Remotely Sensed Soil Moisture Assimilation in the Distributed Hydrological Model Based on the Error Subspace Transform Kalman Filter

Abstract

The data assimilation of remotely sensed soil moisture observations provides a feasible path of improving river flow simulation. In this work, we studied the performance of the error subspace transform Kalman filter (ESTKF) assimilation algorithm on the assimilation of remotely sensed soil moisture from SMAP, including the improvement of soil moisture and river flow in the hydrological model. Additionally, we discussed the advantages and added value of ESTKF compared to the ensemble Kalman filter (EnKF) in a hydrological model. To achieve this objective, we solved the spatial resolution gap between the remotely sensed soil moisture and the simulated soil moisture of the hydrological model. The remotely sensed soil moisture from SMAP was assimilated into the first layer soil moisture in the distributed hydrological model. The spatial resolution of the hydrological model was 600 m, while the spatial resolution of the SMAP remotely sensed soil moisture was 9 km. There is a considerable gap between the two spatial resolutions. By employing observation operators and observation localization based on geolocation, the distributed hydrological model assimilated multiple remotely sensed soil moisture values for each grid, thereby ensuring the consistent updates of soil moisture in the model. The results show the following: (1) In terms of improving soil moisture, we found that both ESTKF and EnKF were effective, and the ubRMSE of ESTKF was lower than that of EnKF. (2) ESTKF improved most cases where open-loop high river flow simulations were too low, but EnKF did not improve this situation. (3) In ESTKF, the relative error of flood volume was reduced on average to 2.52%, but the relative error of flood peak did not improve. The results provide evidence of the value of ESTKF in the hydrological model.

1. Introduction

In the hydrological cycle, soil moisture characterizes land surface processes and controls the complex relationship between precipitation and river flow [1,2,3]. Hydrological models are effective methods for simulating soil moisture and are used to predict river flow. Initial soil moisture conditions introduce uncertainty into hydrological models [4] and control the response time to precipitation [5]. The inaccurate temporal and spatial distribution of precipitation can lead to soil moisture anomalies in hydrological models [6,7]. Different initial soil moistures can affect the hydrological response to precipitation [8,9]. Soil moisture is shown to be the main contribution to model uncertainty [10]. In order to obtain reasonable modeling results of the hydrological model, soil moisture should be constrained during the hydrological model simulation. Therefore, the accurate estimation of soil moisture is crucial to reducing hydrological model uncertainty and improving river flow simulation.

It has been long recognized that remote sensing observations of soil moisture can be used to improve the simulation results of hydrological models [11,12,13]. There are few in situ observations of soil moisture in most regions, but remotely sensed soil moisture provides global surface soil moisture estimates at tens of kilometers of spatial resolution [14]. The most widely used remotely sensed soil moisture systems include the Soil Moisture Active and Passive (SMAP), the Soil Moisture and Ocean Salinity (SMOS), Advanced Scatterometer (ASCAT), the Advanced Microwave Scanning Radiometer-Earth Observing System (AMSR-E), and the Advanced Microwave Scanning Radiometer 2 (AMSR2) [15].

There is often a mismatch in spatial resolution between remote sensing data and hydrological models. Because remotely sensed soil moisture has spatial resolutions of tens of kilometers, a spatial downscaling to several kilometers or even hundreds of meters is required for hydrological models [16]. How to downscale remotely sensed soil moisture in hydrologic models need to be addressed. The downscale schemes can be achieved through satellite, geo-information data, and model based approaches [17]. Satellite-based methods combine active and passive microwave retrievals [18], while geo-information data-based methods retrieve soil moisture at high spatial resolutions from land surface parameters, such as vegetation indexes [19], land surface temperatures [20], meteorological data [21], and terrain attributes [22]. Model-based methods use data assimilation to update fine-scale model soil moisture states with coarse-scale remotely sensed soil moisture observations, which can be considered as downscaled soil moisture [23].

Assimilating remotely sensed soil moisture into hydrological models has shown promising results in reducing model prediction uncertainty and improving model forecasting skills [24,25,26,27,28,29,30]. One of the early demonstrations came from Scipal et al. [31]. They explored the relationship between soil moisture and runoff at the catchment scale with coarse resolution remote sensing data. The observed high correlations between basin-averaged soil moisture and runoff time series (R² > 0.85) demonstrated that the seasonal change from low runoff during the dry season to high runoff during the wet season is well captured by the ERS scatterometer. Brocca et al. [32] investigated the optimal soil moisture monitoring scheme at different scales both for the validation of remotely sensed soil moisture and for application in rainfall-runoff models. Sahoo et al. [33] and Lievens et al. [34] both used data assimilation algorithms to assimilate remotely sensed soil moisture into models. Sahoo et al. used ensemble Kalman filters (EnKF) to assimilate AMSR-E soil moisture into a land surface model. The EnKF algorithm better preserved spatial coherence and accelerate moisture redistribution compared to model integrations without assimilation. Lievens et al. assimilated SMOS soil moisture into a hydrologic model using EnKF. The assimilation improved antecedent wetness conditions and lead to the improved predictions of associated water fluxes.

Many of the related studies that assimilate remotely sensed soil moisture into hydrological models use the ensemble Kalman filter (EnKF) as an assimilation algorithm. Other assimilation algorithms used in relatively fewer studies include the extended Kalman filter (EKF) [35], the variation assimilation (VAR) [36], and the ensemble transform Kalman filter (ETKF) [37]. Recently, a newer data assimilation algorithm called the error subspace transform Kalman filter (ESTKF) was introduced, which is a variant of the EnKF that uses error subspace transformation to improve the estimation of model state variables [38]. This subspace transformation helped to reduce the dimensionality of the state space. It has a smaller computational cost due to the reduced dimensionality of the state space [39]. The ESTKF has been applied in data assimilations for atmospheric [40], oceanic [41,42], and sea ice [43] examples, but there are few related studies in the field of hydrology. The performance of the ESTKF is currently uncertain when assimilating the latest remotely sensed SMAP soil moisture into hydrological models.

The performance of assimilating remotely sensed soil moisture into hydrological models depends on the spatial correlation of soil moisture dynamics. High observation coverage results in a better assimilation performance, depending on its spatial distribution. It has been shown that the spatial correlation of soil moisture was helpful for improving surface soil moisture characterization, especially for the uncovered grid of remotely sensed soil moisture [37]. One of the proposed solutions is observation localization [44,45]. For the local analysis domains of each model grid, observations within a certain defined radius are considered to be observation domains. The localization weighting is equal to one at the local analysis domains. It gradually reduces towards zero as the distance increases in observation domains, and it is zero beyond the observation domains. The observation domains are usually larger than the local analysis domains, which ensures some smoothness of the state analysis estimate [46]. Therefore, it is necessary to consider setting the appropriate spatial corrections and radius of observation domains in data assimilation.

In summary, this study aimed to analyze the following: (1) the performance of the ESTKF algorithm on the assimilation of remotely sensed soil moisture, including the improvement of soil moisture and river flow in the hydrological model, and (2) the advantages and added value of ESTKF compared to EnKF in the hydrological model.

In order to achieve these objectives, a basin in the Qilian Mountains, Northwest China, named Babao River basin, was selected as the study area. Babao River basin is a well-monitored and semi-arid basin with an area of 2484 km². The remotely sensed soil moisture is SMAP-enhanced L2 radiometer surface soil moisture with a spatial resolution of 9 km. The study time encompassed the flood periods from July to September 2015–2017. This paper is organized as follows: Section 2 outlines the study area and the details of data assimilation materials and methods. Section 3 and Section 4 show the results of remotely sensed soil moisture assimilation and the discussion, respectively. The Section 5 contains the conclusions.

2. Materials and Methods

2.1. Study Area

The study area Babao River basin is located in the Qilian Mountains, Northwest China (Figure 1). The altitude of the Babao River basin ranges from 4750 m to 2690 m, and the basin area is 2484 km². The Babao River originates in the southeast of the basin and flows through mountain valleys from east to west. The temperature difference between day and night is significant. The river flow in the basin comes from precipitation supplemented by glacier meltwater in winter and spring. The Babao River basin has complete hydrological observations, including continuous river runoff, soil moisture gauges, and meteorological gauges.

2.1.1. River Flow Observations

The observation of river runoff is located at the outlet of the Babao River Basin. The river flow observations were a daily series and ranged from 2014 to 2017. The data source is the Annual Hydrological Report of the People’s Republic of China. Results were calculated using the relationship curve method established by the relationship line of water level and flow, and the standard deviation of the main curve was 3.0–4.6%.

2.1.2. In Situ Soil Moisture Network

The soil moisture gauges in the Babao River basin are shown in Figure 1. The data source was the hourly soil moisture dataset observed by the eco-hydrological sensor network in the upper reaches of Heihe River (2013–2017) [47,48,49]. The observation period was from July 2013 to December 2017. The soil moisture sensor used was the Hydra Probe II. The sensor locations were 4 cm and 20 cm below the surface, with an observation frequency of 1 h.

2.1.3. Meteorology Observation Network

The meteorology data source was the Heihe Watershed Allied Telemetry Experimental Research (HiWATER) [50]. The observation period was from 1 January 2014 to 31 December 2017. Observation items included air temperature (°C), relative humidity (%), air pressure (hPa), precipitation (mm/hour), and wind speed (m/s). The original data frequency was 10 min, which was uniformly processed into a frequency of 1 h.

2.2. Remotely Sensed Soil Moisture

This study used the enhanced L2 radiometer surface soil moisture from SMAP (SPL2SMP_E) with a spatial resolution of 9 km. SPL2SMP_E was retrieved from SMAP interpolated antenna temperatures by the SMAP onboard radiometer at two local times, 6:00 a.m. and 6:00 p.m., which are descending and ascending half-orbit passes, respectively. Optimal Backus–Gilbert interpolation techniques extracted the maximum amount of information from SMAP antenna temperatures and converted it to brightness temperatures, which were posted to the 9 km grid in a global cylindrical projection. These 9 km brightness temperatures were then used to retrieve the surface soil moisture posted on the 9 km grid as SPL2SMP_E [51,52].

2.2.1. Observation Accuracy

SMAP soil moisture data had an expected unbiased root mean square error (ubRMSE) of 0.04 cm³/cm³ [25]. Figure 2 shows the relatively coarse spatial grids of SPL2SMP_E (9 km) with respect to the Babao River basin. Some studies have evaluated the accuracy of remotely sensed SMAP soil moisture in the Babao River basin. Wang et al. [53] evaluated SMAP-enhanced level 3 soil moisture products using a distributed ground observation network. SMAP L3 products behave well when capturing the soil moisture variation with the averaged ubRMSE value of 0.03 cm³/cm³. Zhang et al. [54] presented an evaluation of SMAP level 2 products, SPL2SMP_E (9 km) and SPL2SMP (36 km), against ground-based observations from in situ soil moisture. The R values were 0.367–0.608, all passing the significance test of 0.001. This indicates that the two SMAP level 2 products performed well when identifying the soil moisture trend. Therefore, we adopted the ubRMSE of SMAP as 0.04 cm³/cm³ in the study area.

2.2.2. Quality Control

This study used the remotely sensed soil moisture from SPL2SMP_E in the original 9 km grid without resampling as an observational input in the assimilation experiment. Given the extent and spatial resolutions of SPL2SMP_E, only the grid data falling inside the study area boundaries were selected. SPL2SMP_E coverage was achieved about once every 2–3 days in the study area. Along with the soil moisture, several ancillary data were provided from SMAP, such as albedo, bulk density, land surface status, vegetation opacity, freeze–thaw fraction, and retrieval quality flags. Low-quality SMAP grids with frozen, water surface, and retrieval-skipped conditions were removed.

2.3. Distributed Hydrological Model

For assimilation, we used the distributed hydrological model named GBHM (Geomorphology Based Hydrological Model) [55]. The model is able to simulate the infiltration and runoff processes of precipitation in the soil and has been previously modeled and applied in eco-hydrology at the location of the study area [56]. It focuses on modeling the soil hydrological process. The hydrological properties of unsaturated soils and the movement of soil moisture are described using the van Genuchten function and Richards function, respectively.

2.3.1. River Flow Generation

We constructed the distributed hydrological model for the Babao River basin with a horizontal spatial resolution of 600 m and a time step of 1 h. Figure 3 displays one of the model grids. The soil profile was divided into 10 layers, each with a thickness of 0.05 m, 0.1 m, 0.15 m, 0.2 m, 0.3 m, 0.5 m, 0.7 m, 1 m, 1.5 m, and 2 m, respectively.

In Figure 3, the precipitation ($P$) reaches the surface soil after being intercepted by vegetation. The surface evaporation rate is $E$. The rate of infiltration ($f_{in}$) on the surface depends on the precipitation ($P-E$) and the saturated hydraulic conductivity ($K_0$) of the soil profile.

$$f_{in}=\min \left(P-E,K_0\right)$$

(1)

The precipitation that does not infiltrate accumulates on the surface soil with a depth of $S$. When the maximum depth of accumulated water ($S_{max}$) on the surface is exceeded, surface-accumulated water will form surface runoff ($q_s$):

$$q_s=\mathit{max}\left(\frac{S-S_{max}}{\Delta t}\times D,0\right)$$

(2)

where $\Delta t$ is the time step, and $D$ is the horizontal spatial resolution. The upper boundary and lower boundary of the soil profile are the infiltration $f_{in}$ and the impermeable boundary, respectively. The soil moisture movement between soil layers can be calculated via the Richards equation:

$$\frac{\partial {\theta }_i}{\partial t}=-\frac{\partial q_{vi}}{\partial z_i}+s_i$$

(3)

where $t$ is the hydrological model time, ${\theta }_i$ is the soil moisture of the i-th soil layer, $z_i$ is the thickness of i-th soil layer, $q_{vi}$ is soil’s water infiltration rate of the i-th soil layer to the (i + 1)-th soil layer, and $s_i$ is the sink term of the i-th soil layer, such as vegetation root water uptake.

In the i-th soil layer, subsurface runoff $q_{mi}$ occurs:

$$q_{mi}=K_{mi}\times \mathit{tan}\left(\alpha \right)\times D$$

(4)

where $K_{mi}$ is the unsaturated hydraulic conductivity, and $\alpha$ is the grid slope.

The surface runoff $q_s$ and subsurface runoff $q_{mi}$ of each grid are regarded as the lateral inflow of the river channel.

$$q_L=q_s+{\displaystyle \sum }_{i=1}^{10}{q}_{mi}$$

(5)

The dynamic wave equation is used to calculate the river flow $Q$:

$$\frac{\partial Q}{\partial x}+\frac{\partial A}{\partial t}=q_L$$

(6)

$$Q=\frac{S_0^{\frac{1}{2}}}{n{P}^{\frac{2}{3}}}\times A^{\frac{5}{3}}$$

(7)

where $x$ is the length of the river channel, $A$ is the cross-sectional area of the river channel, $S_0$ is the river channel slope, $P$ is the wetted circumference of the river channel, and $n$ is the river channel roughness.

2.3.2. Soil Profile Parameters

Unsaturated soils have different soil moistures and unsaturated hydraulic conductivities than saturated soils. In unsaturated soils, the hydraulic conductivity is generally lower than in saturated soils because the presence of air in the soil pores creates resistance to the movement of water. The van Genuchten function is often used to describe the unsaturated hydraulic conductivity of unsaturated soils [57]:

$$\mathsf{\Theta }=\frac{\theta -{\theta }_r}{{\theta }_s-{\theta }_r}$$

(8)

$$h=-\frac{1}{\alpha }{\left({\mathsf{\Theta }}^{-\frac{1}{m}}-1\right)}^{\frac{1}{n}}$$

(9)

$$K_m=K_s\frac{{\left\{1-{\left(\alpha h\right)}^{n-1}{\left[1+{\left(\alpha h\right)}^n\right]}^{-m}\right\}}^2}{{\left[1+{\left(\alpha h\right)}^n\right]}^{\frac{m}{2}}}$$

(10)

where $\theta$ is the soil moisture, ${\theta }_r$ is the residual soil moisture, ${\theta }_s$ is the saturated soil moisture, $\mathsf{\Theta }$ is the effective saturation, $h$ is the pressure head, $K_m$ is the unsaturated hydraulic conductivity, and $K_s$ is the saturated hydraulic conductivity. The $\alpha$, m, and n are parameters of the van Genuchten function. The parameter m is usually:

$$m=1-\frac{1}{n}$$

(11)

The soil parameters used in the hydrological model for the study area are derived from the China Dataset of Soil Hydraulic Parameters [58], including the saturated hydraulic conductivity $K_s$, the saturated soil moisture ${\theta }_s$, the residual soil moisture ${\theta }_r$, and the parameters $\alpha$ and n of the van Genuchten function.

2.3.3. Meteorology Forcing Data

The hydrological model in the Babao River basin is driven by meteorology forcing data. The meteorology forcing data includes precipitation, air temperature, relative humidity, wind speed, and atmospheric pressure. Precipitation is the most forcing data, as it serves as the water input for the soil profile. To integrate the meteorological data into the hydrological model grids, the inverse distance weighting (IDW) method was employed. IDW assumes that the unknown value at a location is proportional to the inverse distance-weighted average of the known values at surrounding locations. IDW has been widely used for generating terrain maps, meteorology forecasting, and other applications.

2.4. Soil Moisture Assimilation Method

In this study, we assimilated the remotely sensed soil moisture from SMAP into the distributed hydrological model using the error subspace transform Kalman filter (ESTKF). Nerger et al. [38] showed the detailed derivation process of the assimilation method ESTKF. Here, we explain how the assimilation method ESTKF was used in combination with our distributed hydrological model.

The ESTKF algorithm has been implemented in the open source data assimilation framework PDAF [59,60]. PDAF was specially developed for high-performance parallel assimilation experiments. In addition to the ESTKF algorithm, there are many assimilation algorithms, including EnKF and its variants. Both PDAF and our distributed watershed hydrological model were written in Fortran code, so that the two can be easily combined into a complete assimilation program, and the watershed hydrological model can be calibrated and validated as needed in the same program. Figure 4 shows the workflow between the hydrological model and the assimilation method.

2.4.1. ESTKF

In the hydrological model, the surface soil moisture ${\theta }_1$ of all $n$ grid cells consists of the state vector $x={\left[{\theta }_1^{\left(1\right)},{\theta }_1^{\left(1\right)},\cdots ,{\theta }_1^{\left(n\right)}\right]}^{\mathrm{T}}$. The superscript $\mathrm{T}$ stands for matrix transpose. The ensemble $\mathrm{X}$ has $N$ members:

$$\mathrm{X}=\left[x_1,x_2,\cdots ,x_N\right]$$

(12)

The mean of ensemble $\mathrm{X}$ is $\overline{\mathrm{X}}$:

$$\overline{\mathrm{X}}=\frac{1}{N}{\displaystyle \sum }_{i=1}^N{x}_i$$

(13)

The ensemble of state vectors represents the distributed hydrological assimilation model uncertainties. The covariance of the ensemble can be given as:

$$\mathrm{P}=\frac{1}{N-1}\left(\mathrm{X}-\overline{\mathrm{X}}\right){\left(\mathrm{X}-\overline{\mathrm{X}}\right)}^{\mathrm{T}}$$

(14)

There are two processes, the forecast and analysis, in each time step of data assimilation.

The forecast state vectors at the time step $t$ are predicted using the analysis state vectors in the previous time step:

$${\mathrm{x}}_t^f=M\left({\mathrm{x}}_{ t-1}^a\right)$$

(15)

where $M$ is the distributed hydrological assimilation model. The superscripts $a$ and $f$ stand for the forecast and analysis, respectively.

The analysis process is used to update the ensemble of state vectors at the time step $t$:

$$X^a=\overline{X^f}+X^f\left(w+W\right)$$

(16)

$$w=TA{\left(H{X}^{f}T\right)}^T{R}^{-1}\left(y_0-H\overline{X^f}\right)$$

(17)

$$\mathrm{W}=\sqrt{N-1}T{\mathrm{A}}^{1/2}{T}^T$$

(18)

$${\mathrm{A}}^{-1}={\gamma }^{-1}\left(N-1\right)I+{\left(H{X}^{f}T\right)}^T{R}^{-1}H{X}^{f}T$$

(19)

where $w$ is the weighting vector that transforms the ensemble mean, and $W$ is the matrix that transforms the ensemble perturbation from the forecast process to the analysis process. $T$ is a projection matrix onto the error subspace. $\gamma$ is the forgetting factor for ensemble inflation, equal to one or less. $y$, $R$, and $H$ are the observation vector, observation accuracy, and observation operator from remotely sensed soil moisture, respectively.

2.4.2. Model State Vector

The state vector of the distributed hydrological assimilation model consists of the grid values of soil moisture across the entire study area. The model encompasses 6899 grids, each of which is composed of up to 10 soil layers that gradually increase in thickness from the surface downward. The thicknesses of these layers range from 0.05 m to 2.0 m, and each layer has a different thickness: 0.05 m, 0.1 m, 0.15 m, 0.2 m, 0.3 m, 0.5 m, 0.7 m, 1.0 cm, 1.5 m, and 2.0 m.

However, since the SMAP remote sensing technology can only measure soil moisture up to a few centimeters below the surface, the state vector primarily focused on the first layer (0.05 m) of soil moisture in the distributed hydrological model. By conducting this, the state vector had a length of $n=6899$. Additionally, a set of $N=48$ assimilation members was established to form the assimilation ensemble $\mathrm{X}$, which had dimensions of $6899\times 48$.

2.4.3. Observation Operator

The observation operator derived the simulated value of SMAP remotely sensed soil moisture from the state variables of the distributed watershed hydrological model. As the SMAP remotely sensed soil moisture has a spatial resolution of 9 km, while the distributed watershed hydrological model has a resolution of only 600 m, each SMAP observation covers approximately 225 state variables. The observation operator combined these variables by taking the arithmetic average to produce an observed value. When multiple SMAP observations were available, the observation vector y consisted of multiple observation values.

2.4.4. Ensemble Generation

To perform data assimilation, it is necessary to create an ensemble that can capture the errors that occur during model propagation. To generate an ensemble with $N=48$ members, the initial model state vector was perturbed using additive random error, which was kept constant across space prior to assimilation. The random error is normally distributed with a mean of zero and a standard deviation of 0.05 cm³/cm³.

The precipitation forcing was subject to perturbations caused by multiplicative random errors. The multiplier was selected from a lognormal distribution with a mean value of 1 and a dimensionless standard deviation of 0.3 [61]. The remaining meteorology forcing data, such as wind speed, relative humidity, and air temperature, were perturbed using either additive or multiplicative random errors with respect to a reference value [62]. The perturbed parameters are listed in

Table 1. The perturbed parameters of meteorology forcing data.

Climatic Forcing Data	Unit	Error Accumulation	Distribution	Standard Deviation
Precipitation	mm/hour	multiplicative	lognormal	0.3
Wind speed	m/s	multiplicative	lognormal	0.3
Relative humidity	%	multiplicative	lognormal	0.1
Air temperature	°C	additive	normal	4

.

2.4.5. Bias Correction of Observation

In data assimilation, merging model predictions and observations requires eliminating systematic differences between the two datasets. Usually, this is achieved by scaling the observations to match the model predictions before assimilation [63]. In this study, we corrected biases in the remotely sensed SMAP soil moisture data using the mean–variance matching method from Nayak et al. [64]. To achieve this, we aligned the mean of the remotely sensed SMAP soil moisture with the monthly mean of the surface layer soil moisture time series from the hydrological model in each SMAP grid.

2.4.6. Observation Localization

In data assimilation, it is necessary to avoid spurious spatial correlation. Without spatial correction, an observation at the local location may produce increments across the study area. This study used observation localization to resolve spurious spatial correlations. For the local analysis domains of each model grid cell in assimilation analysis, observations within the defined radius were considered observation domains. In this study, we set the default radius $r$ of the observation domains to 30 km. The localization weighting was equal to 1 at the local analysis domains. It decreased exponentially with observation domains and was 0 beyond the observation domains:

$$\mathrm{weighting}=\left\{\begin{array}{ll}{e}^{-\left(\frac{d}{r}\right)}& when d<r\\ 0& when d\ge r\end{array}\right.$$

(20)

where $r$ is the default radii of the observation domains, and $d$ is the distance from observation domains.

2.5. Experimental Design

In this study, we designed experiments to assimilate remotely sensed soil moisture in the study area using the ESTKF.

Firstly, we calibrated and validated the hydrological model. The aim of calibration was to adjust the model parameters to improve its simulation performance. The calibration period was from May to September 2014, while the validation period was from May to September between 2015 and 2017.

In the calibration period, we used the daily river flow at the outlet of the study area for calibration. The calibration indicators were the Nash–Sutcliffe Efficiency (NSE), Kling–Gupta Efficiency (KGE), percent bias (Pbias), root mean square error (RMSE), RMSE-observations standard deviation ratio (RSR), and the coefficient of determination (R²) of the observed and simulated river flow [65]. The NSE, Pbias, RSR, and KGE are defined as follows:

$$\mathrm{NSE}=1-\frac{{{\displaystyle \sum }}_{i=1}^N{\left(Q_{sim}-Q_{obs}\right)}^2}{{{\displaystyle \sum }}_{i=1}^N{\left(Q_{obs}-\overline{Q_{obs}}\right)}^2}$$

(21)

$$\mathrm{Pbias}=\frac{{{\displaystyle \sum }}_{i=1}^N(Q_{sim}-Q_{obs})}{{{\displaystyle \sum }}_{i=1}^N{Q}_{obs}}\times 100$$

(22)

$$\mathrm{RSR}=\frac{\sqrt{{{\displaystyle \sum }}_{i=1}^N{\left(Q_{sim}-Q_{obs}\right)}^2}}{\sqrt{{{\displaystyle \sum }}_{i=1}^N{\left(Q_{obs}-\overline{Q_{obs}}\right)}^2}}$$

(23)

$$\mathrm{KGE}=1-\sqrt{{\left(1-r\left(Q_{obs},Q_{sim}\right)\right)}^2+{\left(1-\frac{\overline{Q_{obs}}}{\overline{Q_{sim}}}\right)}^2+{\left(1-\frac{\sigma \left(Q_{obs}\right)}{\sigma \left(Q_{sim}\right)}\right)}^2}$$

(24)

where $Q_{sim}$ is the simulated river flow series, $Q_{obs}$ is the observed river flow series, $\overline{Q_{sim}}$ is the average simulated river flow, $\overline{Q_{obs}}$ is the average observed river flow, $N$ is the number of series, $r\left(·\right)$ is the Pearson correlation coefficient, and $\sigma \left(·\right)$ is the standard deviation. The NSE, Pbias, RSR, KGE, RMSE, and R² approaching 1, 0, 0, 1, 0, and 1 showed better performance, respectively.

In the validation period, we also calculated the unbiased RMSE (ubRMSE) and percent bias (Pbias) of the simulated top layer soil moisture and the observed surface soil moisture of the in situ soil moisture network.

$$\mathrm{ubRMSE}=\sqrt{\frac{{{\displaystyle \sum }}_{i=1}^N{\left(\left(Q_{sim}-\overline{Q_{sim}}\right)-\left(Q_{obs}-\overline{Q_{obs}}\right)\right)}^2}{N}}$$

(25)

We used the results from the validation period as the baseline for the open-loop simulation without assimilation, and we compared the assimilation results with the open-loop simulation to evaluate the improvements by assimilation. The metrics at the flood event scale were the relative error (RE) of flood volumes and peaks:

$${\mathrm{RE}}_{volume}=\frac{V_{sim}-V_{obs}}{V_{obs}}\times 100$$

(26)

$${\mathrm{RE}}_{peak}=\frac{P_{sim}-P_{obs}}{P_{obs}}\times 100$$

(27)

where $V_{sim}$ is the simulated flood volume, $V_{obs}$ is the observed flood volume, $P_{sim}$ is the simulated flood peak, and $P_{obs}$ is the observed flood peak.

Secondly, we assimilated the remotely sensed soil moisture from SMAP using the ESTKF from May to September 2015–2017. We compared the indicators with the open-loop simulation to evaluate the effectiveness of the ESTKF. Thirdly, we compared the ESTKF assimilation results with the EnKF to evaluate the advantages and additional value of the ESTKF. Lastly, we here discuss the factors that impacted the assimilation effect, including the assimilation frequency of remotely sensed soil moisture, the radius of observation localization, and the influence of assimilation on the hydrological process.

3. Results

3.1. Calibration and Open-Loop Performance of Hydrological Model

The period of daily outlet river flow observations at the study area was from 2014 to 2017. The period from May to September 2014 was used for the calibration of hydrological model parameters. The open-loop simulations were conducted without assimilation from May to July between 2015 and 2017.

Figure 5 shows the observation, calibration, and open-loop river flow series and their scatter plots. The calibration and open-loop river flow series comprised 153 and 459 daily values, respectively.

Table 2. The performance of river flow during calibration and open-loop periods.

	NSE	KGE	Pbias	RSR	RMSE	R2	The Number of River Flow Series
Calibration	0.51	0.70	5.83	0.70	11.01	0.51	153
Open-loop	0.45	0.73	12.12	0.74	12.57	0.45	459

shows the performance metrics of the river flow simulation during the calibration and open-loop series. Compared to the calibration period, the NSE decreased, the KGE slightly increased, and the RSR, RMSE, and R² became worse during the open-loop simulation. The Pbias of the open-loop simulation increased from 5.83% of the calibration series to 12.12%. This indicates that the hydrological model overestimated the runoff during the open-loop simulation period. Some markers in Figure 5c appear on the upper left of the 1:1 black dotted line, and most of them are medium to high river flows.

3.2. Improving Soil Moisture Simulation with ESTKF

The comparative experiments of remotely sensed soil moisture assimilation were performed during the open-loop period. The assimilation methods were ESTKF and the EnKF. The observations of the soil moisture were derived from 27 in situ soil moisture gauges 4 cm below the surface.

Figure 6a,b shows the percentage change of soil moisture ubRMSE with ESTKF and EnKF relative to the open-loop series at in situ soil moisture gauges, respectively. In Figure 6a of ESTKF, there are 21 gauges where the ubRMSE decreased. In Figure 6b of EnKF, there are only 13 gauges where the ubRMSE decreased.

Table 3. The performance of soil moisture in ESTKF and EnKF during open-loop periods.

	ubRMSE (×10−2)		Pbias		The Number of Soil Moisture Gauges
	Average	Interquartile Range (IQR)	Average	Interquartile Range (IQR)
Open-loop	12.91	7.86	25.98	59.29	27
ESTKF	12.56	7.70	22.47	54.49	27
EnKF	12.83	7.37	20.41	54.54	27

shows the statistics of soil moisture in the open-loop, ESTKF, and EnKF. The statistics include the average and interquartile range of simulated soil moistures, ubRMSE and Pbias. The ubRMSE and Pbias of ESTKF and EnKF were smaller than those of the open-loop period, indicating that assimilating remotely sensed soil moisture can improve the simulation effect of soil moisture in the hydrological model. ESTKF exhibited a smaller average ubRMSE than EnKF. Although the average Pbias of EnKF was smaller than that of ESTKF, the interquartile range of Pbias in EnKF was larger than in ESTKF.

3.3. Improving River Flow Simulation with ESTKF

The assimilation of remotely sensed soil moisture had a smaller ubRMSE and Pbias of soil moisture than open-loop series in the hydrological model. As soil moisture plays a significant role in river flow, we assessed EnKF and ESTKF in order to determine their effectiveness in improving the simulation of river flow.

Figure 7 shows the line plots and scatter plots of river flow with the open-loop, EnKF, and ESTKF. In Figure 7c, the river flow of ESTKF is obviously closer to the 1:1 black dashed diagonal line than the river flow of the open-loop series. In Figure 7d, the river flow of EnKF does not improve compared with the open-loop series at medium and low river flows, and great deviation occurs at high river flows.

Table 4. The performance of river flow in ESTKF and EnKF during open-loop periods.

	NSE	KGE	Pbias	RSR	RMSE	R2	The Number of River Flow Series
Open-loop	0.45	0.73	12.12	0.74	12.57	0.45	459
ESTKF	0.58	0.75	−1.28	0.65	11.01	0.58	459
EnKF	0.43	0.72	3.66	0.75	12.77	0.43	459

shows the statistics of river flow in open-loop, ESTKF, and EnKF. The statistics include the NSE, KGE, Pbias, RSR, RMSE, and R² of simulated river flows. The RSR, RMSE, and R² of river flows with ESTKF were better than those in open-loop series. In particular, NSE improved from 0.45 in the open-loop series to 0.58. The improvement of KGE was small, only increasing from 0.73 in the open-loop series to 0.75. NSE was more affected by the simulation effect of high river flow compared to KGE. EnKF reduced Pbias only from 12.12% in the open-loop series to 3.66%, and other statistics of river flow in EnKF were worse than the open-loop series. Thus, ESTKF is superior to EnKF in terms of improving river flow in this experiment.

3.4. Improving Flood Events Simulation with ESTKF

The simulation of flood events has received more attention compared to the simulation of a continuous river flow. Therefore, we evaluated the improving effects of ESTKF and EnKF on the simulation of flood events. The flood events were selected with a complete flood process, and precipitation was characterized by more than 10 mm/day precipitation.

Table 5. The flood events and their volumes and peaks in observation.

Flood Id	Start Date	End Date	Observation
			Flood Volume (×106 m3)	Flood Peak (m3/s)
1	1 July 2015	7 July 2015	22.97	52.20
2	7 July 2015	15 July 2015	37.79	67.30
3	25 July 2015	31 July 2015	16.65	42.90
4	31 July 2015	6 August 2015	24.46	68.00
5	10 August 2015	20 August 2015	25.76	37.00
6	25 August 2015	31 August 2015	13.10	28.40
7	1 September 2015	6 September 2015	15.67	40.30
8	7 September 2015	13 September 2015	28.34	65.10
9	14 September 2015	20 September 2015	19.38	35.90
10	24 May 2016	17 June 2016	43.61	27.90
11	18 June 2016	3 July 2016	34.63	54.00
12	5 July 2016	18 July 2016	59.88	123.00
13	20 July 2016	24 July 2016	12.49	34.20
14	25 July 2016	28 July 2016	10.32	40.60
15	3 August 2016	10 August 2016	11.78	23.70
16	12 August 2016	17 August 2016	30.73	72.10
17	18 August 2016	28 August 2016	77.81	115.00
18	15 September 2016	25 September 2016	35.56	44.20
19	27 May 2017	1 June 2017	9.05	24.10
20	3 June 2017	21 June 2017	30.71	27.90
21	1 July 2017	15 July 2017	26.67	45.30
22	22 July 2017	30 July 2017	24.92	48.30
23	1 August 2017	5 August 2017	9.71	38.80
24	6 August 2017	9 August 2017	9.23	48.50
25	19 August 2017	26 August 2017	27.60	68.30
26	3 September 2017	7 September 2017	17.22	53.60
27	12 September 2017	24 September 2017	52.93	61.00

presents 27 flood events, listing the start date, end date, flood volume, and flood peak of each flood event in observation.

Table 6. The performance of flood events in ESTKF and EnKF during open-loop periods.

	Relative Error of Flood Volume (%)		Relative Error of Flood Volume (%)		The Number of Flood Events
	Average	Interquartile Range (IQR)	Average	Interquartile Range (IQR)
Open-loop	19.90	33.99	8.57	38.37	27
ESTKF	2.52	29.69	−8.89	30.43	27
EnKF	8.17	31.52	0.65	39.11	27

shows the average and interquartile range (IQR) of the relative errors in flood volume and flood peak for open-loop, ESTKF, and EnKF. The assimilation of remotely sensed soil moisture greatly reduced the relative error of the flood volume, from 19.90% in the open-loop series to 2.52% in ESTKF and 8.17% in EnKF. ESTKF had a smaller interquartile range of relative error for flood volume than EnKF. However, the assimilation of remotely sensed soil moisture had different performances in the improvement of the flood peak. The relative error of peak flow in ESTKF changed from 8.57% in the open-loop series to −8.89%. The relative error of the peak flow in EnKF was only 0.65%, but its interquartile range became larger than the open-loop series. Most of these flood events lasted for several days to over a week. Remotely sensed soil moisture observations were taken once every 2–3 days, which may not fully capture the flood process, especially during the short-term flood peak. The assimilation frequency of remotely sensed soil moisture may be a limiting factor in improving flood peaks.

Figure 8a shows the variation of relative error in flood volume for different months. The relative error of simulated flood volume during the open-loop period was excessively large for each month. From July to September, the assimilation algorithms ESTKF and EnKF reduced the relative error of the flood volume by assimilating remotely sensed soil moisture. ESTKF provided better results compared to EnKF. However, in May and June, the relative error of the flood volume for both ESTKF and EnKF changed from positive values in the open-loop to negative values, and the assimilation did not improve the simulation effect in the flood volume.

Figure 8b shows the variation of relative error in flood peak for different months. It can be observed that the relative error of the simulated flood peak was high for each month during the open-loop period. From July to September, the assimilation algorithms ESTKF and EnKF decreased the relative error of the flood peak. However, in May and June, both ESTKF and EnKF significantly increased the relative error of the peak. Therefore, the assimilation of remotely sensed soil moisture did not significant improve the flood peak simulations over the entire study period (Table 6).

3.5. The Features of Grid Runoff with Assimilation

In both the open-loop simulation and remotely sensed soil moisture assimilation, the distributed hydrological model generated the river flow in the outlet and the runoff distribution of each grid in the study area. The grid runoff is a sum of flow generated by terrain slope within one grid spatial resolution. Grid runoff flows to river channels and the outlet in the study area. Remotely sensed soil moisture assimilation improved river flow during flood events compared to the open-loop simulation. Grid runoff can represent the distribution of improvements in the study area.

Figure 9 shows the changes of average grid-based runoff between the assimilation using ESTKF and open-loop. There are 6899 grids with a spatial resolution of 600 m by 600 m in the study area. With the use of ESTKF, the grid runoff in most of the study decreased, while only a small portion of grid runoff in the north-western study area showed an increase. Figure 10 shows the scatter plot of soil moisture and grid-based runoff changes. As the soil moisture increased, the decrease in average grid runoff became increasingly significant. Compared to the low soil moisture, the high soil moisture was more greatly affected by assimilation.

4. Discussion

The main objective of the study was to explore the performance of the ESTKF assimilation algorithm on the assimilation of remotely sensed soil moisture, including the improvement of soil moisture and river flow and the advantages and added value of ESTKF compared to EnKF in the hydrological model. To achieve these objectives, we needed to solve the spatial resolution gap between the remotely sensed soil moisture and simulated soil moisture of the hydrological model.

Remotely sensed soil moisture from SMAP was assimilated into the first layer of soil moisture in the distributed hydrological model. The spatial resolution of the hydrological model was 600 m, while the spatial resolution of the SMAP remotely sensed soil moisture was 9 km. There was a considerable gap between the two spatial resolutions. By employing observation operators and observation localization based on geolocation, the distributed hydrological model assimilated multiple remotely sensed soil moisture values for each grid, thereby ensuring the consistent updates of soil moisture in the model. Without the observation localization method, an observation at the local position would produce spurious correlations and false updates of assimilation across the entire study area [66]. The predetermined observation localization radius was utilized in the experiment. As the spatial correlation among observations across different dates and locations may vary, there were other methods to determine the observation localization radius. For example, Han et al. [37] used semivariogram models to improve the surface soil moisture data characterization, especially for uncovered grid cells in the study area. Wang et al. [67] investigated how to set the horizontal localization radius in an ocean model. A varying radius with the latitude following a bimodal Gaussian function was found to fit the model well. Rasmussen et al. [66] studied the performance of the adaptive localization method. The required ensemble size can be reduced with the use of adaptive localization compared to the distance-based localization.

The performance metrics part is important for the evaluation of results. We calculated the widely used performance metrics, including NSE, KGE, Pbias, RSR, RMSE, and R² for river flows; ubRMSE, and Pbias for soil moisture; and the relative error of volumes and peaks for flood events. Moriasi et al. [65] suggested using NSE, PBIAS, and RSR to evaluated models. Burgan et al. [68] also used the quantiles in calculating the performance metric for simulating the upper, middle, and lower parts of the observed river flows. A model simulation can generally be considered satisfactory if the NSE is greater than 0.50, the RSR is less than 0.70, and the Pbias falls within +/−25% for river flows [65]. During the calibration period, the hydrological model performed well when simulating river flow. However, during the open-loop period, the NSE and RSR of the river flow decreased and increased, respectively. This was mainly observed in the form of high river flows, with more frequent occurrences during the open-loop period, as depicted in Figure 5c. Many high river flow simulation values were overestimated and located on the left side of the 1:1 diagonal line. Soil moisture during the open-loop period may be greater than the actual soil moisture values. The evaluation indicator Pbias of the open-loop soil moisture at observation sites in Table 3 confirms this speculation. Therefore, the high flow simulation accuracy requires improvement. This can be achieved by correcting soil moisture.

In terms of improving the ability to simulate soil moisture, we found that both ESTKF and EnKF are effective. At the 27 locations of the soil moisture gauges, the ubRMSE of ESTKF was lower than that of EnKF, indicating better assimilation performance. Furthermore, it is possible to improve the river flow simulation accuracy through assimilating remotely sensed soil moisture. ESTKF and EnKF exhibited different results. ESTKF improved most of the cases where the open-loop high river flow simulations were too low, but EnKF did not improve this situation. The possible reason for this is that the perturbed observations in EnKF corrected the too low spread of the ensemble but introduced additional sampling errors into the filter [69,70]. In contrast to EnKF, ESTKF built the ensemble by writing each posterior member as a linear combination of prior members without perturbed observations and computed the correction in the error-subspace spanned by the ensemble [38].

Assimilation was also effective for improving flood volume in flood events. To avoid the possible impacts of specific seasons and years, we selected 27 flood events during the 3-year open-loop period. In ESTKF, the relative error of the flood volume was reduced on average to 2.52%, but the relative error of flood peak did not improve. The frequency of assimilating soil moisture is an important factor that limits assimilation performance. Increasing assimilation frequency can reduce errors for state variables [71]. Remotely sensed soil moisture observations are taken once every 2–3 days, which may not fully capture the flood process, especially the short-term flood peak. However, too frequent of an interval does not give the model enough time to adjust to the increments and can potentially introduce imbalances and result in insertion noise in models [71]. The optimum temporal acquisition strategy strongly depends on flood morphology and flood wave arrival timing [72].

We also tested the variation of relative error in flood volume and peak for different months with the assimilation. From July to September, the assimilation reduced the relative error of the flood volume and flood peak. However, in May and June, the relative error of the flood volume and peak increased. This situation may be related to the soil moisture conditions in May and June that are different from those in other months in the hydrological model. Figure 11 shows the ubRMSE and Pbias of soil moisture in open-loop and with the use of assimilation. In Figure 11b, we found that the 25th percentiles of Pbias in May and June was around −20%, while in other months, it was only around −5%. The parameters of the hydrological model were calibrated only with the river flow observation in calibration periods, and there was seasonal priori bias of soil moisture in open-loop periods (Figure 11b). However, the success of assimilation depends on unbiased model predictions and unbiased observations [72]. One widely used method to address observation biases is rescaling the observations prior to match that of the model, but the bias in the model itself can substantially complicate the process [73]. Bosilovich et al. [74] estimated and corrected a constant time mean bias for the model. Qin et al. [75] fed the time series of both the analyzed soil moisture and the innovation into a likelihood function from data assimilation to adjust the values of both model parameters and correction coefficients through an optimization algorithm. These results suggest the need for correcting the bias in the model.

The different soil moisture conditions affected the effectiveness of assimilation in hydrological models. Figure 10 shows the variability of the influence on grid-based runoff estimation, highlighting the greater impact of assimilation on wetter soil as opposed to drier soil. In Figure 11a, assimilation significantly reduced the ubRMSE of soil moisture in the middle of the flood season (June to August) compared to the months at the beginning and end of the flood season (May and September) in the open-loop period. Therefore, the assimilation method and remote sensing data used in this study are more effective for wetter soil moisture and high runoff conditions. The impact of assimilating soil moisture on hydrological models has also been discussed in other studies. For example, Mao et al. [61] analyzed the factors that restricted the improvement of river flow by assimilating remotely sensed soil moisture and found that assimilating remote sensing surface soil moisture can improve slow flow underground, but the improvement of the surface fast flow is small. Han et al. [28] found that the assimilation did not produce as much a significant improvement in river flow predictions compared to soil moisture estimates in the presence of large precipitation errors and the limitations of the hydrological model mechanism.

Despite our experiments being case specific and depending on variables such as study area characteristics, hydrological model setting, assimilation methods, and remotely sensed soil moisture, the results provide evidence of the value of ESTKF in the hydrological model and address the advantages and added values of ESTKF compared to EnKF.

5. Conclusions

The data assimilation of remotely sensed soil moisture observations provides a feasible way to improve the ability of river flow simulations. In the mismatch of spatial resolution between remotely sensed soil moisture and the distributed hydrological model, this study employed observation operators and observation localization based on ESTKF, and the distributed hydrological model assimilated multiple remotely sensed soil moisture values for each grid.

We built a distributed hydrological model in the Babao River basin. The comparative experiments of remotely sensed soil moisture assimilation were performed during an open-loop period, and the assimilation methods were ESTKF and EnKF. The hydrological model overestimated the river flows during the open-loop simulation period. The ubRMSE and Pbias of ESTKF and EnKF were smaller than those seen during the open-loop period, indicating that assimilating remotely sensed soil moisture can improve the simulation effect of soil moisture in the hydrological model. ESTKF exhibited a smaller average ubRMSE than EnKF. River flows with ESTKF were better than those seen during the open-loop period. In particular, the NSE improved from 0.45 in the open-loop series to 0.58. The improvement of KGE was small, only increasing from 0.73 in open-loop series to 0.75. EnKF reduced the Pbias from 12.12% in the open-loop series to 3.66%, and the other results of the river flow in EnKF were worse than the open-loop series. The assimilation of remotely sensed soil moisture greatly reduced the relative error of the flood volume, from 19.90% in the open-loop series to 2.52% in ESTKF and 8.17% in EnKF. ESTKF had a smaller interquartile range of relative error for the flood volume than EnKF.

In summary, these results provide evidence of the value of ESTKF in the hydrological model. Remotely sensed soil moisture assimilation is a challenging task. The effects of assimilation with different models and assimilation strategies are uncertain. Therefore, more studies are needed to evaluate the potential of remotely sensed soil moisture to improve river flow in the future.