It is challenging to assimilate the evapotranspiration product (EP) retrieved from satellite data into land surface models (LSMs). In this paper, a perturbed ensemble Kalman filter (PEKF) and à trous wavelet transform (AWT) integrated method are proposed to implement the evapotranspiration assimilation. In this method, the AWT is used to decompose the EPs into multiple channels since it is very powerful in fusing high frequency spatial information of multisource data, and then the Kalman filter is performed in the AWT domain. The proposed method combines the advantages of the PEKF that is capable of accommodating model error and observation error, and the AWT can effectively perform multiresolution fusion. Assimilation experiment conducted with the Noah model and the EP retrieved from the MODIS data shows that the proposed method performs better than the traditional ensemble Kalman filter (EnKF) and PEKF methods. The analysis results fit well with the evapotranspiration observation at two field sites with different land surface conditions. These indicate that the proposed method is promising for assimilating regional scale satellite retrieved EP into LSMs.
Evapotranspiration (ET) is an important component of the water and energy exchanges between the atmosphere and land surface. It is crucial to accurately estimate ET for studying global or regional water and energy balances. Hence, good quality of spatial and temporal ET production (EP) can help to improve comprehension of water and energy cycle. However, this kind of EP is generally difficult to obtain in both dimensions of space and time because ET is influenced by many factors, such as air and skin temperatures, soil moisture, vegetation fraction, and horizontal advection. Up to now, there are two approaches to estimate the ET. One is site observations or remote sensing retrievals. Site observations have high spatial resolutions, but can only provide the EP for limited spatial locations [
DA provides a framework for improving the LSMs by updating the state variables of the LSMs (SVLs) with observations and can combine the high spatial resolution of the observation with the high temporal resolution of the LSM. DA can be realized by two kinds of schemes: continuous assimilation and sequential assimilation. In continuous assimilation, the SVLs are modulated to be close to the observations. Continuous assimilation methods may cause the abrupt change of the SVLs before and after the DA, which will make the subsequent simulation of the LSMs to easily produce obvious errors. In sequential assimilation, the SVLs are updated according to some forecast principle. The update is usually completed by adding the product of the gain that is estimated from model errors and observation errors and the difference between the observations and the outputs of the LSM to the SVLs. Sequential assimilation methods can produce a statistically optimal and dynamically consistent state estimate of the land surface by considering observation errors and model errors. Among sequential assimilation methods, Kalman filter- (KF-) derived methods yield accurate and consecutive ET estimate and have been applied widely in recent years [
Up to now, many KF-derived DA methods have been proposed. The KF method explicitly computes the error covariances through an additional matrix equation that propagates error information from one update time to the next, subject to possibly uncertain model dynamics [
Qin et al. [
The main purpose of the DA is to combine the complementary information from measurements and models of the LSMs into an optimal estimate of the geophysical fields of interest [
The AWT was introduced by Holdschneider et al. in 2002 using an à trous (holes) algorithm and can preserve the translation invariance; that is, a translation of the original signal necessarily implies a translation of the corresponding wavelet coefficients [
The aim of using the AWT to ET assimilation is to improve ET prediction and hence to indirectly correct the heat flux predictions, meanwhile achieving physically correct soil moisture and temperature estimates through DA. Hence, this study tests the application of the AWT to assimilate remote sensed EP for producing better heat fluxes. Seldom papers could be found about the use of the AWT in DA [
In Section
Experimental results conducted in Section
The PEKF can be understood as a purely statistical Monte Carlo method, where the ensemble of the SVLs evolves in state space with considering the mean of the ensemble as the best estimate and the spreading of the ensemble as the error variance [
Given the
In (
The forecast error can be analyzed as
Finally, the assimilation is performed as
In (
The PEKF method uses the ensemble to describe the covariance matrix of the SVLs and avoids the explicit calculation of covariance matrix needed in the EKF. A modest misspecification of the initial ensemble normally does not influence the results very much over time. The PEKF allows for a wide range of noise models and one is not restricted to using Gaussian distributed noise.
It can be found from (
From the perspective of data fusion, the assimilation process can be considered as constructing one coefficient with both the same temporal response as the simulation of the LSM and the same spatial response as the observation at a particular grid location. With the development of the AWT, we expect much room for improvement over the traditional single-grid-based assimilation scheme to merge the observation and the simulation of the LSM in the AWT domain since the wavelet filter can consider the interrelation between the grids during the detail extraction of the observation.
The AWT can provide good localization in both frequency and space domains in terms of decomposing the data with finite energy into multiple channels, each one of them with a different degree of resolution [
This algorithm consists basically in the application of consecutive convolutions between the data under analysis and a scaling function at distinct degradation levels [
If the original data is represented by
To carry out the data synthesis from a degradation level
By manipulating independently the approximation components and the wavelet planes of the data, the AWT shows excellent performance in integrating the spectral information of the multispectral data and the spatial information of the panchromatic data. For the DA, the purpose is to integrate the high spatial resolution information of the observation into the simulation of the LSMs. Hence, the purposes of the data fusion and the DA have similarity.
It can be seen from (
After the forecast of the LSM using the ensemble, see (
In (
Apply the AWT to
In (
Then, (
Then the gain matrix
In (
Finally, the assimilation is performed as
Compared with (
For assimilating the MDRE into the EP of the LSM, the DA procedure based on the AWT-PEKF method comprises the following steps (Figure run the LSM in open time loop; given the Gaussian probability density function, produce the ensemble of the SVLs according to their probability characteristic. Run the LSM during ensemble loop; integrate the LSM to get the simulation result over the simulation range, namely, complete range loop; end ensemble loop. Obtain the mean EP by averaging the simulated EPs as
perturb the MDRE apply the AWT on the MDRE and EP. Respectively, obtain the decomposition results as
calculate the gain at decomposition level innovate the EP of the LSM as
integrate the LSM until ending open time loop.
The schematic flowchart of the AWT-PEKF method.
In this section, the AWT-PEKF method is tested through assimilating the EP of the Noah model with the MDRE retrieved using one operational two-layer method (OTLM) [
Cern was established in 1988 for studying the problems related with Chinese ecology. The network provides continuous observations of ecosystem level exchanges of biogeochemical, water, energy, and momentum at diurnal, monthly, seasonal, and annual time scales. Cern currently comprises forty-six sites, which distribute over the whole of China. The data of the site observations used in this study are obtained from the Yucheng and Lawn sites. At the two sites, ET has been continuously measured by eddy correlation system which is composed of a 3D sonic anemometer and an open path CO2/H2O analyzer since 2001. The ET data that were obtained by averaging the original data of 10 Hz over 30 min are used in the experiment as the validating data.
The Yucheng site was installed on Huabei plain of China. The field site is located near the Yucheng county, Shandong province, China, and the geographical coordinate is approximate 36.96°N, 116.63°E, and the altitude is 20 m above sea level. The climate of the area is warm and semi-humid with annual mean temperature 13.1°C, mean precipitation 600 mm per year, and the average monthly relative humidity 66.44%. The soil is mainly classified as sandy-clay loam and sandy loam. The characteristic of this site is typical of agricultural plot. The farm has been continuously alternated between wheat and corn. Lawn site was temporally installed about 54 kilometers from the Yucheng site in 2005. The latitude and longitude are 36.46°N and 116.13°E. The underlying vegetation of this site is grass. The leaf area index is about 2.0. The average canopy height is about 20–40 cm. The area is 10 km2. Figure
The schematic positions of Yucheng site (⚫), Lawn site (■), and research range (□).
The two sites represent the main land covers of the studied area. The underlying surfaces of the two sites are relatively homogeneous. Therefore, the observation data of the ET from the two sites can represent the ET statuses of the two homogeneous areas around the two sites. The two areas correspond at least to nine grids of the MDRE and the EP of the Noah model at the spatial resolution of 1 km. Hence, it is sufficient to use these observations as the validating data.
In order to retrieve the MDRE, the OTLM proposed by Zhang et al. [
The OTLM has several advantages. First, the PCACA is based on vegetation fraction and ground temperature trapezoid relation theory and initiates a new method of decomposing mixed pixels. Second, it is very convenient because only single angle remote sensing data are required which can be obtained from most of the satellite sensors. Third, the core of the LESA is Bowen-ratio energy balance method, which reduces the uncertainties in surface energy partition based on the Beer law. Fourth, key nonremote sensing parameters that influence regional ET can also be obtained by using the OTLM. The OTLM has been successfully applied to Huabei area, China, [
Because the MODIS is easily influenced by the clouds, the data are not good enough to retrieve the MDRE every day. Figure
The available MDREs retrieved from the Aqua data at 14:30 local solar time in May, 2005. (a) The MDRE on May 2, 2005; (b) the MDRE on May 7, 2005; (c) the MDRE on May 14, 2005; (d) the MDRE on May 19, 2005; (e) the MDRE on May 25, 2005; (f) the MDRE on May 31, 2005.
On the contrary, the better the assimilated results are, the more there are the MDREs. Hence, it first requires to temporally extend the MDRE by the available MDREs and the EPs of the Noah model. The series of the extended MDRE is then used as the synthetic observations. In order to extend the MDRE, an intensity modulation method is employed because this method is very simple and can be easily implemented into the Noah model to perform high speed real-time MDRE extension for a long temporal span. This method can interpolate the absent MDRE with the fidelity to the evolvement trend of the Noah EP. This unavailable
In (
The Noah model is originated from a physically based land surface-vegetation-atmosphere-transfer scheme. During the past ten years, it underwent substantial upgrades, including modifications to the formulations of canopy conductance, bare soil evaporation, vegetation phenology, ground heat flux, and so forth. These model enhancements significantly improve its performance, and are physically more faithful to nature and thus most likely the route for more improvements in the future [
The Noah model was chosen for assimilation test for several reasons. It can simulate many states of the land surface, including soil temperature, skin temperature, and the energy and water fluxes of the land surface energy and water balances. In various coupled and uncoupled assessments, it has been proven to have the ability to reproduce the observed land surface energy, and water budgets effectively [
The Noah model contains four vertical soil layers: a thin 10 cm top layer, a second root zone layer of 30 cm, a deep root zone of 60 cm, and a subroot zone of 100 cm. It can be run for 13 vegetation covers (2 of which use the same parameter values) and nine different soil types (two of which also use the same parameters). It has 33 parameters: 10 related to vegetation and 23 that describe soil properties. It also has 16 initial states (when run with four root layers). The model uses a local greenness fraction from the normalized difference vegetation index (NDVI) to establish seasonality in the model for each of the 13 vegetation types.
In this assimilation experiment, the Noah model was configured to have dimensions of 250 × 250 at 1 km × 1 km spanning a domain bounded by 35.77°N to 38.26°N, 114.81°E to 117.3°E. Time step is 10800 seconds. Assimilation time span is between 1 January, 2005 and 31 May, 2005 after the Noah model is run from 1 June, 2004 because the MODIS data in this span are less affected by clouds, and it is beneficial for retrieving the MDRE. On this grid, the elevation was derived from the 1 km digital elevation of the GTOPO30 database [
The EPs of the Noah model at 15:00 solar time closest to the available MDREs in Figure
The data from the two sites are first used to evaluate the ET values of the available MDREs and the Noah model at the corresponding locations before DA. For validation, we utilize root-mean-square error (RMSE) as the estimation index. RMSE is defined as
In (
The RMSE results of the MDRE and the Noah EP using the two site observations.
Yucheng site | Lawn site | |||
---|---|---|---|---|
MDRE | EP | MDRE | EP | |
RMSE | 18.74 | 51.58 | 23.47 | 68.69 |
It can be found from Table
When performing the AWT-PEKF method, the SVLs including skin temperature, soil temperature and volumetric liquid soil moisture for the first soil layer, canopy water content, and the MDRE are perturbed fifty times according to their error Gaussian distributions, respectively. The four SVLs are perturbed because they are the key variables in the calculation of the ET in the Noah model. For comparisons, the EnKF, and PEKF methods are also performed. The assimilated EPs from the EnKF, PEKF, and proposed methods are, respectively, shown in Figures
The assimilated EPs from the EnKF method at 15:00 solar time. (a) The assimilated EP on May 2, 2005; (b) the assimilated EP on May 7, 2005; (c) the assimilated EP on May 14, 2005; (d) the assimilated EP on May 19, 2005; (e) the assimilated EP on May 25, 2005; (f) the assimilated EP on May 31, 2005.
The assimilated EPs from the PEKF method at 15:00 solar time. (a) The assimilated EP on May 2, 2005; (b) the assimilated EP on May 7, 2005; (c) the assimilated EP on May 14, 2005; (d) the assimilated EP on May 19, 2005; (e) the assimilated EP on May 25, 2005; (f) the assimilated EP on May 31, 2005.
The assimilated EPs from the AWT-PEKF method at 15:00 solar time. (a) The assimilated EP on May 2, 2005; (b) the assimilated EP on May 7, 2005; (c) the assimilated EP on May 14, 2005; (d) the assimilated EP on May 19, 2005; (e) the assimilated EP on May 25, 2005; (f) the assimilated EP on May 31, 2005.
In order to evaluate the three methods, the two site observations are employed to validate the DA results by the RMSE. Table
The RMSE results of the three methods using the two site observations.
EnKF | PEKF | AWT-PEKF | |
---|---|---|---|
Yucheng site | 28.75 | 22.54 | 16.81 |
Lawn site | 25.18 | 21.97 | 18.16 |
The RMSE histograms of the three methods using the two site observations.
The RMSE reveals the accurate degree of the EPs produced by each method. The lower the RMSE is, the better the assimilation effect is, and vice versa. It can be seen from Table
As seen from Tables
By combining the quantitative estimation results and the intercomparison, it can be seen that the AWT-PEKF method gives the EPs closer to the measured EPs than the EnKF and PEKF methods when the assimilated EPs are compared with the observations from the Yucheng and Lawn sites.
Though the AWT-PEKF method outperforms the EnKF and PEKF methods in the experiment, three points are needed to be studied further. The first is that the validating data are sparse. In validation, there are only two available field sites. Because it is difficult to get the ET observation, it needs to confirm whether or not the assimilated EPs obtained by the AWT-PEKF method are also close to the ET observation at other locations. The second is that it needs to test whether or not the AWT-PEKF method is also effective in assimilating other variables. Other variables, such as skin temperature and soil moisture, are forecasted differently from the ET and are also influenced by many factors. Hence, the extension of the AWT-PEKF method to other variables is a big job. The third is that the assimilation of the ET cannot influence the consecutive simulation of the Noah model.
As for the first point, the potential solution is to perform the AWT-PEKF method in the area where the field sites are available, meanwhile the MDREs can be retrieved from cloud-free MODIS data. As for the second point, the AWT-PEKF method can be easily extended to assimilate skin temperature and soil moisture products only if the two products retrieved from the MODIS data are prepared in advance. As for the third point, in the following study, we will introduce another novel assimilation method, in which the SVLs relative with the ET are simultaneously updated in order to transfer the assimilation effect into the consecutive simulation of the LSM. The assimilation idea will be presented in another research paper.
In this paper, we study the hybrid use of the AWT and PEKF methods for assimilating the MDRE into the EP of the Noah model in order to improve the consecutive simulation of the Noah model. The AWT is used to decompose the MDRE for injecting its detail information represented by wavelet planes, while the PEKF is used to complete the assimilation by the model and observation uncertainties. The AWT-PEKF method retains the respective advantages of the AWT and PEKF. Firstly, it is based on multigrid, and the interrelation between grids is considered using the wavelet filter during the filtering procedure. Secondly, according to the gain derived using the PEKF from the model and observation uncertainties, the details and textures of the MDRE are modulated into the EPs of the Noah model using the AWT from image fusion viewpoint.
The performance of the proposed method is compared with those of the EnKF and PEKF methods using one assimilation experiment. Intercomparison results of the RMSE confirm the effectiveness of the AWT-PEKF method in improving the spatial accuracy of the EPs. Overall evaluation shows that the AWT-PEKF method is promising and superior to the traditional EnKF and PEKF methods. Several issues are unresolved, such as the validation of the assimilated results, the effects of ensemble size, initial perturbation fields on assimilated results, and the actual performance of this new method in real other variable assimilations. These aspects require further investigation.
The authors thank the anonymous reviewers for their sincere suggestions which helped to improve the paper. This work is supported jointly by the Project of Natural Science Fund of China (41101329) and National Basic Research Program of China (2010CB950904 and 2010CB428403).