal storm scale varies as a function of basin size and is on the order of 5–8 days for medium-scale (i.e., 500–10 000 km2) basins. Crow et al. (2019) illustrate that the BATS and SIM TOP NoahMP cases are associated with relatively higher values of SRCS and the SIM GW and FD NoahMP cases with relatively lower values.
Naturally, there are many potential sources of systematic error in LSMs and our SFTDA results may be sensitive to the exact nature of the error sources we choose to consider. At a minimum, it is important that the range of SRCS values captured within our selected NoahMP configurations does not include any unrealistic extremes. Therefore, our guiding strategy for selecting NoahMP configurations is approximating the range of water state–water flux coupling found across existing LSMs. In strict terms, storm-scale SRCS values cannot be directly sampled from the grid-based analysis applied here; therefore, we instead sample mean SRCS values from the earlier basin-based analyses in Crow et al. (2018, 2019). Across multiple existing LSMs, results in Crow et al. (2018) show a range of spatial-mean SRCS values between 0.30 and 0.80 for medium-scale basins in the south-central United States. Within a comparable spatial domain, the four NoahMP SRCS configurations discussed above yield a spatially mean SRCS range of 0.24–0.82. Therefore, the intra-NoahMP parameterization range applied here is consistent with the expected inter-LSM range of SRCS values. Note that, for consistency with earlier results in Crow et al. (2018, 2019), SRCS is defined in terms of total runoff Q, as opposed to only surface runoff Qs. See Table 1 for a summary of these four SRCS configurations and their spatially averaged SRCS values within the SGP and SE domains.
NoahMP parameterizations and domain-averaged SRCS values for the four NoahMP SRCS configurations applied in our SFTDA experiments. For the case of varying SRCS, 16 different SFTDA scenarios are generated by considering all possible permutations of the four presented configurations—as either the assimilation or nature LSM. Parenthetical values capture spatial standard deviations within each domain.
An analogous procedure is applied to generate a set of NoahMP simulations associated with various levels of SSM versus ET coupling strength (SECS; defined as the Spearman rank correlation between weekly surface SSM and ET during the growing season). Following Dong et al. (2020a), a range of NoahMP SECS levels is generated by assigning different λ values to parameterize soil resistance to evaporation [see Eqs. (1)–(4) in Dong et al. 2020a]. While theoretically a function of soil texture, λ is typically held constant at a unitless value of 5 by NoahMP. Nevertheless, Dong et al. (2020a) illustrate that NoahMP SECS estimates are highly sensitive to λ via its role in regulating the partitioning between bare soil evaporation E and canopy transpiration T. (A larger fractional contribution of E to ET is associated with increased SECS because of the low water-storage capacity of SSM on which it directly depends.) There are, of course, other ways of synthetically inducing SECS variations into an LSM; however, NoahMP SECS is uniquely sensitive to λ via its control on the E/T ratio (Dong et al. 2020a).
Here we apply λ values of 1, 3, 5, and 10—with larger λ values associated with higher SECS. See Table 2 for a summary of these four SECS configurations and their spatially averaged SECS estimates within our SGP and SE domains.
As with Table 1, we wish to validate the realism of the inter-NoahMP SECS ranges described in Table 2. Within the SGP domain, spatially averaged SECS across these four NoahMP configurations in Table 2 range from 0.33 to 0.86. This prescribed NoahMP SECS range is somewhat wider than the approximate domain-average SECS range (from ∼0.50 to ∼0.80) found across multiple existing LSMs within the SGP region by Lei et al. (2018). However, all four LSMs examined in Lei et al. (2018) were also found to contain positive SECS bias relative to an observation-based estimate of SECS calculated using remote sensing and a triple collocation analysis. Such bias artificially suppresses the range of SECS values captured by existing LSMs. In contrast, our larger prescribed SGP SECS range here (i.e., 0.33 to 0.86) spans both the approximate upper edge (0.80) of LSM-based SECS values as well as the lower edge (0.25) of observation-based SECS values found in Lei et al. (2018). As such, it approximates the range of both modeled and observed SECS values and therefore captures a relevant range of LSM cases.
We use the two sets of NoahMP configurations defined in Tables 1 and 2 to complete two different sets of SFTDA experiments. In each set (i.e., one set for SECS configurations and one set for SRCS configurations), we conduct SFTDA scenarios for all possible permutations of assigning each NoahMP configuration as either the nature or the assimilation LSM. In this way, we systematically introduce SRCS and SECS biases into our SFTDA experiments and examine the impact of these biases on the subsequent ensemble Kalman filter (EnKF) water state and flux analysis. Reflecting their default use in NoahMP simulations, λ = 5 and the SIM GW Q package are chosen as baseline cases for the SRCS and SECS SFTDA experiments, respectively.
Past work has also identified general bias tendencies for LSMs. For example, Crow et al. (2022) use observational data to argue that true SRCS is typically near unity in the southeastern United States, while LSMs commonly predict significantly lower values in the same region. Likewise, as noted above, Lei et al. (2018) identify a region of the south-central United States where LSM SECS predictions are typically biased high. Therefore, SFTDA results are analyzed below considering the regional tendency for existing LSMs to overestimate SECS in our SGP domain and underestimate SRCS in our SE domain.
d. Data assimilation
All data assimilation is performed using the internal EnKF module built into the NASA LIS system (Kumar et al. 2008). The EnKF is implemented using a 20-member ensemble to update all four layered NoahMP SM states (i.e., 0–10, 10–40, 40–100, and 100–200 cm) using 0–10-cm SSM observations.
Model error is based on additive, hourly, mean-zero Gaussian perturbations applied directly to the NoahMP surface, second, and third SM layers. Reflecting relatively drier RZSM conditions in the SGP region, and our desire to keep OL errors roughly comparable in both domains, slightly larger direct SM perturbations are applied in the SE domain (i.e., standard deviations of 0.0030, 0.0015, and 0.0012 m3 m−3 in the top three layers, respectively) than in the SGP domain (i.e., comparable standard deviations of 0.0020, 0.0010, and 0.0008 m3 m−3). State perturbations are perfectly correlated vertically and temporally correlated with an autocorrelation time scale of 3 h.
Additional model error is provided by multiplicative perturbations applied to forcing rainfall and incoming solar radiation inputs that are sampled from mean-unity, lognormal distributions with (unitless) standard deviations of 0.50 and 0.20, respectively. Likewise, air temperature and incoming longwave radiation inputs are perturbed with mean-zero, additive Gaussian noise with standard deviations of 0.5°C and 30 W m−2, respectively. Cross correlation of these forcing perturbations is based on the approach described in Reichle et al. (2002). Statistically identical model state and forcing perturbations are applied to generate the nature run and during the implementation of the EnKF to assimilate observations into the assimilation model.
Observation error is based on serially white, mean-zero, Gaussian additive perturbations with a standard deviation of 0.03 m3 m−3 applied to true daily (0000 UTC) SSM values generated by the nature run. Despite the possibility that actual satellite based SSM retrievals may not be available under certain conditions, continuous daily SSM retrievals are assimilated, and no masking is performed for dense vegetation, active rainfall, snow cover, or frozen soil conditions.
Reflecting current best practices for LDAS, synthetic SSM observations are rescaled before being assimilated back into the assimilation LSM. This step is particularly relevant for the SFTDA experiments considered here, since systematic differences have been intentionally introduced into the assimilation LSM relative to the nature LSM. Rescaling is based on the sampling of long-term cumulative density functions (CDFs) for SSM estimates generated by both the assimilation LSM and synthetically generated SSM observations based on the nature run. Using this CDF pair, synthetically generated SSM observations are transformed to have the same CDF as the assimilation LSM prior to their assimilation. To account for the impact of SSM rescaling on the magnitude of observation error, the prescribed observation error standard deviation (see above) is multiplied by the ratio of the standard deviation of the LSM SSM CDF divided by the standard deviation of the synthetically generated SSM observations.
Due to the potentially significant role of SSM rescaling on the sensitivity of SFTDA results to systematic LSM errors, we also consider two additional SFTDA cases of no rescaling and a monthly CDF rescaling approach where a different CDF pair is sampled for each calendar month. Given the (approximate) 2-yr length of our analysis, such monthly rescaling entails the sampling of separate CDFs for each calendar month using ∼60 data pairs (i.e., ∼30 days per month for ∼2 months) of corresponding assimilation LSM SSM estimates and synthetic SSM observations.
Our overall strategy is to generate SFTDA experiments that are systematically degraded only by the presence of SECS and SRCS biases. To this end, all other aspects of the SFTDA experiments are designed to maximize the quality of the EnKF state analysis. For example, SFTDA experiments are conducted assuming that LSM and observation error statistics for both the assimilation LSM and the assimilated observations are perfectly known; that is, perturbation statistics used to degrade the assimilation LSM and generate synthetic observations are statistically identical to applied synthetic noise used to generate the EnKF forecast ensemble and calculate the Kalman gain. Likewise, the observation error standard deviation for SSM retrievals is set to a, relatively low, value of 0.03 m3 m−3.
In addition, applied SSM rescaling procedures are designed to be as effective as possible. For example, in a synthetic experiment, one methodological consideration is whether to use the OL or the perturbed version of the assimilation LSM to generate the assimilation model CDF. (Note that, in our SFTDA experiments, the OL is defined as the perturbation-free integration of the assimilation LSM.) In general, we find slightly better EnKF SSM and RZSM performance in our SFTDA experiments when the assimilation LSM SSM CDF (i.e., the rescaling target) is sampled from the perturbed version of the assimilation LSM. This is not surprising since this perturbed version of the assimilation LSM is generated via the application of noise that is statistically consistent with noise added in the nature run. (Note that the nature run is used to generate true results for the evaluation of all SFTDA experiments.) Therefore, the SSM CDF of the perturbed assimilation LSM is a better match to the SSM CDF for nature-run results and, consequently, provides a superior rescaling target.
e. Analysis evaluation
Like most LSMs, NoahMP provides separate estimates of both Qs—based on a characterization of quick rainfall/runoff partitioning at the land surface—as well as baseflow describing the slower subsurface transport of water to stream channels following infiltration at the land surface. However, given the key role of Qs for high-flow events and hazard forecasting, as well as previous work emphasizing the role of Qs in determining the overall success of LDAS streamflow applications (Mao et al. 2019; Massari et al. 2018), our SFTDA flux evaluation results are based on NoahMP Qs and ET estimates.
All EnKF and OL estimates are evaluated via comparisons with the LSM nature run (i.e., our assumed synthetic truth). The evaluation of EnKF SFTDA SSM and RZSM estimates is based on temporal precision measured via the Pearson’s correlation (R) between multilayer EnKF and OL SM results versus the nature-run baseline. Note that the length of our experiment (slightly over 2 years) is too short to calculate a seasonal climatology and seasonal anomalies; therefore, our focus is on total, and not anomaly, R.
Our focus on the precision-based evaluation of SSM and RZSM reflects several considerations. First, land applications that directly use SSM and RZSM estimates are generally based on the analysis of relative temporal variations in SM states and, therefore, are not impacted by the presence of additive or multiplicative SM bias (Crow et al. 2005). Second, even within densely instrumented validation sites, the upscaling of ground SM measurements to remotely sensed grid scales is associated with significant bias (Chen et al. 2019). Consequently, we generally lack large-scale SM climatology information to serve as a baseline for coarse-scale SM bias calculations. Third, SM estimates provided by land surface models are most accurately interpreted as a relative soil-water proxy (Koster et al. 2009) and generally contain large, model-dependent SM biases. As a result of these factors, the information content in SSM and RZSM time series is generally assumed to be limited to their relative temporal dynamics (i.e., their precision).
However, the situation is fundamentally different for water fluxes such as Qs and ET. Flux-based applications like NWP and hydrologic forecasting are greatly impacted by the presence of Qs and/or ET bias. Likewise, absolute values of Qs and ET flux estimates must be consistent with overall water- and energy-balance considerations. Reflecting these differences, we instead evaluate water flux results using a mean absolute error (MAE) metric that provides a broader assessment of both flux bias and precision.
The relative improvement of the EnKF analysis versus the OL baseline is assessed using the relative performance metric (MEnKF − MOL)/MOL where M is a particular evaluation metric (i.e., R or MAE) applied to evaluate water state or flux estimates provided by the EnKF and OL cases.
3. Results
Figures A1–A4 (see appendix A) describe SFTDA OL state and flux evaluation metric results for permutations of the four NoahMP SRCS and SECS configurations summarized in Tables 1 and 2 and across both the SE and SGP domains. NoahMP configurations are generally ordered such that higher levels of SECS (in both the assimilation and nature LSM) are generally found as you move upward or to the right within each subfigure. As defined in section 2, OL results reflect differences between the nature LSM and the mean of the nonupdated assimilation LSM ensemble. Evaluation metrics are based on sampling across time and then spatial averaging within each domain. OL results presented along the diagonal of each subfigure capture only the impact of random error added as part of the synthetic experiment. In contrast, off-diagonal results reflect the additional impact of systematic errors with regard to either SECS or SRCS in the assimilation model. Therefore, OL estimates tend to be better along the diagonal of each subfigure. Also note that OL results are largely diagonally symmetric. That is, they are sensitive only to the combination of SECS or SRCS configurations and not which configuration is assigned to the assimilation versus nature LSM. Subsequent SFTDA EnKF results presented below focus on relative improvement versus this OL baseline.
a. Surface water states
SE (top row) and SGP (bottom row) SECS results in Fig. 1 are equivalent to OL results in Fig. A1 of appendix A except for the case of a normalized relative EnKF performance metric [i.e., 100 × (REnKF − ROL)/ROL] describing the net impact of SSM assimilation on the precision (i.e., correlation R versus synthetic truth) of SSM (left column) and RZSM (right column) state estimates. Positive values of the performance metric, reflecting net improvement achieved during EnKF SSM data assimilation relative to baseline OL results, are shaded in blue. As with the OL results described above, the x and y axis of each subfigure in Fig. 1 reflect the selection of each of the four SECS configurations in Table 2 as the nature or assimilation LSM. Therefore, each configuration permutation serves as the basis for either the nature or assimilation LSM in a single SFTDA experiment. As discussed above in section 2d, long-term CDF matching is applied to SSM observations prior to their assimilation. Figure 2 is equivalent to Fig. 1 except instead for the case of SRCS variations.
For the case of varying SECS, normalized relative change in EnKF R results vs an OL baseline [i.e., 100 × (REnKF − ROL)/ROL] for (left) SSM and (right) RZSM within the (a),(b) SE and (c),(d) SGP domains. Within each panel, the nature-run LSM configuration (see Table 2) varies discretely along the x axis and the assimilation LSM configuration varies discretely along the y axis, with SECS increasing upward or to the right in the panel. Parenthetical values represent mean SECS values associated with the four NoahMP configurations listed in Table 2.
Citation: Journal of Hydrometeorology 25, 1; 10.1175/JHM-D-23-0069.1
As in Fig. 1, but for the case of varying SRCS across the four NoahMP configurations listed in Table 1.
Citation: Journal of Hydrometeorology 25, 1; 10.1175/JHM-D-23-0069.1
The range of SECS and SRCS values for NoahMP parameterizations presented in Tables 1 and 2 approximates the range of observed and LSM-based SECS and SRCS values reported in the literature (see section 2c). Therefore, the range of SFTDA experiments considered for water state results in Figs. 1 and 2, as well as water flux results presented later in Figs. 3–6, are relevant for real data LDAS applications.
For the case of varying SECS, normalized relative change in EnKF MAE results vs an OL baseline [i.e., 100 × (MAEEnKF − MAEOL)/MAEOL] for (left) Qs and (right) ET within the (a),(b) SE and (c),(d) SGP domains. Within each panel, the nature-run LSM configuration (see Table 2) varies discretely along the x axis and the assimilation LSM configuration varies discretely along the y axis, with SECS increasing upward or to the right in the panel. Parenthetical values represent mean SECS values associated with the four NoahMP configurations listed in Table 2.
Citation: Journal of Hydrometeorology 25, 1; 10.1175/JHM-D-23-0069.1
As in Fig. 3, but for the case of varying SRCS across the four NoahMP configurations listed in Table 1.
Citation: Journal of Hydrometeorology 25, 1; 10.1175/JHM-D-23-0069.1
For the case of varying SECS in the SE domain, normalized relative change in EnKF MAE results vs an OL baseline [i.e., 100 × (MAEEnKF − MAEOL)/MAEOL] for (left) Qs and (right) ET achieved (a),(b) with and (c),(d) without rescaling of synthetic SSM observations prior to their assimilation. Within each panel, the nature-run LSM configuration (see Table 2) varies discretely along the x axis and the assimilation LSM configuration varies discretely along the y axis, with SECS increasing upward or to the right in the panel. For the SE domain, parenthetical values represent mean SECS values associated with the four NoahMP configurations listed in Table 2.
Citation: Journal of Hydrometeorology 25, 1; 10.1175/JHM-D-23-0069.1
As in Fig. 5, but for the case of varying SRCS across the four NoahMP configurations listed in Table 1 (for the SE domain only).
Citation: Journal of Hydrometeorology 25, 1; 10.1175/JHM-D-23-0069.1
EnKF SSM and RZSM results in Figs. 1 and 2 generally show relative positive increases in SM R (following assimilation and relative to the OL) of 10%–30%. Note that widespread improvement is found for both SSM and RZSM results and across all examined SFTDA experiments. Specifically, there is no clear tendency for improvement to be maximized when the assimilation and nature LSMs are identically configured (i.e., the synthetic cases found along the diagonal of each subfigure in Figs. 1 and 2). That is, the precision of SSM and RZSM state estimates is generally improved regardless of whether the assimilation LSM accurately represents true state–flux coupling. In this way, SFTDA results in Figs. 1 and 2 are consistent with a range of previous LDAS studies that demonstrate the ability of SSM data assimilation to consistently improve SSM and RZSM across a broad range of existing assimilation LSMs.
The single exception is for EnKF RZSM results when the assimilation LSM overestimates SECS within the SGP domain. In this case (captured in the top-left corner of Fig. 1d), EnKF RZSM estimates show degraded precision relative to the OL. This exception underscores that, while EnKF SSM and RZSM results tend to be relatively robust to SECS or SRCS bias in the assimilation LSM, there exists a point beyond which systematic errors become too profound to support RZSM improvement—see additional discussion of this case in section 4c below.
b. Surface water fluxes
In contrast to water state results in Figs. 1 and 2, EnKF Qs and ET flux accuracies are more sensitive to the presence of SECS or SRCS bias in the assimilation LSM. This is seen in Fig. 3 (for SECS scenarios) and Fig. 4 (for SRCS scenarios) where the relative advantage of SSM assimilation for both Qs and ET MAE [i.e., 100 × (MAEEnKF − MAEOL)/MAEOL] falls sharply for SFTDA results that lie off the diagonal in each subfigure—the sole exception being ET when varying SRCS (see Figs. 4b,d). As noted above, plotted MAEEnKF and MAEOL values are based on sampling across time and then averaging over space.
Since Qs tends to be a smaller component of the surface water budget than ET, the presence of SRCS bias (i.e., bias in the coupling strength between SSM and Q) does not feed strongly enough via water balance considerations back into RZSM estimates to significantly degrade ET—particularly in the more arid SGP domain with less Q. Therefore, SFTDA ET results are relatively robust to the presence of SRCS bias in Fig. 4. However, in all other water flux cases shown in Figs. 3 and 4, off-diagonal SFTDA results, reflecting either SECS or SRCS bias, are associated with a reduction in water flux accuracy.
As a result, a range of SECS and SRCS bias scenarios exist that can damage water flux accuracy (Figs. 3 and 4) while having little, or no, analogous impact on water state precision (Figs. 1 and 2). When cross-comparing water state and flux figures, note how the color bar has been flipped in Figs. 3 and 4 relative to Figs. 1 and 2 to accommodate the change in evaluation metric from R to MAE and our desire for blue shading to consistently indicate positive EnKF SSM data assimilation results (i.e., a reduction in MAE or an increase in R versus the OL).
Interestingly, the negative impact on water flux accuracy is not always diagonally symmetric with respect to coupling strength bias. This break in symmetry is not present in analogous OL results (see appendix A) and therefore must arise during data assimilation. Most notably, the worst off-diagonal SFTDA results in Fig. 3 are found for cases where SECS is overestimated by the assimilation LSM—note the tendency for red shading to cluster in the top-left corner of subfigures in Fig. 3. Unfortunately, this worst-case scenario for SECS bias aligns with the suspected tendency for LSMs to overestimate SECS in the SGP domain (see section 2c). This overlap is discussed further in section 4a. In contrast to SECS results in Fig. 3, the impact of SRCS bias produces a more symmetrical response in Fig. 4 and there is no comparable asymmetry regarding the impact of negative versus positive SRCS bias in the assimilation LSM.
c. Role of rescaling
The ubiquity of systematic differences between LSM and observed/retrieved SSM products is well known and typically addressed through the rescaling of SSM observations prior to their assimilation into an LSM (Reichle et al. 2004). Therefore, it is useful to consider the role of such rescaling on the (generally poor) off-diagonal water flux accuracy results reported in Figs. 3 and 4—all of which are obtained following the application of a CDF-based SSM rescaling approach.
To start, consider the case of biased SECS examined in Fig. 5 within the SE spatial domain. Figures 5a and 5b replot the SE-domain SFTDA results shown earlier in Figs. 3a and 3b and obtained using CDF-based SSM rescaling. Figures 5c and 5d plot identical results except lacking any rescaling of SSM observations prior to their assimilation. For this new case of no rescaling, SSM assimilation uniformly moves LSM SSM and RZSM results toward their corresponding true absolute values. Since, in Fig. 5, systematic errors in the assimilation LSM are limited to only SECS (i.e., the relationship between SM and ET), the assimilation LSMs are uniformly capable of correctly linking SM with Qs. Therefore, irrespective of any SECS bias present in the assimilation LSM, consistent improvement in the absolute accuracy of SSM and RZSM estimates will directly translate into more accurate Qs. Such uniform improvement is clearly seen in Qs results shown in Fig. 5c for the case lacking SSM rescaling.
However, for off-diagonal SFTDA cases, uniformly good Qs MAE results in Fig. 5c are sharply degraded when SSM observations are first rescaled to match the SSM climatology of the assimilation LSM—compare good off-diagonal results in Fig. 5c for the no scaling case with much worse off-diagonal results in Fig. 5a for the scaling case. Since the assimilation LSM is impacted by SECS bias, which feeds back into SSM and RZSM calculations via water balance considerations, rescaling observed SSM to the assimilation LSM will add bias into the EnKF SSM and RZSM analyses relative to their corresponding true values. Furthermore, since the SM versus Qs relationship is assumed to be correct in Fig. 5, the introduction of these SM errors leads to a comparable loss of accuracy in Qs. Therefore, in the case of biased SECS (and unbiased SRCS) shown in Fig. 5, SSM rescaling has a net negative impact on Qs accuracy for off-diagonal SFTDA cases.
In contrast, SSM rescaling has a net positive impact on ET accuracy—compare Fig. 5d with Fig. 5b. In this case, SSM rescaling operates more in line with our prior expectations. That is, SECS bias causes a disruption in the relationship between SSM and ET, such that the assimilation of true SSM values may cause a degradation in ET fluxes. Improved ET can therefore be obtained by scaling SSM observations away from their true absolute values and toward the LSM-specific SSM climatology of the assimilation LSM.
Figure 6 shows comparable cases within the SE domain, but for the case of varying SRCS bias instead of SECS. Results here are analogous to Fig. 5 in that, for off-diagonal SFTDA cases with SRCS bias, rescaling is a net positive for ET but a net negative for Qs.
Summarizing Figs. 5 and 6, we see that, in the presence of coupling strength biases in the assimilation LSM between SSM and a particular water flux type (i.e., ET or Qs), SSM rescaling does indeed help for that particular flux type; however, it also causes off-diagonal degradation for the other flux type. That is, SSM rescaling can degrade Qs in the case of SECS bias and ET in the case of SRCS bias. Therefore, in terms of addressing SECS or SRCS bias, the overall effectiveness of SSM rescaling is mixed and cases can be constructed where such rescaling has a net negative impact on the accuracy of the subsequent water flux analysis. Note that all results in Figs. 5 and 6 are taken from the SE regional domain. However, comparable results (see Figs. B1 and B2 in appendix B) are found in the more arid SGP domain, suggesting that the tendencies seen in Figs. 5 and 6 are robust to variations in climate and land cover.
4. Discussion
a. Implications for SSM rescaling strategies
The SFTDA flux accuracy results in Fig. 3 suggest a diagonal asymmetry whereby the case of positive SECS bias is relatively more damaging for water flux accuracy during SSM data assimilation than an analogous negative SECS bias. This break in symmetry is concerning because it suggests that SECS bias is most damaging for the exact case that appears most prevalent in existing LSMs (see discussion in section 2c). Subsequent SFTDA results presented in Fig. 5 suggest that this asymmetry arises from SSM rescaling. Note how, in Fig. 5c, SFTDA results lacking such rescaling do not exhibit any comparable asymmetry; rather asymmetry develops only after the introduction of SSM rescaling in Fig. 5a. As discussed above, for the synthetic case examined in Fig. 5 (i.e., biased SECS but perfect SRCS in the assimilation LSM), rescaling tends to draw LDAS SSM results away from SSM time series values associated with good Q estimates. Naturally, the negative impact of this SSM deviation is most pronounced in SFTDA scenarios where the assimilation LSM is most sensitive to underlying SSM levels (i.e., SFTDA cases found along the top half of each subfigure in Fig. 5). Therefore, in the case of biased SECS (and unbiased SRCS), it is the asymmetrical impact of SSM rescaling that produces the diagonally asymmetric response in Qs accuracy.
It is also worth considering the exact type of SSM rescaling approach applied. Above, all rescaling is based on the calculation of a single, long-term SSM CDF for both the assimilation LSM as well as the assimilated SSM observations. An alternative strategy is calculating a separate CDF for each calendar month. This stronger form of rescaling can be used to correct for seasonal SSM climatological differences existing between the assimilation LSM and assimilated SSM observations. This additional capability is notable because LSM SECS and SRCS errors often manifest as seasonally varying conditional bias in RZSM and/or ET (Crow et al. 2020). To examine the impact of different rescaling strategies, we regenerated the water flux MAE results in Figs. 5 and 6 for the case of constructing different SSM CDFs, and therefore different nonlinear SSM rescaling relationships, separately for each calendar month of the year. However, SFTDA results shown in the appendices (see Figs. C1 and C2 in appendix C) reveal no qualitative change versus Figs. 5 and 6 in response to this variation in rescaling strategy. Nevertheless, given the overall importance of SSM rescaling on the development of asymmetrical responses to LSM SECS and SRCS biases, it seems likely that flux accuracy results will retain some sensitivity to the methodological details of SSM rescaling, which can vary significantly (Yilmaz et al. 2016). Therefore, additional work is needed to clarify the robustness of the asymmetrical patterns seen in Figs. 3 and 5.
A more robust conclusion is that the general practice of SSM rescaling, while clearly needed in certain LDAS cases, is not always an effective strategy for improving the accuracy of LDAS water flux estimates. Specifically, Figs. 5 and 6 reveal its limitations in the presence of systematic errors associated with LSM SECS or SRCS biases. When coupling bias exists in the assimilation LSM between SSM and a particular type of water flux (e.g., ET or Qs), rescaling will only help for that specific flux and may have a net negative impact for other flux types. Therefore, the benefits of SSM rescaling are not comprehensive or robust in the case of LSM state–flux coupling biases.
b. Implications for LDAS best practices
Results suggest that ignoring the presence of systematic errors in an assimilation LSM, or assuming that such errors can be effectively mitigated through SSM rescaling, is not a robust strategy and contributes to the relatively poor track record of applying LDAS to forecasting applications reliant on the accurate estimation of surface water fluxes. Considered in context with previous results underscoring the loss of information accompanying SSM rescaling (Nearing et al. 2018), theoretical problems with the multiplicative aspect of rescaling (Yilmaz and Crow 2013), and the tendency for SSM rescaling approaches to obscure missing land surface processes in LSMs (Kumar et al. 2015), the (flux-based) rescaling limitation noted above adds additional weight to existing criticism of SSM rescaling approaches for LDAS.
Instead, additional direct assessment and calibration strategies are required to correct systematic LSM errors prior to, or in parallel with, state-based data assimilation. For example, past studies have demonstrated that useful SECS and SRCS diagnostics can be generated using existing remotely sensed SSM and ET datasets—see, e.g., Crow et al. (2018) or Lei et al. (2018). This provides an opportunity to detect, and perhaps mitigate, SECS and SRCS biases in an assimilation LSM before they can degrade an LDAS water flux analysis. Therefore, it is our view that best practices for LDAS should include an analysis of SECS or SRCS biases in assimilation LSM OL predictions, particularly when applying LDAS to flux-dependent problems like hydrological forecasting and NWP.
However, additional work is still needed to evaluate the sufficiency of remotely sensed metrics like SECS and SRCS for the robust correction of systematic LSM biases. For example, systematic biases in assimilation LSM SM predictions are likely to persist even following the complete correction of LSM SECS and SRCS biases. Nevertheless, it may be possible to correct SECS and SRCS biases within a broader calibration framework designed to reduce systematic errors in an assimilation LSM prior to SSM data assimilation—and thus minimize the need for additional SSM rescaling (Zhou et al. 2022). This would almost certainly result in a more robust LDAS analysis. It may also be possible to obtain comparable calibration results using a joint state/parameter data assimilation approach (see, e.g., Moradkhani et al. 2005). While challenges remain, these approaches are motivated by mounting evidence that existing L-band, satellite-based products now generally provide improved SM climatological information versus comparable LSM estimates (Dong et al. 2020b).
c. Factors leading to the degradation of water state estimates
Relative to water fluxes, EnKF water state estimates are more robust to the presence of SECS or SRCS bias in the assimilation LSM. However, there are limits to this lack of sensitivity. Specifically, Fig. 1d indicates an unusual loss of SGP RZSM precision in the case of positive SECS bias within the assimilation LSM. A close examination of this case is useful for determining the limits of LDAS water state robustness to systematic LSM errors.
As discussed in section 2c, LSM SECS bias is introduced into our SFTDA experiments through variations in the NoahMP λ parameter. These variations are closely linked to NoahMP predictions regarding the relative importance of surface evaporation E versus transpiration T as a contributor to total ET. In particular, higher values of λ produce higher E/T ratios (Dong et al. 2020a). This link has two important implications. First, it means that, due to the unique ability of E to quickly reduce SSM levels, larger λ (and thus SECS) is associated with decreased SSM serial autocorrelation. Within our SFTDA experiments, this means that SECS biases will lead to time-scale memory differences between assimilated SSM observations and the estimated background SSM provided by the assimilation LSM—a condition that is known to degrade the quality of SM estimates in an LDAS (Qiu et al. 2014). Second, larger values of E/T also reduce the temporal cross correlation between concurrent NoahMP SSM and RZSM estimates. Therefore, our SECS SFTDA results partially reflect the presence of vertical SSM/RZSM coupling bias within the assimilation LSM. This is notable since bias in vertical SM coupling is known to degrade the quality of EnKF RZSM estimates (Kumar et al. 2009).
These two factors are behind the unusual degradation of EnKF RZSM results in Fig. 1d. For comparison purposes, Figs. 7a and 7b replicate earlier SGP SSM and RZSM SECS bias results shown in Figs. 1c and 1d. In addition, Fig. 7c plots innovation autocorrelation [lag = 1 (day)] values for all SGP SECS SFTDA configuration pairs. Innovations reflect the difference between assimilated SSM observations and background assimilation LSM SSM predictions immediately prior to SSM assimilation. Serially white innovations is a necessary and sufficient condition for an optimal EnKF analysis. Therefore, positive innovation autocorrelation values, associated with systematic differences in the serial memory of SSM observations versus background assimilation LSM SSM estimates (see above), plotted in the off-diagonal portions of Fig. 7c suggest a potential for degraded water state estimates.
For the case of varying SECS in the SGP domain, normalized relative change in EnKF R results vs an OL baseline [i.e., 100 × (REnKF − ROL)/ROL] for (a) SSM and (b) RZSM. For the same SFTDA case (i.e., varying SECS in the SGP domain), also shown are (c) lag 1-day autocorrelation of EnKF innovations, and (d) LSM bias (i.e., assimilation LSM minus nature-run LSM) in SSM vs RZSM temporal cross correlation.
Citation: Journal of Hydrometeorology 25, 1; 10.1175/JHM-D-23-0069.1
However, there are qualitative differences between the patterns of positive innovation autocorrelation in Fig. 7c and the degradation of EnKF RZSM estimates in Fig. 7b. Most notably, Fig. 7c shows positive autocorrelation for the cases of both positive and negative SECS bias in the assimilation LSM, whereas RZSM degradation in Fig. 7b is limited only to the case of positive SECS bias. As a result, some additional factor must be responsible for the break in diagonal symmetry seen in Fig. 7b. This additional factor is the sign of vertical SSM/RZSM overcoupling bias plotted in Fig. 7d. As discussed above, positive SECS bias in the assimilation LSM is associated with a corresponding negative bias in vertical SSM/RZSM coupling. That is, overestimating SECS, and therefore E/T, also causes the assimilation LSM to underestimate the vertical coupling between SSM and RZSM. For the case of assimilating SSM, the underestimation of vertical coupling in an assimilation LSM is known to produce a relatively stronger reduction in the quality of the EnKF RZSM analysis versus the corresponding case of overestimating vertical coupling (Kumar et al. 2009).
Therefore, RZSM degradation seen in the top-left portion of Fig. 1d appears to arise from a combination of known innovation autocorrelation (Fig. 7c) and vertical coupling bias (Fig. 7d) effects associated with positive SECS bias within the SGP region. The fact that these effects are stronger in the SGP region than in the SE region is likely due to the increased importance of ET, and thus SECS, in the relatively arid SGP region. Interestingly, the RZSM degradation seen in the top-left portion of Fig. 1d (and Fig. 7b) is improved when SSM rescaling is removed (results not shown), suggesting that, analogous to the water flux accuracy results shown in Figs. 5 and 6, there also exist LSM systematic error cases whereby water state precision results are also negatively impacted by SSM rescaling. However, additional work is needed to better define and understand such cases.
Considered more broadly, the results in Fig. 7 underscore how SFTDA experimental results are relatively sensitive to the exact type and magnitude of systematic errors reflected in their setup. For example, Kumar et al. (2009) utilized a similar SFTDA approach to examine the impact of vertical water coupling errors within an assimilation LSM and found generally higher off-diagonal degradation in RZSM precision than results here. This contrast is likely due to the nature of the systematic errors considered in each analysis. Kumar et al. (2009) focused on systematic errors affecting the vertical coupling between SSM and RZSM. In contrast, we focus here on systematic errors affecting water state–flux coupling. Therefore, to be relevant to real-world LDAS applications, SFTDA experiments must be based on systematic errors known to be present in LSMs. For this analysis, such relevance is based on the observed tendency for LSMs to predict SECS or SRCS values that differ substantially between LSMs and contain bias relative to existing observation-based SECS and SRCS baselines—see earlier discussion in section 2c. As a result, we believe our SFTDA experiments provide relevant insight into the current inability of LDAS to robustly constrain surface water fluxes and, thereby, contribute to important hydrological and NWP forecasting applications.
5. Conclusions
Here, we describe a series of SFTDA experiments that illustrate the sensitivity of LDAS surface water state precision and water flux accuracy results to the magnitude of state–flux coupling bias (i.e., bias in SECS or SRCS) present in an assimilation LSM. The experiments illustrate that while the precision of LDAS water state estimates is generally robust to the presence of coupling strength biases, see Figs. 1 and 2 and discussion of the exception to this rule in section 4c, the accuracy of water flux estimates is sharply degraded by realistic levels of such bias (see Figs. 3 and 4). This suggests that the successful application of LDAS to constrain water flux estimates (i.e., ET and Qs) makes more stringent demands on the overall reliability of assimilation LSMs than comparable activities aimed at only improving SSM or RZSM temporal precision. We suggest that these additional demands represent a major reason why the application of LDAS to improve water state precision (targeting, e.g., agricultural drought applications) has significantly outpaced comparable forecast applications dependent on improved surface water flux accuracy.
Systematic differences between assimilation LSMs and observations are typically resolved by rescaling observations to match climatological aspects of the assimilation LSM. However, such rescaling does not always address the systematic LSM biases examined here. In certain cases (e.g., the case of estimating Qs in the presence of coupling strength biases in the relationship between SM and ET), SSM rescaling can even degrade flux accuracy. This underscores the theoretical limitations of observation rescaling as a preprocessing step in an LDAS and provides further motivation for attempts to develop alternative approaches for addressing such differences (e.g., LSM calibration prior to any LDAS activity). Fortunately, recent advances in water state and flux remote sensing have made it possible to assess LSM SECS and SRCS predictions across broad geographic regions. Such assessments should be a recommended best practice for the development of any LDAS application, particularly those focused on the accuracy of surface water and energy flux estimates.
Although SFTDA experiments presented here all focus on the assimilation of SSM, there is no reason to expect that key conclusions would change if lower-level observations (e.g., brightness temperature) that form the basis of satellite-based SSM retrievals were assimilated instead.
Acknowledgments.
This research was supported by a NASA SMAP science team grant (80HQTR21T0054: “Using SMAP soil moisture products to improve streamflow forecasting in ungauged basins”) awarded to author W.T. Crow. The USDA Agricultural Research Service is an equal opportunity employer.
Data availability statement.
All datasets used to generate and evaluate the SFTDA experiments described here are publicly available and can be accessed via the NASA LIS webpage.
APPENDIX A
Summary of OL SFTDA State and Flux Results Obtained Prior to Data Assimilation
Figures A1–A4 describe SFTDA OL state and flux evaluation metric results for permutations of the four NoahMP SRCS and SECS configurations summarized in Tables 1 and 2 and across both the SE and SGP domains.
For the case of varying SECS, OL R results for (left) SSM and (right) RZSM within the (a),(b) SE and (c),(d) SGP domains. Within each panel, the nature-run LSM configuration (see Table 2) varies discretely along the x axis and the assimilation LSM configuration varies discretely along the y axis, with SECS increasing upward or to the right in the panel. Parenthetical values represent mean SECS values associated with the four NoahMP configurations listed in Table 2.
Citation: Journal of Hydrometeorology 25, 1; 10.1175/JHM-D-23-0069.1
As in Fig. A1, but for the case of varying SRCS across the four NoahMP configurations listed in Table 1.
Citation: Journal of Hydrometeorology 25, 1; 10.1175/JHM-D-23-0069.1
For the case of varying SECS, OL MAE results for (left) Qs and (right) ET within the (a),(b) SE and (c),(d) SGP domains. Within each panel, the nature-run LSM configuration (see Table 2) varies discretely along the x axis and the assimilation LSM configuration varies discretely along the y axis, with SECS increasing upward or to the right in the panel. Parenthetical values represent mean SECS values associated with the four NoahMP configurations listed in Table 2.
Citation: Journal of Hydrometeorology 25, 1; 10.1175/JHM-D-23-0069.1
As in Fig. A3, but for the case of varying SRCS across the four NoahMP configurations listed in Table 1.
Citation: Journal of Hydrometeorology 25, 1; 10.1175/JHM-D-23-0069.1
APPENDIX B
The Impact of SSM Rescaling on Water Flux MAE Results in the SGP Domain
Figures B1 and B2 provide SGP-domain results for comparison with earlier SE-domain results shown in Figs. 5 and 6.
As in Fig. 5, but for the SGP spatial domain.
Citation: Journal of Hydrometeorology 25, 1; 10.1175/JHM-D-23-0069.1
As in Fig. 6, but for the SGP spatial domain.
Citation: Journal of Hydrometeorology 25, 1; 10.1175/JHM-D-23-0069.1
APPENDIX C
The Impact of Month-by-Month SSM Rescaling on Water Flux MAE Results
SFTDA results for the case of month-by-month CDF rescaling are shown in Figs. C1 and C2.
As in Fig. 5, but for separate CDF rescaling applied for each calendar month.
Citation: Journal of Hydrometeorology 25, 1; 10.1175/JHM-D-23-0069.1
As in Fig. 6, but for separate CDF rescaling applied for each calendar month.
Citation: Journal of Hydrometeorology 25, 1; 10.1175/JHM-D-23-0069.1
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