The GLDAS, developed by the NASA Goddard Earth Sciences Data and Information Services Center (GES DISC), integrates multi-source observational data with LSMs to generate global land surface variable products (Rodell et al., 2004). GLDAS consists of three major versions: GLDAS-2.0, GLDAS-2.1, and GLDAS-2.2. It provides s simulations from several LSMs, including Noah, CLM, VIC, Mosaic, and Catchment.

Among them, GLDAS-2.1 is driven by a combination of meteorological forcing datasets, including atmospheric analysis fields from the NOAA Global Data Assimilation System (GDAS), daily precipitation data from the Global Precipitation Climatology Project (GPCP), and radiation data from the Air Force Weather Agency's (AFWA) Agricultural Meteorology modeling system (AGRMET). This version spans from 2000 to the present and is available at spatial resolutions of 1° × 1°, and 0.25° × 0.25°, with a temporal resolution of 3 h. The Noah model in GLDAS-2.1 provides four-layer soil moisture simulations corresponding to depth ranges of 0–10, 10–40, 40–100, and 100–200 cm.

In this study, we use the top three soil moisture layers from the Noah model in GLDAS-2.1, covering the period from June to August 2022. Since the spatial resolution of the CoLM used in this study is 1.4°, the GLDAS data are resampled to the model resolution using bilinear interpolation to ensure spatial consistency between datasets.

It should be noted that resampling the 0.25° GLDAS data onto the lower-resolution 1.4° model grid using bilinear interpolation may introduce a smoothing effect. In regions with highly heterogeneous land surface conditions, this interpolation may weaken local soil moisture extremes and blur small-scale spatial gradients. However, this compromise is mainly constrained by the spatial resolution of the current land surface model. To ensure consistency in spatial scale between the observation field and the model background field, resampling is a necessary preprocessing step. In future applications with higher-resolution land surface models, the impact of this smoothing effect is expected to be substantially reduced.

This study uses high-quality in situ soil moisture observations from two sources. The first is the automatic soil moisture observation network maintained by the China Meteorological Administration (CMA), and the second is the International Soil Moisture Network (ISMN).

The CMA's observation network was initiated in 2009 and has been gradually put into operational use since 2011, forming a nationwide system for routine soil moisture monitoring. The network employs Frequency Domain Reflectometry (FDR) technology and deploys high-precision sensors at standardized depth intervals (0–10, 10–20, 20–30, 30–40, 40–50, 50–60, 70–80, and 90–100 cm) to measure volumetric soil water content at high temporal resolution (Wang et al., 2018). Observations are recorded hourly and reported as 10 min averages preceding each hour. This dataset offers advantages in both temporal resolution and spatial coverage. In this study, we use 10 cm depth soil moisture observations from 2878 stations for the period June to August 2022.

ISMN initiated by the Vienna University of Technology, is the world's largest open-access in situ soil moisture database. It supports the validation of LSMs and the calibration of satellite soil moisture products (Dorigo et al., 2021). ISMN integrates soil moisture data from 71 independently operated networks across 58 countries, comprising over 2800 stations, with records dating back to 1952. All data are standardized and provided in hourly volumetric soil water content. In this study, we select 10 cm soil moisture observations from the ISMN network located in the United States for the period from June to August 2022, including 148 stations used for validation.

This study uses a high-precision station–satellite merged precipitation analysis product developed by Shen et al. (Shen et al., 2014) as the precipitation observational dataset. The product is generated using the Probability Density Function–Optimal Interpolation (PDF-OI) algorithm, which merges hourly precipitation observations from over 30 000 automatic weather stations operated by the CMA with satellite-based precipitation estimates from the Climate Prediction Center Morphing technique (CMORPH) developed by NOAA's Climate Prediction Center. The resulting gridded dataset has a temporal resolution of one hour and provides high-quality precipitation fields suitable for LSM applications.

CoLM was developed by Dai et al. (2003) based on the LSM from the National Center for Atmospheric Research (NCAR LSM) (Bonan et al., 2002), the Biosphere–Atmosphere Transfer Scheme (BATS) (Dickinson et al., 1993), and the IAP94 model from the Institute of Atmospheric Physics, Chinese Academy of Sciences (Dai and Zeng, 2007). The model is designed to simulate the exchange of energy, carbon, and water between the land surface and the atmosphere. It comprehensively incorporates biophysical, biogeochemical, ecological, and hydrological processes, enabling realistic simulation of soil temperature, soil moisture, surface heat fluxes, and other land surface variables.

At present, two major versions of CoLM have been released, CoLM2005 and CoLM2014. Compared to CoLM2005, CoLM2014 introduces several important updates across multiple modules. In runoff parameterization, CoLM2014 replaces the original BATS-based scheme with the SIMTOP runoff model derived from TOPMODEL (Niu et al., 2005). It also implements a new-generation multi-layer lake model, replacing the simpler two-layer scheme used in CoLM2005 (Dai et al., 2018).

In this study, the offline mode of CoLM2014 is employed. The model is run at a horizontal resolution of 1.4° × 1.4°, with 10 soil layers and up to 5 snow layers in the vertical dimension. Atmospheric forcing variables required by the model include downward shortwave and longwave radiation, surface air temperature, specific humidity, near-surface wind speed, surface pressure, and precipitation rate. These inputs are obtained from the near-surface reanalysis dataset provided by the European Centre for Medium-Range Weather Forecasts (ECMWF), covering the period from 1979 to 2022. Before conducting the experiments, the model is driven cyclically for 360 years using the forcing data to bring the system to equilibrium.

The curvelet transform, proposed by Candès and Donoho (2000), is a signal and image processing technique designed for multi-scale geometric analysis. It was developed to overcome the limitations of wavelet transforms in representing geometric features such as edges and curves in two-dimensional or higher-dimensional signals. By decomposing two-dimensional data into basis functions at multiple scales and orientations, the curvelet transform efficiently identifies and extracts curvilinear structures and localized variations. At scale j, the curvelet basis function φj is constructed using a radial window function W and an angular window function V, and is defined as follows: 1φj(ω)=2-3j4W(2-j|ω|)V2j2θω2π

The normalization factor 2-3j/4 ensures that the basis functions are properly scaled in the L2 space. The radial window function W facilitates multi-scale decomposition of the signal, while the angular window function V determines directional selectivity and resolution. The scale parameter j defines the effective support of the curvelet basis functions in the frequency domain, with an approximate frequency width of 2j and frequency length of 2j/2. Directional basis functions are then generated through rotation operations, and spatial localization is achieved via translation, ultimately forming the complete curvelet frame: 2φj,k,l(x)=φjRθj,lx-bk(j,l) where φj is the mother curvelet, j denotes the scale, k represents the translation parameter, l denotes the orientation, and x is the coordinate of a sample grid point in two-dimensional space. Rθj,l is the rotation matrix with angle θj,l, and bk(j,l) is the translation parameter. For a discrete two-dimensional signal f(x), the curvelet transform represents it as a linear combination of curvelet functions at different scales, orientations, and positions. The curvelet coefficients are calculated as follows: 3cj,k,l=∑x∈Ωf(x)φj,k,l(x) where Ω denotes the set of grid points in two-dimensional space, φj,k,l(x) is the curvelet function, and cj,k,l is the curvelet coefficient corresponding to scale j, position k, and orientation l. The coefficients obtained from the curvelet transform can be reconstructed back to the original space through the inverse transform, which is expressed as follows: 4f(x)=∑j=1J∑l=1Lj∑k=1Kj,lcj,k,lφj,k,l(x) where J is the total number of scales, Lj denotes the number of orientations at scale j, and Kj,l denotes the number of positions at scale j and orientation l. Equation (4) shows that the original field can be reconstructed by summing the curvelet coefficients over all scales, orientations, and positions.

Similar to the PCA method commonly used in meteorological studies, curvelet analysis can also decompose the variable into several components with different spatial characteristics. PCA usually extracts leading modes according to the magnitude of variance contribution in time series, and these modes reflect the dominant spatial structures of the variable. In contrast, curvelet analysis uses predefined basis functions at different scales and orientations and projects the variable field at a given time onto these basis functions, thereby obtaining components with different scales and orientations. A single mode in curvelet decomposition can be interpreted as a spatial fluctuation feature within a certain orientation and scale range, and can therefore represent soil moisture variations at a specific spatial scale. In general, lower-order modes correspond to low-frequency signals and mainly reflect large-scale spatial distribution features, whereas higher-order modes correspond to higher-frequency components and describe smaller-scale spatial details. As subsequent modes are gradually introduced, curvelet analysis can resolve increasingly finer spatial structures.

The curvelet-derived spatial features are grouped according to the meteorological scale convention, from planetary-scale to large-scale and mesoscale features. These categories correspond, respectively to broad background patterns, regional structures, and finer variations associated with local heterogeneity. Figure 1 presents the spatial structure of global soil moisture fields reconstructed using curvelet transform modes at different scales. Figure 1a shows the original soil moisture distribution simulated by CoLM, exhibiting a clear latitudinal gradient in which soil moisture decreases from low to high latitudes over the Americas and Eurasia. When the reconstruction is performed using only the first curvelet mode (Fig. 1b), the image retains only the planetary-scale spatial features, capturing broad global patterns such as humid zones in the high latitudes of the Northern Hemisphere and arid regions in the tropics. However, much of the regional detail is lost. For example, over the Sahara Desert, the reconstruction yields only a single intensely arid center, with the influence of dryness gradually diminishing outward toward the desert margins. When the number of reconstruction modes increases to two (Fig. 1c), additional spatial features across multiple scales begin to emerge, and the internal structures of some large-scale systems become more distinctly defined. Taking the Sahara Desert again as an example, the previously unified arid center evolves into three distinct northeast–southwest-oriented moisture gradient zones, allowing for a more accurate delineation of the core arid region and its transitional boundaries. When three modes are included (Fig. 1d), the reconstructed image captures most of the key spatial structures from the original field, including the complex patterns over Central Asia, land–ocean contrasts in Australia, and the detailed structure of extreme aridity within the Sahara. Using the first four curvelet modes (Fig. 1e), the reconstructed field closely resembles the original image, not only accurately recovering large-scale patterns but also preserving mesoscale regional features. For example, the pronounced spatial heterogeneity over South America and the monsoon-influenced soil moisture gradients over East Asia are well represented. These results demonstrate that the curvelet transform effectively decomposes complex spatial fields into scale-separated structural components, providing a basis for incorporating observational spatial structure into data assimilation frameworks.

In contrast to traditional assimilation approaches that concentrate on modifying variable magnitudes, image assimilation techniques focus on recalibrating spatial characteristics across multiple scales. The core difficulty resides in disentangling variables into clearly defined spatial structural attributes. Because variable magnitudes fluctuate with meteorological systems and temporal progression, achieving objective separation of structural features relying exclusively on magnitude values presents considerable challenges. Yet when variables are transformed from observation space to spectral space, areas of structural variation map directly onto domains with elevated spectral coefficients. Accordingly, this investigation implements curvelet analysis utilizing anisotropic basis functions to transition variable fields into spectral space, wherein spectral coefficients at distinct frequencies correspond to structural constituents at separate scales. Land surface assimilation thereby becomes directly redefinable as spectral coefficient assimilation. A significant advantage of the curvelet transformation stems from its capacity for precise inverse representation through analytical expressions, securing complete reversibility during data conversion between observational and spectral domains. Following assimilation procedures in spectral space, the synthesized results can therefore be reconstructed into the original observation space while maintaining informational completeness. The orthogonal nature of the curvelet transform across scales further permits autonomous handling of individual spatial modes, successfully eliminating cross-scale interference among fluctuating components.

To distinguish the proposed framework from the conventional EnKF, which updates the full ensemble state in physical model space, we refer to it as an “EnKF-like” assimilation framework. Unlike the EnKF, this method does not estimate state-variable error covariances directly in physical space. Instead, the forecast ensemble is mapped into curvelet spectral space, where the error covariance of the spectral coefficients is dynamically estimated from the ensemble samples. Taking advantage of the approximately independent properties of the curvelet basis functions, the analysis update is then performed in spectral space using the standard Kalman filter equation. For computational simplicity, the framework does not include a full ensemble update after the analysis step; therefore, it is termed “EnKF-like”.

This study employs the EnKF method to improve assimilation of different spectral coefficients in spectral space. The EnKF estimation formula for the analysis field Xa is: 5Xa=Xf+K(y-H(Xf)) where Xf represents the model background field, and H is the observation operator matrix that maps the model state to observation space. The Kalman gain matrix K is obtained through: 6K=BHT(HBHT+R)-1 where B represents the background error covariance matrix of model states, and R is the observation error covariance matrix.

In practical assimilation applications, error estimation of spectral coefficients for both observations and background fields are the most critical component. Assimilation in spectral space requires estimating errors of spectral coefficients corresponding to different spatial modes. Using the ensemble-based error estimation approach of EnKF and the accurate decomposition capability of the curvelet transform, we can directly transform variable errors from observation space to spectral coefficient errors.

Leveraging the mathematical exactness of curvelet analysis, we can transform not only the structure of variables in physical space into spectral coefficients, but also the corresponding errors into coefficients in spectral space. In this EnKF-like assimilation framework, following the error estimation strategy of the EnKF, we first estimate the background error of soil moisture at each grid point in physical space, random perturbations with zero mean and a standard deviation corresponding to the physical-space error are added to the background soil moisture field, thereby generating an ensemble of perturbed members. Let the ith ensemble member in physical space be denoted as xi (i=1,2,…,N), where N is the ensemble size. In this study, the ensemble size is set to N=50. By applying the curvelet transform operator C, these members are mapped one by one into spectral space, yielding the corresponding curvelet coefficients with errors: 7ci=C(xi)

The background error covariance matrix B in spectral space is dynamically estimated directly from the ensemble samples in spectral space: 8B=1N-1∑i=1Nci-c‾ci-c‾T where c‾ denotes the ensemble mean of the curvelet coefficients, and the superscript T indicates matrix transpose. The observation error covariance matrix R is transformed in a similar manner. The standard deviations of the random perturbations are prescribed as 0.15 m3m-3 for the background soil moisture field and 0.1 m3m-3 for the observation field. In fact, owing to the orthogonality of different basis functions, the background error covariance B and observation error covariance R are diagonal in the spectral space. Thus, the unrealistic divergence of error impacts can be avoided, making the error localization and inflation procedure unnecessary.

After B and R are constructed, the assimilation system performs the EnKF analysis step in spectral space to compute the updated curvelet coefficients ca. Finally, the inverse curvelet transform operator C-1 is applied to map the updated curvelet coefficients back to physical space, thereby obtaining the final analyzed field: 9xa=C-1(ca)

To provide a clearer illustration of the overall architecture of the curvelet-transform-based EnKF-like image assimilation system, we present a schematic flowchart in Fig. 2. As shown in Fig. 2, one assimilation cycle of the framework consists mainly of four key steps. First, in the physical space, the initial ensemble is generated by adding random perturbations to both the background field and the observation field. Second, the forward curvelet transform operator is applied to map the ensemble members from physical space to spectral space. Third, in spectral space, the background and observation error covariances are dynamically estimated from the ensemble samples, and the Kalman filter update is performed to obtain the analysis field of curvelet coefficients. Finally, the inverse curvelet transform is applied to convert the updated curvelet coefficients back to the physical space, thereby optimizing the spatial structure of the model state variables as a whole.

Figure 3 shows the estimated background error for the first mode based on curvelet transform. Figure 3a displays the spatial distribution of the global soil moisture background field, consistent with Fig. 1a, serving as the reference for subsequent error analysis. Figure 3b shows the difference between the original background field and the reconstructed field obtained through first-mode curvelet inverse transformation after adding error perturbations to the background field. The difference field exhibits distinct regional patterns. In transition zones with steep soil moisture gradients, such as western Sahara and northern Arabian Peninsula, Fig. 3b shows notably large errors. Eastern China, characterized by plains with relatively uniform soil moisture, displays smaller errors in Fig. 3b. This spatial pattern reflects the error structures are closely related to background field characteristics. This demonstrates that the curvelet transform, through its multiscale decomposition capability, enables background errors to preserve structural features during the transformation from physical space to spectral space.

This study focuses on global simulation applications using CoLM with a spatial resolution of 1.4° × 1.4°. Regions with strong land-atmosphere coupling include the Amazon Basin, Sahel transition zone, Tibetan Plateau, and North American Great Plains (Barton et al., 2025; Koster et al., 2004; Seneviratne et al., 2010), where soil moisture variations serve as important signals for climate prediction. Among these regions, arid and semi-arid areas exhibit complex terrain and strong spatial heterogeneity in soil moisture, with surface energy and water budgets playing crucial feedback roles in the global climate system.

The assimilation period runs from 2 June to 1 August 2022, followed by a simulation period from 1 to 31 August 2022. Two experiments are designed. The first experiment employs the image assimilation method for soil moisture data assimilation (DA), conducting assimilation four times daily at 6 h intervals (00:00, 06:00, 12:00, and 18:00 UTC), assimilating only the GLDAS 0–10 cm surface soil moisture data. The second experiment serves as a control run (CTL) without data assimilation, running continuously from 2 June to 31 August 2022.

Regarding data preprocessing, considering that GLDAS soil moisture products have undergone rigorous quality control procedures, this study does not apply additional quality control to GLDAS data. For bias correction, we choose not to perform traditional bias correction because the primary purpose of image assimilation is to improve spatial structure characteristics of the background field. Bias correction processes statistically adjust the numerical distribution of observation fields, which can unavoidably affect or even distort the authentic spatial structure information contained in the observations (Shen et al., 2024; Wang and Tian, 2024; Zhou et al., 2020).

The study first verifies the assimilation experiments to assess the effectiveness of the soil moisture image assimilation system. Figure 4 presents the spatial characteristics of the background and analysis fields for surface soil moisture at 06:00 UTC on 2 June 2022. As shown in Fig. 4a, tropical rainforest regions that include the Amazon Basin, the Congo Basin, and Southeast Asia generally exhibit values above 0.36 m3m-3. Arid and semi-arid regions that comprise the Sahara Desert, the Arabian Peninsula, central Australia, and the Atacama Desert show values below 0.12 m3m-3. Midlatitude temperate zones fall between 0.20 and 0.28 m3m-3. This arrangement accords with global patterns of precipitation and evapotranspiration. The analysis field in Fig. 4b preserves the background structures across most areas, yet within continental interiors it introduces clear adjustments at multiple scales. The analysis increment in Fig. 4c indicates that the improvements from image assimilation arise chiefly in two types of regions. One consists of extensive plains such as the North American Great Plains, the East European Plain, and the outer margins of the West Siberian Plain. The other includes areas near major high terrain, for example the eastern slopes of the Rocky Mountains, the western flanks of the Andes, and the northern edge of the Tibetan Plateau. Because terrain effects amplify structural model errors in these transition belts, sizable assimilation increments emerge. More specifically, positive increments are concentrated over the central North American plains and the temperate grasslands of Eurasia, whereas negative increments appear in eastern Canada and along the northern boundary of the Brazilian Highlands.

To examine the fine-scale features of the image assimilation effect, Fig. 5 presents soil moisture before and after assimilation for two representative regions, the United States and China, together with GLDAS fields at the same time for comparison. As shown in the left column of Fig. 5, over China the background field indicates extensive aridity across the northwest under strong topographic influence. This dry zone connects directly to the Central Asian arid belt and lacks a clear transitional humidity gradient. Meanwhile, wetter conditions in the background concentrate near the boundary between Guangdong and Guangxi. The GLDAS data likewise display a pronounced soil moisture transition zone along the northwestern border of China and high soil moisture over the middle and lower reaches of the Yangtze River, consistent with Meiyu-season rainfall. After image assimilation, the analysis field reproduces both the northwestern transition belt and the humid Yangtze region well, and its spatial pattern agrees closely with GLDAS. In terms of magnitude, the analysis lifts the background's underestimated soil moisture in the northwestern transition zone from about 0.12 to roughly 0.20 m3m-3, close to GLDAS, and it increases the Yangtze Basin maximum from 0.32 m3m-3 in the background to nearly 0.40 m3m-3, approaching the GLDAS value.

In the United States region, the background field indicates widespread dryness across the western domain, particularly along the Cordillera mountain ranges, and relatively higher soil moisture levels appear in the central and eastern plains. The GLDAS dataset presents more detailed spatial patterns. In the west, soil moisture gradually decreases from north to south. In the east, it decreases outward from the central moist zone of the Great Plains. The analysis field reproduces these spatial features, including the north-to-south drying gradient in the west and the concentrated wet zone in the east. The spatial agreement between the analysis field and the GLDAS reference is generally consistent. In terms of magnitude, the assimilation system adjusts the underestimated soil moisture in both the northern mountainous regions in the west, where values increase from approximately 0.12 to about 0.20 m3m-3, and in parts of the eastern plains, where values increase from around 0.24 m3m-3 to nearly 0.32 m3m-3. These adjustments reduce the discrepancies between the analysis and the GLDAS reference. The results suggest that the image-based assimilation system based on the curvelet transform improves the spatial representation of soil moisture. The analysis field better reflects multiscale spatial patterns that are consistent with independent observations.