
The community Noah LSM with multiparameterization options (NoahMP) was used (Niu et al., 2011). NoahMP is a fully augmented Noah LSM version with multiple options for the key land–atmosphere interaction process. Some improved biophysical realism (land memory processes) has been added, including the separate vegetation canopy and ground temperature, a multilayer snowpack, an unconfined aquifer model for groundwater dynamics, and an interactive vegetation canopy layer.
The model state includes the soil moisture and soil temperature in four layers, skin temperature (bare soil or vegetation), canopy water storage, and a variety of storage variables related to snow processes (Niu et al., 2011). NoahMP has been validated over a variety of underlying surfaces and climate regimes (Cai et al., 2014). The 2m depth of soil is divided into 4 layers at the depths of 0.10, 0.40, 1.0, and 2.0 m. The parameterization scheme includes the Ball–Berry stamatal resistance scheme (Ball et al., 1987), soil resistance scheme (Sakaguchi and Zeng, 2009), and Monin–Obukhov surface transfer coefficient scheme. Surface albedo is diagnosed from the surface solar radiation budget.

The procedure in EnKF includes three steps: first, an initial ensemble is generated; second, the forecast covariance is calculated between the state variables and measurement predictions; finally, each ensemble forecast member is updated at the measurement time.
In a pure Monte Carlo implementation, the ith analysis member x_{i}^{a}(t) is obtained by using a randomly perturbed vector of new observations y_{i}^{o} and a member of a corresponding ensemble of background estimates x_{i}^{f}(t) as in Eq. (1) (Zhang et al., 2010):
$$ {{x}}_i^{\rm a}{\rm{(}}t{\rm{) = }}{{x}}_i^{\rm f}{\rm{(}}t{\rm{) + }}{{K}}[{{y}}_i^{\rm o}  H{{x}}_i^{\rm f}{\rm{(}}t{\rm{)]}},\;\;\;\;\;\;\;\;\;\;\;\;i = {\rm{ }}1,{\rm{ }}2,{\rm{ }} \ldots ,N. $$ (1) The Kalman gain K is defined as:
$${{K}} = {{{P}}^{\rm{f}}}{H^{\rm{T}}}{(H{{{P}}^{\rm{f}}}{H^{\rm{T}}} + {{R}})^{  1}}, $$ (2) where
$$ \hspace{38pt} {{{P}}^{\rm f}}{H^{\rm{T}}} = \frac{1}{{N  1}}\sum\limits_{i = 1}^N {({{x}}_i^{\rm f}  \overline {{{x}}_{}^{\rm f}})} {(H{{x}}_i^{\rm f}  \overline {H{{x}}_{}^{\rm f}})^{\rm{T}}}, $$ (3) $$\!\!\! \!\!\!\!\! \!\!\!\!\! H{{{P}}^{\rm f}}{H^{\rm{T}}} = \frac{1}{{N  1}}\sum\limits_{i = 1}^N {(H{{x}}_i^{\rm f}  \overline {H{{x}}_{}^{\rm f}})} {(H{{x}}_i^{\rm f}  \overline {H{{x}}_{}^{\rm f}})^{\rm{T}}}, $$ (4) $$ \hspace{144pt} \overline {{{{x}}^{\rm f}}} = \frac{1}{N}\sum\limits_{i = 1}^N {{{x}}_i^{\rm f}} , $$ (5) where H is the forward observation operator, superscript T is the matrix transpose, P^{f} is the background error covariance, and R is observation error covariance.

SepKF is implemented in two steps: first, a standard EnKF is used to derive a biasblind state estimate (stage 1, hereafter named state filter); and second, a bias filter similar to that in Dee and Da Silva (1998; hereafter DD98) is used to estimate the bias (stage 2).
As in DD98, a forecast model M is used to transport the ith unbiased estimation of the state member
${\tilde { x}}_i^{\rm a}(t)$ to the next analysis time as in Eq. (6):$$ {{x}}_i^{\rm f}(t + 1) = M[{\tilde { x}}_i^{\rm a}(t)],\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;i = {\rm{ }}1,{\rm{ }}2,{\rm{ }} \cdots , N, $$ (6) and an unbiased analysis member
${\tilde { x}}_i^{\rm a}(t)$ is obtained by Eq. (7):$$ \begin{align} {\tilde { x}}{_i^{\rm a}}(t) = & \left({{{x}}_i^{\rm f}(t)  {{{b}}^{\rm a}}(t)} \right) + {{K}}\left[ {{{y}}_i^{\rm o}  H\left({{{x}}_i^{\rm f}(t)  {{{b}}^{\rm a}}(t)} \right)} \right], \\ i = & {\rm{ }}1,{\rm{ }}2,{\rm{ }} \cdots , N, \end{align}$$ (7) where b^{a}(t) is a bias vector at analysis time t; the state Kalman gain K is calculated by using the same procedure as in Section 2.2.1.
Different from DD98, we use a persistence model for the bias propagation:
$${{{b}}^{\rm f}}{\rm{(}}t{\rm{ + 1) = }}{{{b}}^{\rm a}}(t),$$ (8) and the bias analysis is given by:
$$  {{{b}}^{\rm a}}{\rm{(}}t{\rm{) = }}  {{{b}}^{\rm f}}{\rm{(}}t{\rm{) + }}{{L}}[{{d}}{\rm{(}}t{\rm{) + }}H{{{b}}^{\rm f}}{\rm{(}}t)], $$ (9) where the innovation vector
${{d}}(t) = {{{y}}^{\rm o}}  {\overline {H{{{x}}^{\rm f}}}} (t)$ with an assumption of no bias in the observation vector y^{o}; the bias error covariance matrix${{{T}}^{\, \rm f}} $ is assumed to be proportional to the forecast error covariance matrix${{{P}}^{\, \rm f}} $ , i.e.,${{{P}}^{\, \rm f}} $ =$\lambda {{{P}}^{\rm f}} $ .As in DD98, Kalman gain for the bias filter L is easily calculated by:
$${{L}} = \lambda {{{P}}^{\rm f}}{H^{\rm{T}}}{\left[ {H{{{P}}^{\rm f}}{H^{\rm{T}}} + {{R}}} \right]^{  1}}, $$ (10) where the free parameter λ is determined empirically.

In OSSE, the atmosphere forcing data, soil and plant parameters are obtained by measurements at Bondville observation station (40.01°N, 88.37°W), which is located in a flat field with the soil texture of silt loam over Great Plains of North America (Meyers and Hollinger, 2004; Brunsell et al., 2011). The land cover is grassland, and the measurements were performed from 10 May to 8 August 1998.
Considering that the modeled soil moisture in deep soil layers takes a long time to reach a hydraulic balance, the NoahMP simulation is spun up for three years, forced by cycling the same observations in 1998. The final spunup results are chosen as a true state, whereas the “observations” in OSSE are obtained by adding random errors with the specified statistics into the true state (Fig.1).
An ensemble with 40 members is adopted in EnKF and is much larger than the model state with only 4layer soil moisture forecasts. To reflect errors from different sources as many as possible, initial soil moisture and plant and soil parameters are all perturbed according to possible uncertainties in these variables (Table 1; Zhou et al., 2006; Zhang et al., 2010).
Variable Nominal value Uncertainty in replicates Humidity; solar radiation; wind speed;
wind directionObservations Temporally uncorrelated multiplicative uniform noise: relative humidity: U (0.9, 1.1), solar radiation: U (0.9, 1.1), wind speed: U (0.7, 1.3), wind direction: U (0.7, 1.3) Air temperature Temporally uncorrelated additive uniform noise: U (–4, 4) Precipitation Temporally uncorrelated multiplicative lognormal noise with a mean of 1.0 and a standard deviation equal to 35% of the nominal value Leaf area index (LAI) Multiplicative uniform noise U (0.85, 1.15) for the LAI without change with time
(i.e., static)Initial soil moisture 0.25 cm^{3} cm^{–3} Vertically uncorrelated additive zeromean Gaussian noise with the variance of 0.1^{2} Soil moisture observation Simulated true value Temporally uncorrelated additive zeromean Gaussian noise with the variance of 0.05^{2} Table 1. Scheme of perturbation in the generation of ensemble members
The assimilated observations are only nearsurface soil moisture observations. Assimilating the observations is carried out every six hours and further tests with other time intervals is presented in Section 4.1. Two kinds of bias models are investigated: one is persistent with a consistent bias at each soil layers to represent a fixed systematic model error, and the other is drifting to reflect the variation of meteorological forcings, defined by:
$$b(t) = A[\alpha + \sin (\frac{{2\pi }}{T}t + \beta)],$$ (11) where b(t) is the model bias at time t with the same unit as the model state, A is the amplitude, T is the period,
$\alpha $ is a regulatory factor, and β is the phase at the initial time.Setup of all experiments including specifications of the key parameters is listed in Tables 2, 3. To quantitatively compare estimations by different approaches, root mean square error (RMSE) and mean absolute error (MAE) are computed as follows:
Experiment Bias (cm^{3} cm^{–3}) Bias correction λ (–) b_{0} (cm^{3} cm^{–3}) Observation error (cm^{3} cm^{–3}) EnKFC1 0.01 No – – 0.01 EnKFC2 0.03 No – – 0.01 SepKFC1 0.01 Yes 0.2 0.005 0.01 SepKFC2 0.03 Yes 0.2 0.005 0.01 SepKFC1A 0.01 Yes 0.8 0.005 0.01 SepKFC2A 0.03 Yes 0.8 0.005 0.01 Table 2. Setup in different OSSEs by using the constant model bias
Experiment Bias correction T (day) A (cm^{3} cm^{–3}) λ (–) b_{0} (cm^{3} cm^{–3}) Observation error (cm^{3} cm^{–3}) EnKFS1 No 48 0.03 – – 0.01 EnKFS2 No 6 0.03 – – 0.01 SepKFS1 Yes 48 0.03 0.2 0.005 0.01 SepKFS2 Yes 6 0.03 0.2 0.005 0.01 SepKFS1A Yes 48 0.03 0.8 0.005 0.01 SepKFS2A Yes 6 0.03 0.8 0.005 0.01 Table 3. Setup in different OSSEs by using the sinusoidal model bias
$${\rm{RMSE = \bigg[}}{M_t^{  1}}\sum\limits_{i = 1}^{M_t} {{{(f  \tilde f)}^2}} {{\rm{\bigg]}}^{1/2}}, $$ (12) $$\hspace{30pt} {\rm{MAE = }}{M_t^{  1}}\sum\limits_{i = 1}^{M_t} {\left {f  \tilde f} \right} , $$ (13) where M_{t} is the number of verification times, f is the state estimate, and
${\tilde f}$ is the truth. To compare the improvement of soil moisture profiles with assimilation approaches relative to the openloop simulation, the percentage RMSE improvement (PRI) is defined as follows:$$ \begin{aligned} {\rm{PRI}} = & {\rm {[RMSE }}\left( {{\rm{Openloop}}} \right){\rm{ \,\, RMSE }}\left( {{\rm{DA}}} \right)]/ \\ & {\rm{ RMSE }}\left( {{\rm{Openloop}}} \right){\rm{ \times 100 \normalfont\text% }}, \end{aligned} $$ (14) where RMSE(Openloop) is the RMSE of soil moisture profiles in the openloop simulation, and RMSE(DA) is the RMSE in the data assimilation experiment.

Observation data from 1 to 30 August 2009 at the SemiArid Climate and Environment Research Observatory (SACOL) are used in the real experiment. SACOL is located in Yuzhong Campus of Lanzhou University (35.94°N, 104.13°E) with an elevation of 1965.8 m (Huang et al., 2008). The soil is mainly quaternary aeolian loess with the soil type of sierozem. The surface is covered mainly by short grass with species of stipa bungeana, artemisia frigida, and leymus secalinu.
There is a 32m observation tower at SACOL. Wind speed, temperature, and relative humidity observation instruments are installed at the 1, 2, 4, 8, 12, 16, and 32m heights, respectively. The surface radiation flux measurement includes the outgoing and incoming longwave radiation. The measurement of soil temperature and soil moisture is at the depths of 5, 10, 20, 40, and 80 cm. In addition, the surface pressure, surface temperature, and soil heat flux are also measured. All the above data are collected every 30 minutes.

Figure 2 shows the soil moisture estimations at the 2nd, 3rd, and 4th layers in SepKFC1. For comparison, the estimations with no bias corrections are also shown. In SepKFC1, the soil moisture estimations quickly approach the true values with the cycling update and then maintain around, and the estimations at the two shallow layers (i.e., the 2nd and 3rd layers) are more accurate. In EnKFC1, the estimations slowly arrive at the truth, but then gradually depart from the truth, showing the large impact of the bias on estimating the soil moisture if the bias is not corrected in the cycling update.
Figure 2. Comparison of the soil moisture estimation in SepKFC1 and EnKFC1 with the truth at the (a) 2nd, (b) 3rd, and (c) 4th layers with the bias of 0.01 cm^{3} cm^{–3}.
To comprehensively compare the performance during the whole time window of data assimilation, RMSE of the soil moisture estimation and the estimated bias are listed in Table 4. If the model bias exists, SepKF produces better soil moisture estimation than EnKF with no bias correction although the bias is not completely corrected; with the increasing model bias, the error in soil moisture estimation in each layer with SepKF increases, but it is still smaller than that with EnKF with no bias correction (Table 5).
Experiment Soil moisture estimation Bias estimation Layer 1 Layer 2 Layer 3 Layer 4 Layer 1 Layer 2 Layer 3 Layer 4 EnKFC1 0.023 0.037 0.103 0.074 – – – – SepKFC1 0.012 0.016 0.017 0.019 0.012 0.014 0.019 0.021 SepKFC1A 0.019 0.021 0.033 0.039 0.012 0.021 0.024 0.026 EnKFC2 0.043 0.057 0.133 0.104 – – – – SepKFC2 0.018 0.021 0.025 0.028 0.019 0.023 0.028 0.035 SepKFC2A 0.032 0.036 0.069 0.072 0.022 0.029 0.033 0.036 Table 4. RMSE (cm^{3} cm^{–3}) of the soil moisture estimation and estimated bias with the constant model bias
Experiment Layer 1 Layer 2 Layer 3 Layer 4 EnKFC1 54.0 53.8 14.2 50.7 SepKFC1 76.0 80.0 85.8 87.3 SepKFC1A 62.0 73.8 72.5 74.0 EnKFC2 14.0 28.8 –10.8 30.7 SepKFC2 64.0 73.8 79.2 81.3 SepKFC2A 36.0 55.0 42.5 52.0 Table 5. PRI (%) for the soil moisture estimation with the constant model bias
To investigate the effect of the parameter λ on the bias correction, λ = 0.2 and 0.8 are tested (Fig. 3). Compared to λ = 0.2, the convergence speed of the estimated bias with λ = 0.8 is faster due to more observation information absorbed, but the fluctuation of the estimated bias is much larger. Using a large parameter λ at the early stage of assimilation to increase the convergence speed, and then using a small parameter λ at the later stage to reduce noise in the bias estimation, are suggested.
Figure 3. Comparison with the true value of the bias estimation using SepKF respectively with λ = 0.2 and 0.8.
To assess the impact of the time interval in the experiments, three additional assimilation experiments are carried out, respectively, with the intervals of 24, 48, and 96 h, and RMSEs of the soil moisture estimations at the four layers during the assimilation period are listed in Table 6. RMSE at the 6h time interval is the smallest at all layers among all assimilation intervals. For example, RMSE increases from 0.019 cm^{3} cm^{–3} at the 6h assimilation interval to 0.074 cm^{3} cm^{–3} at the 96h assimilation interval at the fourth layer, but the RMSE difference of the estimations for different time intervals is not large at the two shallow layers. RMSE increases with the increase of depth for all time intervals, showing that the deeper the soil layer is, the more difficult the soil moisture estimation is.
Time interval (h) Layer 1 Layer 2 Layer 3 Layer 4 6 0.012 0.016 0.017 0.019 24 0.015 0.018 0.021 0.025 48 0.018 0.021 0.027 0.029 96 0.023 0.037 0.054 0.074 Table 6. RMSE (cm^{3} cm^{–3}) for the soil moisture estimation by SepKFC1 with the assimilation intervals of 6, 24, 48, and 96 h
Unlike the atmospheric state variables, soil moisture shows a very slow diurnal variation at the shallow layer only and it almost does not vary at the deep layers without rainfall (Fig. 4), suggesting that it does not change drastically even using more frequent assimilating intervals such as once every six hours. Under this condition, the update does not need much time to finish. This is because magnitudes of the covariance are very small, especially those between the near surface and deep layers. Nonetheless, it will significantly improve the soil moisture estimation, especially at the deep layer.

To simulate a near real condition, a sinusoidal model bias is used. For a 48day period, variations of the model bias and soil moisture estimations at the 2nd, 3rd, and 4th layers are plotted in Fig. 5, including the estimations with EnKF for comparison. Since SepKF can correct the model bias with the cycling assimilation, especially the phase change, the soil moisture estimation with SepKFS1 is much consistent with the true value while that with EnKF does not.
Figure 5. Comparison of the soil moisture estimation in EnKFS1 and SepKFS1 with the truth by using the sinusoidal model bias with T = 48 days and λ = 0.2 at the (a) 2nd, (b) 3rd, and (c) 4th layers. The bias estimation in SepKFS1 are also compared to the truth at the (d) 2nd, (e) 3rd, and (f) 4th layers.
If the temporal variability of the model bias increases within the period of six days, the error in the soil moisture estimation in SepKFS2 also increases although it is still more consistent with the truth compared to EnKFS2. Similarly, the model bias in SepKFS2 is not well corrected compared with the low temporal variability of the model bias although the phase change has been better estimated (Fig. 6).
Figure 6. (a) The 2ndlayer soil moisture estimation by SepKFS2 with T = 6 days and λ = 0.2, and EnKFS2 compared with the truth; (b) the 2ndlayer bias estimation with SepKFS2 compared with the truth.
RMSE between the soil moisture estimation with SepKF or EnKF and true value within the 90day assimilation window is listed in Table 7. First, RMSE for the soil moisture estimation is analyzed. For EnKF, RMSE increases with depth and further does with the temporal variability of the model bias becoming quick. For SepKF, RMSE is very small for the low temporal variability of the bias with a reduction of 62.0%, 68.8%, 71.7%, and 76.7%, respectively, from the 1st to 4th layers; but for the high temporal variability, RMSE increases but is still smaller with a reduction of 10.0%, 18.8%, 36.7%, and 41.3% in comparison with Openloop (Table 8).
State Bias Layer 1 Layer 2 Layer 3 Layer 4 Layer 1 Layer 2 Layer 3 Layer 4 EnKF_S1 0.042 0.063 0.084 0.132 – – – – SepKF_S1 0.019 0.025 0.034 0.035 0.004 0.015 0.021 0.025 SepKF_S1A 0.033 0.044 0.052 0.065 0.015 0.023 0.025 0.035 EnKF_S2 0.074 0.093 0.112 0.134 – – – – SepKF_S2 0.045 0.065 0.076 0.088 0.015 0.026 0.032 0.034 SepKF_S2A 0.023 0.025 0.030 0.035 0.002 0.012 0.017 0.029 Table 7. RMSE (cm^{3} cm^{–3}) for the soil moisture and bias estimation with/without the bias correction for the sinusoidal bias
Experiment Layer 1 Layer 2 Layer 3 Layer 4 EnKF_S1 16.0 21.3 30.0 12.0 SepKF_S1 62.0 68.8 71.7 76.7 SepKF_S1A 34.0 45.0 56.7 56.7 EnKF_S2 –48.0 –16.3 6.7 10.7 SepKF_S2 10.0 18.8 36.7 41.3 SepKF_S2A 54.0 68.8 75.0 76.7 Table 8. PRI (%) for the soil moisture estimation in different experiments
Second, the bias estimation is investigated. For the low temporal variability of the bias, except for the first soil layer, the bias is not well estimated with a relative error even larger than 50%, and slightly becomes better if the smaller λ is used (i.e., 0.2 rather than 0.8). For the high temporal variability of the bias, RMSE of the bias estimation increases for the smaller λ (0.2) in comparison with the low temporal variability and becomes smaller for the larger λ (0.8). Therefore, it is suggested that a large (small) λ should be used for the high (low) temporal variability of the model bias.

To evaluate the performance of SepKF in the real situation, assimilating pointscale observations on the natural grass terrain over Northwest China was performed with an updating frequency of once every six hours. Based on the results in OSSE, λ was set to 0.2 in the first 15day assimilation period and decreased to 0.1 in the later 15day period. The initial model bias b_{0} is set to 0.001 cm^{3} cm^{–3}. The observation error is 0.01 cm^{3} cm^{–3}. The perturbation scheme is the same as that in OSSE. RMSE and MAE are also used to assess the performance by different approaches.
The soil moisture estimation at the two middle layers during the final 30 days is plotted in Fig. 7. For comparison, the ensemble forecast (Openloop), observation (Obs), and estimation with EnKF without the bias correction (EnKF) are also shown. The soil moisture estimation with EnKF at all layers is closer to the observations than that from Openloop. SepKF further improves the soil moisture estimation, especially at the third layer. For the whole assimilation period, RMSE is the largest at all layers in Openloop and smallest with SepKF (Table 9). For example, RMSE of the soil moisture estimation relative to EnKF has a reduction of 40.4% and 13.5% with Openloop, respectively, at layers 2 and 3, while that with SepKF reduces 51.9% and 71.2% (Table 10).
Figure 7. Comparison of the soil moisture estimation by Openloop (star line), EnKF (cross line), and SepKF (solid line) with observations (circle line) at the (a) layer 2 and (b) layer 3 during the final 30 days at SACOL.
Layer 1 Layer 2 Layer 3 Layer 4 Openloop 0.038 0.052 0.104 0.060 EnKF 0.018 0.031 0.093 0.051 SepKF 0.014 0.025 0.032 0.042 Table 9. RMSE (cm^{3} cm^{–3}) for the soil moisture estimation in different experiments at SACOL
Layer 1 Layer 2 Layer 3 Layer 4 EnKF 52.6 40.4 13.5 16.7 SepKF 63.2 51.9 71.2 33.3 Table 10. PRI (%) for the soil moisture estimation in different experiments at SACOL
Variable  Nominal value  Uncertainty in replicates 
Humidity; solar radiation; wind speed; wind direction  Observations  Temporally uncorrelated multiplicative uniform noise: relative humidity: U (0.9, 1.1), solar radiation: U (0.9, 1.1), wind speed: U (0.7, 1.3), wind direction: U (0.7, 1.3) 
Air temperature  Temporally uncorrelated additive uniform noise: U (–4, 4)  
Precipitation  Temporally uncorrelated multiplicative lognormal noise with a mean of 1.0 and a standard deviation equal to 35% of the nominal value  
Leaf area index (LAI)  Multiplicative uniform noise U (0.85, 1.15) for the LAI without change with time (i.e., static)  
Initial soil moisture  0.25 cm^{3} cm^{–3}  Vertically uncorrelated additive zeromean Gaussian noise with the variance of 0.1^{2} 
Soil moisture observation  Simulated true value  Temporally uncorrelated additive zeromean Gaussian noise with the variance of 0.05^{2} 