1. Introduction
2. Study Area and Data
3. Problem Formulation and Methodology
 Calibration of the deterministic HBV hydrological model based on observations and measurements from the hydrometeorological monitoring network of the IMWM—NRI;
 Removal of bias from the satellite precipitation product, ${P}_{\mathrm{SAT}}$: ${P}_{{05}_{24}}$, and from satellite soil moisture observations, ${H}_{14}$: ${H}_{14}^{1.00}$ and ${H}_{14}^{2.89}$, using the BC method in all phases (e.g., probability distribution fitting, validation and correction);
 Simulation of the HBV model with and without the updating procedure. The HBV model was updated using three methods:
 Bias correction ${H}_{14}^{1.00\mathrm{BC}}$ or ${H}_{14}^{2.89\mathrm{BC}}$ without assimilation (the biascorrected satellite observations replaced the proper state variables of the HBV),
 Assimilation of the uncorrected satellite soil moisture data, i.e., ${H}_{14}^{1.00}$ or ${H}_{14}^{2.89}$, using EnKF, and
 Assimilation of the uncorrected satellite soil moisture data, i.e. ${H}_{14}^{1.00}$ or ${H}_{14}^{2.89}$, with the bias correction of the perturbed background prediction of soil moisture, for the creation of an unbiased ensemble of model states using EnKFBC.
These three methods use ${P}_{{05}_{24}}^{\mathrm{BC}}$ as the precipitation input (i.e., a forcing data),  Simulation in forecast mode in the form of an ensemble (interval forecast) using the hindcast method, in which the input (forcing data) of the hydrological model is the historical data instead of the meteorological forecast data, e.g., ${P}_{\mathrm{OBS}}$, observed groundbased precipitation and ${T}_{\mathrm{OBS}}$, temperature.
3.1. Calibration of the Deterministic HBV Hydrological Model
3.2. Bias Correction of GroundBased Observations and Satellite Products—The Distribution Derived Transformation Method
3.3. Assimilation of Satellite Products Using the Ensemble Kalman Filter
3.3.1. Model Error and the Perturbation of Error Factors within the EnKF Filter Algorithm
 ${{\theta}^{\prime}}_{ji}$ was the perturbed ${\theta}_{j}$, i.e., the soil moisture storage for the ith ensemble member, for $j=1,\dots N$, where $N$ is the number of simulation steps, and for $i=1,2,\cdots ,n$, where n is the size of the ensemble assumed in the procedure,
 ${\theta}_{j}$ is the soil moisture storage in the j^{th} step,
 ${\theta}_{j1}$ is the soil moisture storage calculated in the previous (j − 1)^{th} step and
 U is the uniform distribution in the range of ±${\xi}_{\theta}\left({\theta}_{j}{\theta}_{j1}\right)$.
 ${{P}^{\prime}}_{ji}$ is the perturbed ${P}_{j}$ precipitation for the i^{th} ensemble member, for $j=1,\dots ,N$ and for $i=1,2,\cdots ,n$,
 ${P}_{j}$ is the input areal average precipitation from the measurement and observation network in the j^{th} step, and
 U is a uniform distribution in the range $\pm {\xi}_{P}{P}_{j}$.
3.3.2. A New Procedure for the EnKF Coupled with the BiasCorrection Scheme Using DistributionDerived Transformation
 ${F}_{\theta}$,${F}_{{\theta}_{1}^{\prime}},\dots ,{F}_{{\theta}_{n}^{,}}$ are the cumulative distribution functions (CDF) of the unperturbed and perturbed background prediction of the soil moisture variable, $\theta $, and
 ${F}_{\theta}^{1}$ is the inverse CDF corresponding to the unperturbed variable, $\theta $.
3.3.3. Scheme of the KF Method Used in the Assimilation Procedure
3.4. Model Performance Comparison
 ${\mathrm{R}}_{0}\mathrm{M}$ showing over (${\mathrm{R}}_{0}\mathrm{M}>1.0$), under (${\mathrm{R}}_{0}\mathrm{M}<1.0$) or perfect (${\mathrm{R}}_{0}\mathrm{M}=1.0$) prediction of the desired parameter [62].$${\mathrm{R}}_{0}\mathrm{M}=\frac{\frac{1}{n}{{\displaystyle \sum}}_{i=1}^{i=n}{P}_{i}}{\frac{1}{n}{{\displaystyle \sum}}_{i=1}^{i=n}{O}_{i}}=\frac{\overline{P}}{\overline{O}}$$
 n is the number of modeled (corrected) values,
 ${O}_{i}$ is the i^{th} observed value,
 $\overline{O}$ is the mean of the observed values,
 $\overline{P}$ is the mean of the predicted values and
 ${P}_{i}$ is the i^{th} predicted value.
 $\mathrm{RMSE}$, a scaledependent measure of accuracy for assessing different models’ ability to predict a single variable [63], was expressed in $\mathrm{mm}$ for satellite precipitation and pseudo in situ observations of soil moisture, and as ${\mathrm{m}}^{3}{\mathrm{s}}^{1}$ for the procedure product,
 the NashSutcliffe Model efficiency index ($\mathrm{EI}$) [64], where $\infty <\mathrm{EI}\le 1.0$, with $\mathrm{EI}=1.0$ representing a perfect match between observed and predicted values, and $\mathrm{EI}\le 0$ representing a prediction no better than the mean of observed values. Values of $\mathrm{EI}\ge 0.5$ were taken to represent a satisfactory model performance.
4. Results and Discussion
4.1. Selecting the Best Probability Distribution Function for $P$ and $\theta $ Based on the AIC
4.2. Assessing the Influence of Bias Correction on Model Accuracy
4.3. Simulating Discharge with or without Updating Using the HBV Model
 Using the best precipitation product, i.e., ${P}_{{05}_{24}}^{\mathrm{BC}}$ as a forcing data without assimilation of the soil moisture observations (the second input option from top in Figure 2),
 Using the best precipitation product, i.e., ${P}_{{05}_{24}}^{\mathrm{BC}}$ and corrected satellite soil moisture product (${H}_{14}^{1.00\mathrm{BC}}$ or ${H}_{14}^{2.89\mathrm{BC}}$) without assimilation, but with updating (replacing the corresponding state variable of the HBV; the third input option from the top in Figure 2),
 Using the best precipitation product, i.e., ${P}_{{05}_{24}}^{\mathrm{BC}}$ and assimilation of the ${H}_{14}^{1.00}$ or ${H}_{14}^{2.89}$ with the EnKF filter (the fourth input option from top in Figure 2), and
 Using the best precipitation product, i.e., ${P}_{{05}_{24}}^{\mathrm{BC}}$ and assimilation of the ${H}_{14}^{1.00}$ or ${H}_{14}^{2.89}$ using the BC scheme to create an ensemble of the unbiased model states using EnKFBC (the fifth input option from top in Figure 2).
4.3.1. Simulating Discharge with BiasCorrected Satellite Precipitation without Assimilation
4.3.2. Simulating Discharge Using Two Methods, with BiasCorrected Satellite Soil Moisture or with the Assimilation Procedure
 (i)
 ${Q}_{\mathrm{OBS}}$, observed flow;
 (ii)
 ${Q}_{\mathrm{HBV}}$, flow simulated by HBV with ${P}_{\mathrm{OBS}}$ precipitation;
 (iii)
 ${Q}_{\mathrm{HBV}\mathrm{BC}\mathrm{BC}}$, flow simulated by HBV with ${P}_{{05}_{24}}^{\mathrm{BC}}$ and ${H}_{14}^{2.89\mathrm{BC}}$;
 (iv)
 Avg. ${Q}_{\mathrm{HBV}\mathrm{BC}\mathrm{EnKF}}$ and using ${P}_{{05}_{24}}^{\mathrm{BC}}$ and assimilating ${H}_{14}^{2.89}$,
 (v)
 And avg. ${Q}_{\mathrm{HBV}\mathrm{BC}\mathrm{EnKF}\mathrm{BC}}$ using ${P}_{{05}_{24}}^{\mathrm{BC}}$ and assimilating ${H}_{14}^{2.89}$ with the creation of an unbiased ensemble of model states.
4.3.3. Examples of Hydrological Updating the HBV Model Using EnKFBC and Simulation of the HBV Model in Forecast Mode for the Sola Basin at Zywiec for Selected Flood Events
 (i)
 ${Q}_{\mathrm{OBS}}$—observed hydrograph (dotted blue line);
 (ii)
 ${Q}_{\mathrm{HBV}}$—forecasted hydrograph without updating (solid green line);
 (iii)
 and avg. ${Q}_{\mathrm{HBV}}$—averaged forecasted hydrograph with updating (solid red line) with the ensemble hydrographs (solid gray lines).
5. Summary and Conclusions
 The most effective transformation function for observed daily precipitation (${P}_{\mathrm{OBS}}$) was the GE distribution. For satellite precipitation (${P}_{{05}_{24}}$), the WE distribution was the most effective.
 The bestsuited transformation function for the HBV soil moisture (${\theta}_{\mathrm{HBV}}$) was the WE distribution in both winter and summer. For satellite soil moisture (${H}_{14}^{1.00}$), the optimum transformation function was GA in both winter and summer. For ${H}_{14}^{2.89}$, the optimum transformation function was WE in the winter and GA in the summer.
 Removing bias from the satellite precipitation improved the modeling accuracy of ${P}_{\mathrm{OBS}}$, both during distribution fitting and the validation of the product. Removing the bias from satellite soil moisture products, however, was only effective in summer during distribution fitting and validation. In winter, negative efficiency index values were observed.
 The tested satellite product, ${P}_{{05}_{24}}^{\mathrm{BC}}$, without the assimilation soil moisture product, did not significantly improve hydrograph performance, especially in the summer. This is likely due to the influence of convective precipitation upon runoff. The quality of the spatial distribution of precipitation, however, was an important factor influencing the quality of the simulated hydrograph, especially in the mountain catchment.
 After taking into account the assimilation of the satellite soil moisture product, the best simulation model included the assimilation of ${H}_{14}^{1.00}$ in summer. In winter, a negligible improvement was observed between the simulated hydrograph using the corrected assimilation ${H}_{14}^{2.89\mathrm{BC}}$ and the assimilation ${H}_{14}^{2.89}$ using EnKF and EnKFBC filters.
 The HBV model with updating in the forecast mode generally performed poorly; however, it improved the forecast compared to the HBV without updating. It should be noted that the forecasts were calculated based on matrix state variables determined in the final step of updating using the assimilation of satellite soil moisture.
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
Appendix A. Choice of the Optimal Theoretical Marginal Distributions Algorithms
Appendix B. Data Assimilation Algorithms for the Ensemble Kalman Filter (EnKF)
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Product  Distribution  AIC  

Type  Coefficients  
A  β  ε  
Precipitation  
${P}_{\mathrm{OBS}}$  WE  0.0091  0.3389  0.0990  9673.47  
GA  0.9979  0.2313  0.0000  9702.60  
GE  1.2140  4.9784  0.0000  9520.16  
${P}_{{05}_{24}}$  WE  0.0130  0.3239  0.0990  8134.90  
GA  0.7193  0.1824  0.0540  21895.00  
GE  1.8952  8.2915  0.0000  15248.30  
Soil Moisture  
Winter Season  ${\theta}_{\mathrm{HBV}}$  WE  34.4556  5.6428  38.00  4549.28 
GA  69.1987  1.0107  0.00  4709.36  
GE  11.3523  0.0885  36.00  5102.45  
${H}_{14}^{1.00}$  WE  20.6861  4.0103  42.40  4212.17  
GA  50.8409  0.8123  20.00  4186.79  
GE  12.1816  0.1415  40.00  4502.38  
${H}_{14}^{2.89}$  WE  40.6269  3.8497  19.80  5178.59  
GA  19.8836  2.8677  0.00  5250.16  
GE  5.9666  0.0631  18.60  5572.46  
Summer Season  ${\theta}_{\mathrm{HBV}}$  WE  33.2265  2.9743  22.80  4954.12 
GA  20.0311  2.6243  0.00  4982.81  
GE  9.6627  0.0820  18.00  5110.21  
${H}_{14}^{1.00}$  WE  17.8183  2.9249  46.50  4173.68  
GA  61.1260  0.7925  14.00  4165.42  
GE  8.2893  0.1502  44.50  4324.50  
${H}_{14}^{2.89}$  WE  26.5781  4.6010  38.80  4253.31  
GA  66.0066  0.8109  9.60  4197.62  
GE  28.7413  0.1320  34.00  4481.33 
Product  Modeling Phase  

Distribution Fitting: from 1 January 2012 to 31 July 2014  Validation: from 1 August 2014 to 30 April 2015  
No Bias Correction  With Bias Correction  No Bias Correction  With Bias Correction  
${\mathbf{R}}_{0}\mathbf{M}$  $\mathbf{R}\mathbf{M}\mathbf{S}\mathbf{E}$$\left(\mathbf{m}\mathbf{m}\right)$  $\mathbf{E}\mathbf{I}$  ${\mathbf{R}}_{0}\mathbf{M}$  $\mathbf{R}\mathbf{M}\mathbf{S}\mathbf{E}$$\left(\mathbf{m}\mathbf{m}\right)$  $\mathbf{E}\mathbf{I}$  ${\mathbf{R}}_{0}\mathbf{M}$  $\mathbf{R}\mathbf{M}\mathbf{S}\mathbf{E}$$\left(\mathbf{m}\mathbf{m}\right)$  $\mathbf{E}\mathbf{I}$  ${\mathbf{R}}_{0}\mathbf{M}$  $\mathbf{R}\mathbf{M}\mathbf{S}\mathbf{E}$$\left(\mathbf{m}\mathbf{m}\right)$  $\mathbf{E}\mathbf{I}$  
Precipitation  
${P}_{{05}_{24}}$  0.908  0.885  −0.045  0.945  0.818  0.108  0.699  0.617  0.126  1.025  0.512  0.397  
Soil Moisture  
winter  ${H}_{14}^{1.00}$  0.815  13.572  −1.728  1.007  12.584  −1.346  0.876  13.673  −2.112  1.000  11.487  −1.197 
${H}_{14}^{2.89}$  0.876  19.580  −4.679  1.002  12.569  −1.340  0.815  19.793  −5.523  1.001  11.953  −1.397  
summer  ${H}_{14}^{1.00}$  1.187  14.012  −0.655  0.999  9.615  0.220  1.188  13.219  −0.419  1.000  9.538  0.261 
${H}_{14}^{2.89}$  1.201  13.702  −0.583  1.001  7.764  0.492  1.201  13.167  −0.408  1.002  7.944  0.487 
Input Product  BC  Filter  Output Product  Season  

EnKF EnKFBC  Summer 1 Aug 2014–31 Oct 2014  Winter 1 Nov 2014–30 Apr 2015  
$${\mathbf{R}}_{0}\mathbf{M}$$

$$\mathbf{R}\mathbf{M}\mathbf{S}\mathbf{E}\phantom{\rule{0ex}{0ex}}\left({\mathbf{m}}^{3}{\mathbf{s}}^{1}\right)$$

$$\mathbf{E}\mathbf{I}$$

$${\mathbf{R}}_{0}\mathbf{M}$$

$$\mathbf{R}\mathbf{M}\mathbf{S}\mathbf{E}\phantom{\rule{0ex}{0ex}}\left({\mathbf{m}}^{3}{\mathbf{s}}^{1}\right)$$

$$\mathbf{E}\mathbf{I}$$
 
${P}_{\mathrm{OBS}}$  —  —  ${Q}_{\mathrm{HBV}}$  1.274  11.806  −0.045  1.065  18.843  −0.185 
—  
${P}_{{05}_{24}}$  √  —  ${Q}_{\mathrm{HBV}\mathrm{BC}}$  1.271  10.134  0.103  0.801  15.537  0.195 
—  
${P}_{{05}_{24}}$  √  —  ${Q}_{\mathrm{HBV}\mathrm{BC}\mathrm{BC}}$  1.232  9.486  0.326  1.314  14.652  0.284 
${H}_{14}^{1.00}$  √  —  
${P}_{{05}_{24}}$  √  —  ${Q}_{\mathrm{HBV}\mathrm{BC}\mathrm{BC}}$  1.158  7.791  0.545  1.139  12.668  0.465 
${H}_{14}^{2.89}$  √  —  
${P}_{{05}_{24}}$  √  √  ${Q}_{\mathrm{HBV}\mathrm{BC}\mathrm{EnKF}}$  1.192  8.575  0.418  1.359  13.702  0.492 
${H}_{14}^{1.00}$  —  —  
${P}_{{05}_{24}}$  √  —  ${Q}_{\mathrm{HBV}\mathrm{BC}\mathrm{EnKF}\mathrm{BC}}$  1.140  6.755  0.658  1.161  11.884  0.529 
${H}_{14}^{1.00}$  —  √  
${P}_{{05}_{24}}$  √  √  ${Q}_{\mathrm{HBV}\mathrm{BC}\mathrm{EnKF}}$  1.180  7.764  0.583  1.160  11.882  0.551 
${H}_{14}^{2.89}$  —  —  
${P}_{{05}_{24}}$  √  —  ${Q}_{\mathrm{HBV}\mathrm{BC}\mathrm{EnKF}\mathrm{BC}}$  1.143  6.838  0.650  1.139  11.392  0.567 
${H}_{14}^{2.89}$  —  √ 
Products Used for Updating  BC  Filter  Updating  Products Used to Forecast  Forecast  

120 h  72 h  
EnKFBC 
$${\mathbf{R}}_{0}\mathbf{M}$$

$$\mathbf{R}\mathbf{M}\mathbf{S}\mathbf{E}$$

$$\mathbf{E}\mathbf{I}$$

$${\mathbf{R}}_{0}\mathbf{M}$$

$$\mathbf{R}\mathbf{M}\mathbf{S}\mathbf{E}$$

$$\mathbf{E}\mathbf{I}$$
 
21–26 September 2014  26–29 September 2014  
${P}_{\mathrm{OBS}}$  —  —  0.860  7.324  0.461  ${P}_{\mathrm{OBS}}$  1.187  7.079  −0.141 
${P}_{{05}_{24}}$  √  —  0.992  2.467  0.839  ${P}_{\mathrm{OBS}}$  1.049  2.557  0.485 
${H}_{14}^{1.00}$  —  √  —  
25–30 March 2015  30 Mar–2 April 2015  
${P}_{\mathrm{OBS}}$  —  —  2.248  17.163  0.188  ${P}_{\mathrm{OBS}}$  1.394  11.678  −0.996 
${P}_{{05}_{24}}$  √  —  1.155  12.959  0.304  ${P}_{\mathrm{OBS}}$  1.031  9.634  0.353 
${H}_{14}^{2.89}$  —  √  — 
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