simultaneous estimation of soil moisture and hydraulic parameters using residual resampling particle...

SCIENCE CHINA Earth Sciences

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• RESEARCH PAPER • doi: 10.1007/s11430-013-4742-y

Simultaneous estimation of soil moisture and hydraulic parameters using residual resampling particle filter

BI HaiYun1,2*, MA JianWen1, QIN SiXian1,2 & ZHANG HongJuan1

1 Institute of Remote Sensing and Digital Earth, Chinese Academy of Sciences, Beijing 100094, China; 2 University of the Chinese Academy of Sciences, Beijing 100049, China

Received April 25, 2013; accepted July 24, 2013

Land data assimilation (DA) has gradually developed into an important earth science research method because of its ability to combine model simulations and observations. Integrating new observations into a land surface model by the DA method can correct the predicted trajectory of the model and thus, improve the accuracy of state variables. It can also reduce uncertainties in the model by estimating some model parameters simultaneously. Among the various DA methods, the particle filter is free from the constraints of linear models and Gaussian error distributions, and can be applicable to any nonlinear and non-Gaussian state-space model; therefore, its importance in land data assimilation research has increased. In this study, a DA scheme was developed based on the residual resampling particle filter. Microwave brightness temperatures were assimilated into the mac-ro-scale semi-distributed variance infiltration capacity model to estimate the surface soil moisture and three hydraulic parame-ters simultaneously. Finally, to verify the scheme, a series of comparative experiments was performed with experimental data obtained during the Soil Moisture Experiment of 2004 in Arizona. The results show that the scheme can improve the accuracy of soil moisture estimations significantly. In addition, the three hydraulic parameters were also well estimated, demonstrating the effectiveness of the DA scheme.

data assimilation, residual resampling particle filter, microwave brightness temperature, soil moisture, hydraulic parameter

Citation: Bi H Y, Ma J W, Qin S X, et al. Simultaneous estimation of soil moisture and hydraulic parameters using residual resampling particle filter. Science China: Earth Sciences, 2013, doi: 10.1007/s11430-013-4742-y

Observation and model simulation are two basic methods in traditional earth science research. Both have some limita-tions in describing complex land surface conditions. Obser-vations are relatively true records of the status of the land surface, but they cannot guarantee consistency and continuity in both time and space. A land surface model (LSM) is able to provide a consistent set of data in both time and space; however, model simulation results often deviate far from the true values owing to uncertainties in model parameters, meteorological forcing data, and model structure (Li et al., 2004). How best to combine model simulations and obser-

vations to extract the benefits of both is a current focus of research. The land data assimilation (DA) method provides a viable path to achieve this aim (Gong, 2009; Li et al., 2010).

Integrating new observations into a LSM by the DA method can correct the predicted trajectory of the LSM and thus, improve the estimation accuracy of state variables (Jia et al., 2009; Shi et al., 2011). Currently, the most commonly used DA methods include the three- and four-dimensional variational methods, the Kalman filter (KF), and the ensem-ble Kalman filter (EnKF) (Shi et al., 2011). The variational method is commonly employed in numerical weather pre-diction. It depends on the backward integration of a tangent linear adjoint model; however, the development of such an

2 Bi H Y, et al. Sci China Earth Sci January (2013) Vol.56 No.?

adjoint model is often a difficult task. The KF is a typical representative of sequential DA methods. It includes not only the analysis and forecast of the state variables, but also the analysis and forecast of the error statistics, which cannot be achieved by the variational method. However, optimal solutions can only be obtained by a KF in linear and Gauss-ian systems, but most LSMs are nonlinear and the error dis-tributions are often non-Gaussian, which greatly limit the use of KF (Ma et al., 2012). Based on the KF, the EnKF was developed to use ensembles to approximate the proba-bility distribution function (pdf) of the state variables and thus, it can be applied to nonlinear systems. Therefore, the EnKF has gradually become one of the most popular DA methods (Shi et al., 2010; Shi et al., 2011). However, the EnKF also invokes the assumptions that the pdf is Gaussian, and the evolution of the filter is governed only by its first- order and second-order characteristics, resulting in a signif-icant loss of information (Moradkhani et al., 2005a). The variational method and the EnKF have been combined in some studies and their results show that the combined method is superior in performance to either the EnKF or variational method individually (Hamii et al., 2000; Hansen et al., 2001). However, the effectiveness and applicability of the combined method remains a subject to be further ex-plored and verified.

Particle filter (PF) is a sequential DA method based on Bayesian importance sampling. The key idea of PF is to approximate the required posterior pdf of the state variable by a set of particles with associated weights (Li et al., 2010). In comparison with KFs, PF has no restrictive assumptions regarding the forms of the pdf of the state variables and error structures; thus, it is applicable to any nonlinear and non-Gaussian state-space model. In addition, PF is capable of representing the entire posterior pdf of the state variables by the Monte Carlo sampling method; hence, it can better reflect the evolution in nonlinear systems (Ma et al., 2012). Therefore, since Moradkhani et al. (2005a) first used PF in land data assimilation and achieved great success, PF has received increasing attention and gradually becomes a hotspot in land data assimilation (Weerts et al., 2006; Noh et al., 2011; Nagarajan et al., 2011; Dechant et al., 2012; Plaza et al., 2012). However, there is a common problem with PF that is the particle degradation. In order to solve this problem, Gordon et al. (1993) proposed the method of resampling. Douc et al. (2005) made a comparative analysis of different resampling algorithms and found that the resid-ual resampling method could effectively reduce the vari-ances of particle weights and was computationally more efficient than other resampling methods. However, while solving the problem of particle degradation, the residual resampling method leads to a dramatic loss of diversity in particles. Zhang et al. (2013) made some improvements to the original residual resampling method. The improved re-sidual resampling method not only solves the problem of particle degradation, but also maintains the diversity of par-

ticles. In this study, soil moisture was estimated by using the improved residual resampling particle filter (RR-PF) method to prove its validity in land data assimilation.

The DA method can not only improve the estimation ac-curacy of state variables, but also reduce uncertainties in LSMs by estimating some model parameters simultaneously and thus improve the prediction accuracy of LSMs (Mo-radkhani et al., 2005a; Salamon et al., 2009). Hydraulic pa-rameters are very important in LSMs and the accurate quan-tification of their values is critical to the accurate prediction of LSMs. According to the study of Montaldo et al. (2007), when key hydraulic parameters in LSMs are estimated poorly, there will be a large deviation in the results of the model simulation from the true values. However, the hy-draulic parameters in many LSMs are often difficult to ob-tain by direct measurement, and they are usually estimated according to soil texture and bulk density by using pedo-transfer functions. Because these functions are usually de-termined from specific experimental data under specific experimental conditions, uncertainties will arise when ap-plying these functions to other LSMs (Nie et al., 2011). The DA method provides a good solution to this problem, be-cause it can be used to estimate model parameters and state variables simultaneously; thus, reducing uncertainties in model parameters and increasing the accuracy of the model simulation (Moradkhani et al., 2005b; Qin et al., 2009; Rings et al., 2010; Nie et al., 2011; Montzka et al., 2011). However, only a few publications are available that have used PFs to estimate state variables and model hydraulic parameters jointly (Qin et al., 2009; Montzka et al., 2011).

In this study, a DA scheme based on the RR-PF method was developed. The semi-distributed macro-scale variance infiltration capacity (VIC) model was chosen as the LSM and a radiative transfer model (RTM) was selected as the observation operator. Microwave brightness temperatures were assimilated into the VIC model to estimate soil mois-ture, and three hydraulic parameters (porosity s, saturated hydraulic conductivity Ks, and shape parameter b) were es-timated simultaneously by the kernel smoothing method. Finally, the DA scheme was validated by a series of com-parative experiments with experimental data obtained during the Soil Moisture Experiment of 2004 (SMEX04) in Arizona.

1 Data assimilation scheme

Generally, a complete land data assimilation scheme com-prises the following four parts: (1) the LSM, (2) the obser-vation operator, (3) the DA algorithm, and (4) the data sets. This section presents details of the LSM, the observation operator, and the DA algorithm.

1.1 Land surface model

The LSM used in this study is the VIC model (Liang et al.,

Bi H Y, et al. Sci China Earth Sci January (2013) Vol.56 No.? 3

1994; Liang et al., 1996) developed by the University of Washington, Princeton University, and University of Cali-fornia at Berkeley et al. As a semi-distributed macro-scale hydrological model, it considers different cover types of bare soil and vegetation in a computational grid and bal-ances both the water and surface energy budgets within the grid cell. Its sub-grid variations are captured statistically. The VIC model has participated in the Project for Inter-comparison of Land Surface Parameterization Schemes, where it has performed well relative to other schemes and to available observations. Additionally, it has been widely used and validated in many DA systems.

In VIC model, soil is defined as three layers. The one- dimensional Richards’s equation, used to describe the ver-tical soil water movement between different layers, is ex-pressed as:

( ) ( ) , 1,2i

i

ii Z

Z

z P R E K D it z

, (1)

2

2

33 2 ( ) ( ) , 3bZ

Z

z z K D Q it z

, (2)

where is the volumetric soil moisture content, z is the soil depth, K is the hydraulic conductivity, D is the soil water diffusivity, P is the precipitation, R is the surface runoff, E is the evaporation, and Qb is the base flow. Among these parameters, K and D are two important hydraulic parame-ters in VIC model. However, parameter D in fact does not work because the soil moisture diffusion process has not been implemented when programming the VIC model. Therefore, the hydraulic conductivity K is the only parame-ter to be estimated in this study. K is calculated in the VIC model by an empirical formula defined by Clapp et al. (1978):

2 3b

ss

K K

, (3)

where denotes the unsaturated soil moisture, s is the sat-urated soil moisture, i.e., the porosity, Ks is the saturated hydraulic conductivity, and b is an empirical shape parame-ter. In this study, the parameters s, Ks, and b were chosen to be estimated simultaneously.

1.2 Observation operator

The observation data used in this study are the microwave brightness temperatures which reflect the radiative charac-teristics of the surface soil moisture. Therefore, a RTM is needed to link them during the DA process (Jackson et al., 1991) which is expressed as:

1 exp 1

1 exp 1 exp ,

Bp p s c c p

c p c

T R T T

R

(4)

where the subscript p represents the vertical or horizontal polarization, TBp is the brightness temperature, Rp is the ef-fective reflectivity of a rough surface, Ts and Tc are the thermodynamic temperatures of soil and vegetation canopy, respectively, and they were presumed equal in this study (Njoku et al., 2003). c is the vegetation optical depth and p is the single-scattering albedo of vegetation.

To calculate the effective reflectivity of a rough surface Rp, the Qp model is used (Shi et al., 2005):

1p p q p pR Q r Q r , (5)

where Qp is the surface roughness parameter, rp and rq de-note the Fresnel reflectivity of a smooth surface, corre-sponding to the horizontal and vertical polarization, respec-tively. They are related to the soil complex permittivity and the system parameters of the microwave sensor and can be derived using the Fresnel formulas. The soil complex per-mittivity can be determined by the Dobson model (Dobson et al., 1985):

1

1 1bs v fw v

s

m m

, (6)

where denotes the soil complex permittivity, b represents the bulk density, s is the solid density and usually takes the value of 2.66, s is the dielectric constant of the solid mate-rial in the soil, and generally s=4.692, is a constant factor with a value of 0.65, mv represents the soil volumetric water content, is an adjustable parameter dependent on the soil texture, and fw is the complex permittivity of free water, which can be computed by the bulk density, temperature, soil texture, and frequency.

1.3 Data assimilation algorithm

1.3.1 Bayesian theory

Bayesian theory is the theoretical basis of DA and it pro-vides a generalized theoretical framework for DA (Li et al., 2010). From the perspective of Bayesian estimation, the purpose of DA is to seek the posterior pdf p(xk|z1:k) of the state xk based on the set of all available observations z1:k. It mainly consists of two steps: the prediction and the update. These two steps are formulated as eqs. (7) and (8), respec-tively:

1: 1 1 1 1: 1 1| | | dk k k k k k kp x z p x x p x z x , (7)

1: 11:

1: 1

| || ,

| | d

k k k kk k

k k k k k

p z x p x zp x z

p z x p x z x

(8)

where the subscript k denotes the time step, 1 1: 1( | )k kp x z

denotes the posterior pdf at the last time step. When the


transition pdf 1( | )k kp x x is acquired from the LSM, the

prior pdf 1: 1( | )k kp x z can be obtained through the predic-

tion step (7). When the new observation zk arrives, the like-lihood pdf ( | )k kp z x is calculated and the posterior pdf

1:( | )k kp x z can be derived through the update step (8).

Most LSMs are very complex, which makes the analyti-cal solutions of (7) and (8) intractable; therefore, some methods of approximation, such as the EnKF and the PF, have to be used in practical applications.

1.3.2 Residual resampling particle filtering

PF is an implementation of Bayesian theory by Monte Carlo sampling. The principle of PF is to represent the posterior pdf of state variables by a set of random particles with asso-ciated weights. With an increase in the number of particles, the real posterior pdf of the state variable is gradually ap-

proached (Li et al., 2010). Suppose N particles 1

,Ni i

k k ix w

are sampled from the posterior pdf 1:( | )k kp x z , where ikx

denotes the state of the particle and ikw represents the as-

sociated weight. The posterior pdf of the state variable can be approximated as:

1:1

| ,N

i ik k k k k

i

p x z w x x

(9)

where is the Dirac function. Since the posterior pdf is usually unknown, and it’s difficult to directly sample from

1:( | )k kp x z , the particles are often generated from a known

important density denoted by 1:( | )k kq x z . In practical ap-

plications, the sequential importance sampling method is often used. It updates the importance weights of particles in a recursive form, as follows:

1

1

1

| |,

| ,

i i ik k k ki i

k k i ik k k

p z x p x xw w

q x x z

(10)

where 1( | , )i ik k kq x x z denotes the importance density. The

transitional prior pdf is usually selected as the importance

density for simplicity, where: 1 1| , |i i i ik k k k kq x x z p x x .

Then the weight-updating formula in eq. (10) becomes:

1 | .i i ik k k kw w p z x (11)

With these particles and associated weights, the final es-timated state ˆ

kx is the weighted mean of these particles, as

follows:

1

ˆ .N

i ik k k

i

x x w

(12)

However, there is a common problem with PF that is the particle degradation, where after a few iterations, most par-

ticles have negligible weights. This means that large com-putational efforts are devoted to updating particles whose contributions to the approximation of the posterior pdf are nearly zero. In this study, the improved residual resampling method, proposed by Zhang et al. (2013), was used to solve the problem of particle degradation. This method uses the Halton sequence and the exponential function to generate new particles instead of directly copying the original parti-cles; thus, it is capable of overcoming the problem of parti-cle degradation while maintaining the diversity of particles. Details about this algorithm can be found in Zhang et al. (2013).

1.3.3 Parameter estimation method

Yang et al. (2007) and Jia et al. (2009) developed the dual- pass DA scheme, which was able to estimate some uncer-tain model parameters. This scheme comprises two phases: the parameter optimization phase, and the DA phase. In the first phase, uncertain model parameters are optimized by using long-term historical observation data in a long time window. In the second phase, the DA method is used to assimilate the observation data into the model with optimal parameters obtained in the first phase to estimate the state variable in a much shorter time window. Experiments have shown that the scheme can estimate model parameters ef-fectively and thus, it can improve the assimilation accuracy of the state variable. However, the scheme requires a long period of historical observation data; therefore, it can be applied only in cases with adequate historical observation data. Additionally, for those parameters with large temporal variability, the optimal values obtained in the first phase may not be appropriate in the second phase. In this study, the parameter estimation method used is to put the state variable and model parameters together to constitute an augmented state variable and then to estimate the parameters and state variable simultaneously during the DA process. This method can avoid the dependence on a large number of historical observation data.

When using the RR-PF method to estimate the state var-iable and model parameters simultaneously, the time evolu-tion of the state variable relies on the LSM. However, there is no model explicitly for the time evolution of the model parameters; therefore, a parameter evolution model has to be built. There are generally two methods to achieve this. One is the parameter perturbation method introduced by Moradkhani et al. (2005a). This method is easy, but will make the covariance of parameters increase over time. The other is the kernel smoothing method, which is adopted in this study (West, 1993; Liu, 2000). It can realize the evolu-tion of parameters and ensure that the variance of parame-ters distribution will not increase over time. Suppose there is a set of particles 1 1 1{ , }i i N

k k iw to approximate the poste-

rior pdf of the parameters 1: 1( | )k kp z , where 1ik denotes

the state of the particle and 1ikw

denotes the associated


weight. The mean and variance of all particles are 1k and

1kV , respectively. West (1993) suggested that the posterior

pdf could be approximated by a weighted mixture of Gaussian densities as follows:

21: 1 1 1 1

1

| | , ,N

i ik k k k k k

i

p z w N m h V

(13)

where 1k

im

and 2

1kh V denote the mean and variance, respec-

tively, and h is a smoothing parameter used to control the change speed of the parameter. To avoid an over-dispersed posterior pdf, West (1993) suggested that:

1 1 1(1 )i ik k km a a 21 .a h (14)

Hence, by using the kernel smoothing method, the time evolution of the parameter can be performed as:

2 2 21 1 1 1| ~ | 1 1 1 , .i i

k k k k k kp N h h h V

(15)

2 Technical process and experimental procedures

Based on the above introduction of the DA scheme, the technical process of the proposed scheme is summarized in Figure 1, where the state variable represents the surface soil moisture and the parameter vector represents the three hy-draulic parameters (s, Ks, and b). According to the tech-nical process, the main experimental procedures are ex-tracted as follows:

(1) Define the state variable as x and the parameter vec-tor as . At the initial time step k=0. Then, generate N parti-cles with equal weights from the prior pdf of the state varia-ble and parameter vector: 0 1 0{ } ~ ( }i N

ix p x , 0 1 0{ } ~ ( }i Ni p ,

0 1{ } 1i Niw N .

(2) Determine the time evolution of the parameter parti-cles by the kernel smoothing method:

2 2 21 1 1{ } ~ | 1 1 1 ,i N i

k i k k k kN h h h V .

(3) Prepare the soil file, vegetation file, and forcing file

Figure 1 Flow chart of the DA scheme.


to run the VIC model. Then, the forecast states of the state

variable particles are obtained: 1 1 1 1{ } ~ | ,i N i i

k i k k kx p x x .

(4) If brightness temperature observations zk+1 are available at the current time step, compute the simulated brightness temperatures through the RTM. Then the likelihood pdf

1 1 1( | , )i ik k kp z x is calculated. Finally, the weights of all

state variable and parameter particles are updated: 1ikw

1 1 1| , ,i i ik k k kw p z x and normalized: 1 1 1

1

.N

i i ik k k

i

w w w

Otherwise, skip to step (6) and the final estimated state is obtained.

(5) If particle degradation occurs, perform residual resampling for all the state variable and parameter particles according to 1

ikw . Then, N new particles with equal weights

are generated: 1 1

Nik i

x , 1 1

Nik i

, 1 1

1Ni

k iw N

. Oth-

erwise, skip to step (6) and the final estimated state is de-rived.

(6) Calculate the final estimated state of the state variable and the parameter vector:

1 1 11

ˆN

i ik k k

i

x w x

, 1ˆk 1 1

1

Ni ik k

i

w .

(7) k=k+1 and then return to step (2) until all desired time steps are finished.

3 Study area and data preparation

3.1 Study area

The study area is located in the southeast of Arizona in the United States, covering an area of 32 km×32 km. It is part of the SMEX04 experimental area and its geographic loca-tion is presented in Figure 2(a). Abundant soil moisture field observation data and meteorological data were ob-tained at the study area during SMEX04. Three soil mois-ture field observation networks: the SCAN (Soil Climate Analysis Network), the AZ (Arizona Regional site), and the RG (Walnut Gulch Watershed Rain Gauge Site), are de-ployed in the study area providing 31 soil moisture observa-tion sites, as shown in Figure 2(b). These soil moisture ob-servation sites can provide daily soil moisture observations throughout the entire experimental period. Moreover, the Polarimetric Scanning Radiometer obtained both C-band (7.32 GHz) and X-band (10.7 GHz) brightness temperatures of the study area during SMEX04 for nine days (5, 8–10, 12–13, and 24–26 August). The Polarimetric Scanning Radiometer has a footprint size of 800 m × 800 m, with both horizontal and vertical polarizations, and the incident angle is 55°.

3.2 Data preparation

Brightness temperatures were available for nine days: 5,

8–10, 12–13, and 24–26 August. Therefore, the assimilation period was chosen as 5–26 August. As brightness tempera-tures can only reflect the radiative characteristics of surface soil moisture, the three soil depths of the VIC model were defined as 5, 30, and 100 cm to maintain physical con-sistency with the brightness temperatures, and only the soil moisture in the top 5 cm was updated. Additionally, the VIC model was run in the water balance mode. The spatial reso-lution was defined as 800 m×800 m to be consistent with that of the brightness temperatures. Therefore, there were 40×40 grids in the study area, and the time resolution was defined as 24 hours. In addition, the meteorological forcing file, soil file, and the vegetation file are required for the operation of the VIC model. The meteorological forcing file was extracted by joint interpolation of the meteorological data provided both by the meteorological stations in the study area and by the North American Land Data Assimila-tion System (NLDAS). The vegetation file was obtained from the global land cover type data provided by the Uni-versity of Maryland, and the vegetation parameters library file was provided by NLDAS. The soil file was determined from the CONUS-SOIL database built by the United States Department of Agriculture. All data were projected to the same coordinate system and resampled to a resolution of 800 m×800 m. Finally, the model run was set to begin on 20 June 2004 to eliminate the spin-up period, but only the data following 5 August 2004 were used for the DA experi-ments.

4 Experiment design

To validate the DA scheme, four comparative experiments were designed. In experiment 1, a forward reference run of the VIC model was performed to simulate soil moisture. In experiment 2, brightness temperatures were assimilated by using a PF method without residual resampling to estimate soil moisture with empirical hydraulic parameters. In ex-periment 3, brightness temperatures were assimilated by the RR-PF method to estimate soil moisture with empirical hy-draulic parameters, and in experiment 4, brightness temper-atures were assimilated by the RR-PF method to estimate both soil moisture and hydraulic parameters simultaneously.

The hydraulic parameters to be estimated in this study include the porosity s, the saturated hydraulic conductivity Ks, and the shape parameter b. Cosby et al. (1984) gave the empirical values of Ks, s, and b of different soil types. When estimating soil moisture only (experiments 2 and 3), the three parameters were assigned the empirical values. When estimating soil moisture and hydraulic parameters simultaneously (experiment 4), the initial ranges of the three parameters were obtained by adding random noise to the empirical values, as the upper and lower bounds, but also guaranteeing that the parameters lie within their physical ranges. In the study area, there are two principal soil moisture


Figure 2 Location of the study area and distribution of all soil moisture observation sites. (a) Location of the study area; (b) distribution of all soil moisture observation sites within the study area.

observation networks: the AZ, and RG, from which two soil moisture observation sites, AZ06 and RG100, were selected as representative sites for the experiment. The empirical values and initial ranges of the three parameters at sites AZ06 and RG100 are presented in Table 1. Considering both the computational efficiency and assimilation accuracy, 100 particles were used in the experiment. We compute the

mean and standard deviation of all observation data at the observation sites and suppose that they are represented as and , respectively. Then, the initial particles are sampled from a Gaussian distribution with mean and standard de-viation . The model error is obtained from the error statis-tics of the model simulation results, based on the presump-tion of ground observations as true values. The observation


Table 1 Initial ranges and default values of the three hydraulic parameters

Station name Parameter Unit Minimum Maximum Empirical value

AZ06

s m3 m3 0.14 0.62 0.40

Ks mm day1 237.59 691.55 451.90

b – 3.12 7.51 4.84

RG100

s m3 m3 0.29 0.68 0.45

Ks mm day1 100.58 548.14 292.03

b – 3.98 8.42 5.30

error is sampled from a Gaussian distribution with mean 0 and standard deviation 3. At the same time, two error statis-tics, the root mean square error (RMSE) and the mean bias error (MBE), are selected to evaluate and compare the as-similation results and the model simulation results quantita-tively. They are defined as:

2

1

1,

N

i ii

RMSE X ObsN

(16)

1

1,

N

i ii

MBE X ObsN

(17)

where N denotes the total number of time steps, Obsi repre-sents the ground observations, and Xi represents the model simulation results or assimilation results.

5 Experimental results and analysis

5.1 Polarization selection

The observation data used in this study are the C-band mi-crowave brightness temperatures, including both horizontal and vertical polarizations. When assimilating brightness temperatures, the polarization selection is always a contro-versial issue. On the one hand, Owe et al. (2001) highlight-ed that the horizontal polarization brightness temperature is more sensitive to soil moisture compared with the vertical polarization. Hence, the horizontal polarization brightness temperature is always used to estimate soil moisture in the field of passive microwave remote sensing. On the other hand, according to the ground-based microwave experi-ments conducted by Fujii (2005), the horizontal polarization brightness temperature is more sensitive to the heterogenei-ty of the land surface. As the heterogeneity cannot be ex-pressed easily in a LSM, Yang et al. (2007) recommended the use of vertical polarization brightness temperature for DA. Therefore, to establish which polarization brightness temperature brings higher assimilation accuracy, two dif-ferent polarization brightness temperatures were used in this study to perform experiments 1 and 3 at sites AZ06 and RG100. The experimental results are compared with ground observations at these two sites (Figure 3). The error statis-tics of the assimilation results of both polarization bright-

ness temperatures are displayed in Table 2. From Figure 3 and Table 2, the following conclusions can be drawn: (1) Comparing the assimilation results of horizontal and verti-cal polarizations, it can be seen that the soil moisture esti-mated from vertical polarization is higher than that by hori-zontal polarization, which is consistent with the conclusions drawn by Tian et al. (2009) and Zhao et al. (2013). (2) In the arid study area, assimilating both horizontal and vertical polarization brightness temperatures significantly improves the accuracy of soil moisture estimations. Both the specific values and overall trends of the assimilation results agree much better with ground observations than the VIC model simulations. However, the assimilation results of the hori-zontal and vertical polarizations are quite similar and do not differ greatly. Finally, the horizontal brightness temperature was chosen as the observation data in this study, and the following experimental results and analyses are all based on the horizontal polarization brightness temperature data.

5.2 Soil moisture estimation

The results of four comparative experiments are compared with ground observations at sites AZ06 and RG100 (Figure 4), and their error statistics are listed in Table 3. From Fig-ure 4 and Table 3, the following conclusions can be drawn: (1) As shown in the results of experiment 1, the simulation results of the reference run deviate far from the observations, and the simulation results are overestimated significantly. This is mainly due to the uncertainty resources in the VIC model, including the model parameters, the meteorological forcing data, and the model structure. (2) By comparing the results of experiments 1 and 3, it can be seen that assimilat-ing brightness temperatures by the RR-PF method reduces uncertainties in the VIC model; thus, improving the accu-racy of soil moisture estimations significantly. Both the specific values and overall trends of the assimilation results are more consistent with the observations in contrast to the simulation results, and the RMSE and MBE values of the assimilation results are also much smaller than those of the simulation results. (3) By comparing the results of experi-ments 2 and 3, it is clear that the soil moisture estimations of the RR-PF method show higher accuracy and better con-formity with ground observations than those of the PF


Figure 3 Comparison of the assimilation results of vertical polarization and horizontal polarization. (a) Comparison results at site AZ06. (b) Comparison results at site RG100. Pre is the precipitation; OBS is the ground observations; VIC is the VIC model simulation results; DA-H and DA-V are the assimila-tion results of horizontal polarization and vertical polarization, respectively; the arrows indicate the time steps when brightness temperature observations are available.

Table 2 Error statistics of the assimilation results of horizontal polariza-tion and vertical polarization

Station name Error statistics VIC DA-H DA-V

AZ06 RMSE 0.131 0.028 0.031

MBE 0.128 0.006 0.006

RG100 RMSE 0.170 0.021 0.023

MBE 0.167 0.010 0.011

method without residual resampling. The RMSE and MBE values of the RR-PF method are smaller than for the PF method without residual resampling. Initially, there is no significant difference in their performance because particle degradation is not serious. However, over time, particle degradation becomes increasingly serious. The RR-PF method solves the problem of particle degradation through the residual resampling procedure, which results in estima-tions of soil moisture consistent with ground observations. Serious particle degradation occurs in the PF method with-out residual resampling and thus, the particles can no longer accurately represent the real pdf of the state variables, which leads to a large deviation of the assimilation results

from the true values. (4) By comparing the results of ex-periments 3 and 4, it is apparent that when estimating soil moisture and hydraulic parameters simultaneously, the as-similation results show higher accuracy and agree better with the observations than those when hydraulic parameters take the empirical values. Additionally, the RMSE and MBE values are also smaller when estimating soil moisture and hydraulic parameter jointly. Hence, conclusions can be drawn that the hydraulic parameters play a significant role in the estimation of soil moisture, and that a false hydraulic parameterization will generate a large deviation in the esti-mation of soil moisture. When estimating soil moisture and hydraulic parameters simultaneously, uncertainties from the hydraulic parameters can be reduced gradually and there-fore, improve the accuracy of estimations of soil moisture. (5) By comparing all the experimental results with ground observations, we can see that although the accuracy of the estimation of soil moisture is improved significantly by us-ing the DA method compared with the VIC model simula-tion results, deviations still exist between the assimilation results and ground observations. The main reasons for this are as following: First, only uncertainties in model parameters are considered in this study, and other sources of uncertainty,


Figure 4 Comparison of the assimilation results of four comparative experiments with ground observations. (a) Comparison results at site AZ06. (b) Com-parison results at site RG100. Pre is the precipitation; OBS is the ground observations; VIC, PF-SM, RR-PF-SM, and RR-PF-DUAL denote the experimental results of experiments 1, 2, 3, and 4, respectively; the arrows indicate the time steps when brightness temperature observations are available.

Table 3 Error statistics of four comparative experiments

Station name Error statistics VIC PF-SM RR-PF-SM RR-PF-DUAL

AZ06 RMSE 0.131 0.030 0.028 0.021

MBE 0.128 0.013 0.006 0.005

RG100 RMSE 0.170 0.028 0.021 0.018

MBE 0.167 0.013 0.010 0.008

such as errors in the meteorological forcing data and model structure are not considered. Second, the region is in a semi-arid climate with little annual rainfall, which results in low soil moisture throughout the study area. Some studies have indicated that brightness temperatures are less sensi-tive to soil moisture in semi-arid regions (Su et al., 2011). Therefore, assimilating brightness temperatures could im-prove the accuracy of the estimation of soil moisture to some extent in this study area, but the result is not satisfac-tory. Furthermore, as the observation data used in this study are the brightness temperatures with a spatial resolution of 800 m×800 m, the assimilation results obtained are the are-al-averaged soil moisture over the area of 800 m×800 m, whereas the ground soil moisture observations are point based. Thus, comparing them directly is bound to bring

some errors because of inconsistent horizontal spatial scales. In addition, the penetration depth of the C-band brightness temperature is generally 1–2 cm (Jackson et al., 1989), whereas the depth of surface soil moisture simulated by the VIC model is 5 cm. Therefore, when assimilating brightness temperatures into the VIC model, the mismatch of the ver-tical scale will also cause some errors.

Only the experimental results at the observation sites AZ06 and RG100 are displayed above. To further demon-strate the effectiveness of the DA scheme across the entire study area, experiments 1 and 4 were performed at all of the 31 soil moisture observation sites. The RMSE and MBE values of the two comparative experiments at all sites are shown in Figure 5(a) and 5(b). As can be seen, the accuracy of soil moisture estimations at all sites is improved greatly


Figure 5 Comparison of MBE and RMSE values of the VIC model simulation results and assimilation results at all 31 sites within the study area. (a) Comparison of MBE. (b) Comparison of RMSE. MBE-VIC and MBE-RR-PF represent the MBE values of the VIC model simulation results and assimila-tion results, respectively; RMSE-VIC and RMSE-RR-PF denote the RMSE values of VIC model simulation results and assimilation results, respectively.

through the process of DA, and the RMSE and MBE values of the assimilation results are significantly lower than those of the VIC model simulation results. The RMSE and MBE values of the assimilation results are mainly concentrated in the range of 0 to 0.05 and 0.05 to 0, respectively, whereas the RMSE and MBE values of the model simulation results are mainly concentrated in the range of 0.1 to 0.15 and 0.05 to 0.15, respectively. These results further demonstrate the effectiveness of the scheme for the entire study area.

On the other hand, to reduce the difference in spatial scale between the point-based ground observations and the grid- based brightness temperatures, as well as to further verify the effectiveness of the DA scheme, brightness temperatures of both horizontal and vertical polarizations of all grids in the study area were averaged. Then, the averaged brightness temperatures were assimilated by the RR-PF method to es-timate the soil moisture. The assimilation results of both polarizations and the VIC model simulation results are compared with the average of all ground observations of all sites within the study area (Figure 6), and the error statistics are presented in Table 4. From Figure 6 and Table 4, we can see that compared with the VIC model simulation results, the assimilation results of the averaged brightness tempera-tures are much closer to the averaged ground observations

of soil moisture. Both their specific values and overall trends are very consistent. This is in accordance with the experimental results of a single grid; thus, further demon-strating the validity of the DA scheme. Additionally, both Figure 6 and Table 4 show that the assimilation results of the vertical polarization are slightly higher than the hori-zontal polarization, but they do not differ greatly. Assimi-lating both horizontal and vertical polarization brightness temperatures can improve the accuracy of soil moisture es-timations, and their estimation accuracy is very similar. These findings are consistent with the conclusions drawn in section 5.1.

5.3 Hydraulic parameters estimation

In this study, the hydraulic parameters estimated simulta-neously include the porosity s, the saturated hydraulic con-ductivity Ks, and the shape parameter b. Suppose that the initial distribution of the three parameters is uniform (Mo-radkhani et al., 2005a; Salamon et al., 2009; Qin et al., 2009), then the three hydraulic parameters can be sampled uniformly in their initial ranges to represent the initial un-certainty of each parameter. With the assimilation of bright-ness temperatures, the uncertainty ranges of the parameters


Figure 6 Comparison of the assimilation results of averaged brightness temperatures of all grids of both horizontal and vertical polarizations with the av-eraged ground observations of all sites. Pre is the precipitation; OBS is the averaged ground observations; VIC is the VIC model simulation results; DA-H and DA-V denote the assimilation results of averaged brightness temperatures of horizontal polarization and vertical polarization, respectively; the arrows indicate the time steps when brightness temperature observations are available.

Table 4 Error statistics of the assimilation results of averaged brightness temperatures of all grids of both horizontal and vertical polarizations

Error statistics VIC DA-H DA-V

RMSE 0.150 0.019 0.020

MBE 0.149 0.014 0.015

decrease gradually and the parameters approach the true values progressively. The estimation results of the three parameters at sites AZ06 and RG100 are presented in Fig-ure 7. This shows that with the assimilation of brightness temperatures, the three parameters begin to converge with the uncertainty bounds gradually becoming smaller. At the initial stage, there is a dramatic reduction in the uncertainty of the parameters. Then, the parameters remain unchanged during the period when there are no brightness temperature observations. Finally, the parameters all converge into a small range. Among the three parameters, s is the most identifiable, showing the fastest convergence with the smallest final uncertainty bound, and the true value is lo-cated within the final convergent range. In comparison, pa-rameters Ks and b are less identifiable with final uncertainty bounds much larger than those of s. The reason is that s is more strongly connected with surface soil moisture and thus, directly affects the brightness temperatures (Yang et al., 2007; Qin et al., 2009; Nagarajan et al., 2011), whereas Ks and b have comparatively smaller effects on surface soil moisture. This is because Arizona is the driest state in the United States, which means that very little rainfall is re-ceived by the study area. As parameters Ks and b influence soil moisture transport process immediately after precipita-tion events, connections between parameters Ks, b, and sur-face soil moisture are stronger during precipitation events and weaker during dry periods (Nagarajan et al., 2011). Therefore, parameters Ks and b have relatively smaller in-

fluence on surface soil moisture and thus, are less sensitive to brightness temperatures. The information which is more closely related to the parameters is included in the observa-tions; the more uncertainties are removed in the final esti-mated results (Qin et al., 2009). Therefore, s has higher estimation accuracy than do Ks and b.

6 Discussions and conclusions

Recently, the land data assimilation method has developed into an important earth science research method because of its capability of combining model dynamics and observa-tions. Integrating new observational data into a LSM through the DA method not only improves the accuracy of the esti-mation of state variables, but also increases the prediction accuracy of the LSM by estimating some uncertain model parameters. Among the various DA methods, the RR-PF method is free from the constraints of system linearity and Gaussian distribution of error structures. Additionally, it is capable of solving the problem of particle degradation ef-fectively and thus, shows great promise for the simultane-ous estimation of both state variables and model parameters.

In this study, a DA scheme was developed based on the RR-PF method. Brightness temperatures were assimilated into a VIC model to estimate soil moisture and three hy-draulic parameters (Ks, s, and b) simultaneously. Then, to validate the scheme, a series of comparative experiments was performed with experimental data obtained in Arizona during SMEX04. The results show that the scheme can im-prove the accuracy of soil moisture estimations, and that the assimilation results are more consistent with ground obser-vations compared with the model simulation results. In ad-dition, as the RR-PF method can address the problem of particle degradation, the assimilation results of the RR-PF method show higher accuracy than those of the PF method


Figure 7 Estimation results of the three hydraulic parameters. (a), (b), and (c) denote the estimation results of Ks, s, and b at site AZ06. (d), (e), and (f)

denote the estimation results of Ks, s, and b at site RG100. Shaded areas correspond to 95%, 90%, 70%, and 10% confidence intervals. White line shows the average and the asterisk denotes the true observed value, and only s has the true observed value.


without residual resampling. Furthermore, the three hydrau-lic parameters in the VIC model are also well estimated by the RR-PF method, and s has the highest estimation accu-racy, followed by Ks and b. Moreover, the simultaneous estimation of hydraulic parameters can reduce uncertainties in model parameters and thus, further improve the accuracy of soil moisture estimations.

There is still much work to do to further improve the DA scheme. First, the parameter estimation method used in this study is to place the state variable and model parameters together to constitute an augmented state variable, and then to let model parameters change synchronously with the state variable during the DA process. However, Yang et al. (2007) identified that the time scale over which the state variable affects the model simulation results is not consistent with that of the model parameters. Specifically, the state variable often affects the model simulation results over a short time scale, whereas the model parameters generally affect the model simulation results over a much longer time scale. Therefore, the strategy of choosing the same time steps for both the state variable and the model parameters and then letting them change synchronously is not very reasonable. Setting different time steps for the state variable and the model parameters and then estimating them separately will be considered in future work. Second, the validation strate-gy used in this study is to compare the assimilation results of a grid with the ground observations at a site located within the grid. This validation strategy is bound to bring some errors because of the mismatch of their spatial scales. To reduce the differences of their spatial scales, brightness temperatures of all grids are averaged and assimilated. Then, the assimilation results of the averaged brightness tempera-tures of all grids are compared with the average of all ground observations of all sites within the study area. However, this method still does not fundamentally solve the problem of the mismatch of spatial scales between the satel-lite brightness temperatures and the ground observations. The multi-scale DA method proposed by Pan et al. (2010) will be adopted to further improve the assimilation accuracy. Finally, uncertainties in the model parameters are consid-ered in this study, and other sources of uncertainty, such as errors in the meteorological forcing data and the model structure, will be taken into account in future work to im-prove the assimilation accuracy comprehensively.

This work was supported by the Institute of Remote Sensing and Digital Earth Chinese Academy of Sciences under the project “High-resolution Optical Image Automatic Target Recognition” (Grant No. Y2YY02101B). We thank all the reviewers for their valuable comments and the National Snow and Ice Data Center (NSIDC) for providing the experimental data of SMEX04.

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