the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
Development and preliminary validation of a land surface image assimilation system based on the common land model
Abstract. Data assimilation is an essential approach to improve the predictions of land surface models. Due to the characteristics of singlecolumn models, assimilation of land surface information has mostly focused on improving the assimilation of singlepoint variables. However, land surface variables affect shortterm climate more through largescale anomalous forcing, so it is indispensable to pay attention to the accuracy of the anomalous spatial structure of land surface variables. In this study, a land surface image assimilation system capable of optimizing the spatial structure of the background field is constructed by introducing the curvelet analysis method and taking the similarity of image structure as a weak constraint. The ERA5_Land soil moisture reanalysis data are used as ideal observation for the preliminary effectiveness validation of the image assimilation system. The results show that the new image assimilation system is able to well absorb the spatial structure information of the observed data and has a remarkable ability to adjust the spatial structure of soil moisture in the land model. The spatial correlation coefficient between model surface soil moisture and observation has increased from 0.39 to about 0.67 after assimilation. By assimilating the surface soil moisture data and combining with the model physical processes, the image assimilation system can also gradually improve the spatial structure of deep soil moisture, with the spatial correlation coefficient between model soil moisture and observation increased from 0.35 to about 0.57. The forecast results show that the positive assimilation effect could be maintained for more than 30 days. The results of this study adequately demonstrate the application potential of image assimilation system in the shortterm climate prediction.

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The requested preprint has a corresponding peerreviewed final revised paper. You are encouraged to refer to the final revised version.

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The requested preprint has a corresponding peerreviewed final revised paper. You are encouraged to refer to the final revised version.
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RC1: 'Comment on egusphere20232473', Anonymous Referee #1, 13 Dec 2023
Summary:
Most current land surface assimilation systems are basically single point assimilation. Single point assimilation can easily break the coherent largescale spatial structures of soil moisture anomaly, which are usually the important land surface factors to influence shortterm climate variations over the land. The manuscript entitled “Development and preliminary validation of a land surface image assimilation system based on the common land model” propose an image assimilation method by using the curvelet transform to denoise the observational data with only the primary structural information to be assimilated. Preliminary results showed that this assimilation method can adjust the structures of model soil moisture based on the observed spatial structure characteristics, increasing the spatial similarity of soil moisture between the model and the observation. This image assimilation method shows potential for improving the forecast of shortterm climate variability related to soil moisture anomalies. However, benefit of the image assimilation is not well evaluated. That is, the paper should show more results regarding the advantages of the image assimilation over traditional single point assimilation. The paper is generally well built up. However, still this manuscript needs to be improved greatly, especially regarding the issues mentioned above. Efforts should be made to improve the readability. I think this paper can be considered for publication after some issues/questions are resolved/explained.
Major issues:
 There are many other image denoising techniques, why use curvelet for land surface images? Curvelets are anisotropic, they have a high directional sensitivity and are very efficient in representing vortex edges. Therefore, the curvelet transform is suitable for geophysical fluids. But what is the argument for choosing it for land surface?
 The argument for choosing the threshold of 0.5 for the curvelet denoising is not convincing enough. Probably different thresholds will lead to different assimilation results. If it is true. How should understand this?
 Don’t understand why there is no error covariance matrixes involved in the term J_1 in Equation (5).
Minor comments:
 No information of the used atmospheric forcing data.
 Figure 9: the correlations generally decline until the middle of July and then increase, how to understand this?
 line 6267: the sentences are ambiguous and hard to follow, please clarify to be concise and accurate.
 Section 3.1: It is indicated in line 230 that resolution of the soil moisture reanalysis data is 31 km, while line 235 states “increased to 9 km”.
 line 252: To demonstrate the benefit of the image assimilation and evaluate its advantages over traditional single point assimilation, if set J_O=0 in equation (5), authors should do one more set of experiments performing single point assimilation. Another option is to first do J(x) = J_B + J_O as conventional assimilation, and then do J(x) = J_B + J_O + J_I to see the benefit of the image assimilation.
Citation: https://doi.org/10.5194/egusphere20232473RC1 
AC2: 'Reply on RC1', Wangbin Shen, 30 Jan 2024
Summary:
Most current land surface assimilation systems are basically single point assimilation. Single point assimilation can easily break the coherent largescale spatial structures of soil moisture anomaly, which are usually the important land surface factors to influence shortterm climate variations over the land. The manuscript entitled “Development and preliminary validation of a land surface image assimilation system based on the common land model” propose an image assimilation method by using the curvelet transform to denoise the observational data with only the primary structural information to be assimilated. Preliminary results showed that this assimilation method can adjust the structures of model soil moisture based on the observed spatial structure characteristics, increasing the spatial similarity of soil moisture between the model and the observation. This image assimilation method shows potential for improving the forecast of shortterm climate variability related to soil moisture anomalies. However, benefit of the image assimilation is not well evaluated. That is, the paper should show more results regarding the advantages of the image assimilation over traditional single point assimilation. The paper is generally well built up. However, still this manuscript needs to be improved greatly, especially regarding the issues mentioned above. Efforts should be made to improve the readability. I think this paper can be considered for publication after some issues/questions are resolved/explained.
Response:
Thanks to your valuable comments and suggestions. Following your suggestions, we have revised the manuscript carefully from the beginning to the end. The pointbypoint response is listed below according to your specific comments.
Major issues:
 There are many other image denoising techniques, why use curvelet for land surface images? Curvelets are anisotropic, they have a high directional sensitivity and are very efficient in representing vortex edges. Therefore, the curvelet transform is suitable for geophysical fluids. But what is the argument for choosing it for land surface?
Response:
As the reviewer highlighted, the basis function of curvelet analysis exhibits anisotropic characteristics, thereby demonstrating its exceptional capability in accurately reproducing the rapidlyevolving properties of earth fluids. Although the soil moisture does not exhibit rapid temporal variations, there are many smallscale spatial structures of soil moisture due to the high spatial heterogeneity of soil. Therefore, the curvelet analysis method is selected as a more effective approach to capture the intricate local variations of soil moisture.
Indeed, a variety of mathematical image analysis techniques are available. For instance, the Fourier decomposition method and wavelet analysis method are commonly employed in meteorological research. However, the Fourier analysis method primarily focuses on the average feature of the sequence at different frequencies, and lacks the ability to accurately describe the regional variations. The Wavelet analysis could provide more detailed variation information in the timefrequency domain, but its basis functions with isotropic characteristic limit the ability to accurately represent the characteristics of smallscale spatial variations.
On the other hand, the curvelet analysis method has been selected to fulfill the requirements of variational data assimilation. The curvelet transform is an observation operator in an image assimilation system. During the process of minimizing the cost function in variational data assimilation, the adjoint function of the observation operator becomes necessary. The adjoint function of curvelet analysis is just its inverse transformation, which proves to be a highly advantageous property for minimizing the cost function in a variational data assimilation system.
 The argument for choosing the threshold of 0.5 for the curvelet denoising is not convincing enough. Probably different thresholds will lead to different assimilation results. If it is true. How should understand this?
Response:
Thanks for your valuable suggestions. Just as the reviewer pointed out, the image assimilation system determines the spatial structural characteristics of assimilation according to the threshold values, and different threshold values could result in certain variations in the spatial structure of assimilation.
Naturally, a higher threshold can capture more spatial structural features of the observed variables, but the presence of observation errors imposes limitations on its continuous increase. More discussions have been added to the revised manuscript in Line 366404 to prove that a threshold of 0.5 can not only capture the spatial structure information of observation data, but also mitigate the impact of observational errors. The specific content comprises the following three aspects:
(1) The definition of the threshold σ has been further elaborated in order to provide a clearer rationale for its selection. This detailed description has been incorporated into line 370371 of the revised manuscript. The specific wording is as follows:
The threshold σ means the modulus of the decomposition coefficient falls within the first 100*σ% percentile. For instance, a value of 0.5 indicates that the mode retaining the top 50% of decomposition coefficient.
(2) By employing the spatial correlation method, we demonstrate that a threshold of 0.5 adequately captures the primary spatial information derived from soil moisture observations, the following discussions have been added to Line 366404 of the revised manuscript:
The image assimilation system finds the spatial structural characteristics of assimilation according to the threshold values, and different thresholds could result in certain variations in assimilated spatial structure. In order to clarify the spatial structure differences corresponding to different thresholds, the spatial correlation method (Daley, 1991) is employed in this study to elucidate the distinctive characteristics of spatial structure corresponding to varying thresholds.
The hourly soil moisture data from ERA5Land from May 1 to 30, 2016 is selected for analysis. The threshold σ means the modulus of the decomposition coefficient falls within the first 100*σ% percentile. For instance, a value of 0.5 indicates that the mode retaining the top 50% of decomposition coefficient. The original image can be reconstructed by selecting different threshold ranges, namely (0,0.01], (0.01,0.03], (0.03,0.05], (0.05,0.1], (0.1,0.2], (0.2,0.3], (0.3,0.4], (0.4,0.5], (0.5,0.6], (0.6,0.7], (0.7,0.8], (0.8,0.9] and (0.9,1.0]. The correlation coefficient between each grid point and its neighboring grid points can be obtained based on the reconstructed time series of each grid point. The spatial structural characteristics of different scales in the reconstructed images could be quantitatively expressed by the average correlation coefficients corresponding to different grid point distances.
The mean correlation coefficient corresponding to grid point distance is illustrated in Figure 4. As can be seen, the variation characteristics of the intergrid correlation coefficient of the original soil moisture represented by the black line with respect to the grid distance. The average correlation coefficient can exceed 0.5 within a radius of 200 km, while maintaining above 0.4 within a radius of 300 km. The distance corresponding to high correlation coefficients represents the characteristics of consistent changes in soil moisture within a similar range, that is, soil moisture has the characteristics of spatial structure at the corresponding scale.
When the threshold value is 0.01, the average correlation curve exhibits a similar change in correlation coefficient of the original variable, thereby indicating that the curvelet coefficient corresponding to this threshold value effectively reproduces the largescale spatial structure. The spatial structure scale represented by the corresponding curvelet transformation reconstruction results decrease as the threshold value increases, leading to a rapid decrease in the correlation coefficient with increasing distance. The curvelet reconstruction results with different threshold intervals represent the structural characteristics of different horizontal scales, while the cumulative threshold can well represent the spatial structural characteristics of soil moisture variables represented by the selected threshold in the assimilation. The average correlation coefficient of the cumulative threshold is depicted in Figure 4b. As can be seen, the top 10% of curvelet coefficients can effectively replicate the spatial correlation characteristics of soil moisture variables. The results also indicate that the variations in threshold values have minimal impact on the assimilated spatial structure when the threshold value exceeds 0.1.
Figure 4: Variation curves of the average correlation coefficient between grid points with the distance in the reconstructed ERA5Land hourly soil moisture image of the study area from May 1 to 30, 2016, which is reconstructed based on the curvelet coefficients of (a) different threshold intervals and (b) cumulative thresholds.
(3) According to the stochastic characteristics of observation errors, we conducted an analysis on the probability distribution properties of the reconstructed residuals and found that a threshold value of 0.5 effectively mitigates the impact of observation errors.
Naturally, a higher threshold can effectively capture more spatial structural features of the observed variables, but the presence of observation errors imposes limitations on its continuous increase. The observational error is typically characterized by stochastic fluctuations. When the discrepancy between the reconstructed results and the original variables exhibits random variation characteristics, it can be inferred that the observation information eliminated by the threshold method primarily consists of observation errors.
To better clarify the statistical characteristics of the reconstruction errors under different thresholds, Figure 5 shows the probability density distribution curves of the reconstruction errors for 100 reconstructed fields at different thresholds. For the error at the threshold of 0.5, the skewness coefficient of the probability density distribution curve is 0.00 and the kurtosis coefficient is 0.38, indicating the curve is close to the standard normal distribution curve (the skewness and kurtosis coefficients are all 0). With the gradual increase of threshold value, although the reconstruction error decreases, the residual error is mainly concentrated in the range of smaller values, and the curve shows a "sharp peak" distribution. Considering that the observation errors are mostly random errors, it is reasonable to believe that the reconstruction errors at the threshold of 0.5 are mainly observation errors, which also implies this threshold is good for the purpose of denoising the observation images.