Adaptive localization method based on ensemble paleoclimate data assimilation framework

Abstract

This invention discloses an adaptive localization method based on an ensemble paleoclimate data assimilation framework. It constructs an observation density field and updates the localization radius of model grid points. If a proxy record is located within its localization radius, its weight is calculated using a Gaspari-Cohn fifth-order polynomial. If the proxy record is located outside its localization radius, the correlation information between the climate model grid points and the proxy record is calculated. If the correlation information meets a correlation threshold, the weight of the proxy record is calculated using the correlation information; otherwise, the weight of the proxy record outside the localization radius is calculated using a Gaspari-Cohn fifth-order polynomial. Combining the statistical correlation information within the reconstruction period, a final mixed weight matrix is ​​obtained, which is then used to adjust the covariance matrix. While ensuring that each model grid point retains a certain amount of observational influence, statistical correlation is used to reduce spurious teleconnections and improve reconstruction results.

Description

Technical Field

[0001] This invention relates to an adaptive localization method based on an ensemble paleoclimate data assimilation framework, belonging to the field of data assimilation technology. Background Technology

[0002] Reconstructing paleoclimate states is crucial for understanding the mechanisms and influences of Earth system evolution. Currently, this field primarily relies on two methods: paleoclimate proxy records and Earth system model simulations. However, both methods have significant limitations in reconstructing past climate states. On the one hand, proxy records (such as tree rings, ice cores, stalagmites, and corals) play an indispensable role in reconstructing climates over the past thousands of years due to their long time spans, but they often face problems such as uneven spatiotemporal distribution, discontinuity, and sparseness. Furthermore, climate signals are easily contaminated by noise during preservation, making it extremely challenging to establish a quantitative relationship between proxy records and actual climate states, and reconstructions of the same climate state from different proxy sources often show inconsistencies. On the other hand, climate models based on physical processes can provide physically consistent, globally complete, and spatiotemporally comprehensive climate fields, effectively capturing large-scale characteristics of the climate system. However, due to the inaccuracy of physical process parameterization and the uncertainty of responses to external forcing, model simulations may exhibit systematic biases, making it difficult to fully reproduce observed climate variability, resulting in limited predictive capabilities. Therefore, to overcome the limitations of single methods, paleoclimate data assimilation (PDA) techniques have emerged as a powerful tool for optimally fusing information from proxy records (sparse, noisy, and indirect indicators of past climate) with the dynamic constraints of climate models. Through this method, PDA produces spatially complete and physically consistent climate field estimates, similar to reanalysis products from the instrumental era, while also quantifying the uncertainties of reconstruction. Among various methods, ensemble Kalman filtering (EnKF) is frequently used in PDA due to its ease of implementation and ability to handle high-dimensional nonlinear systems.

[0003] Despite the immense potential of PDAs, they face unique challenges distinct from modern meteorological data assimilation, primarily due to the considerable temporal and spatial sparseness of proxy records, which typically represent time-averaged climate signals (e.g., annual averages) rather than instantaneous observations. Furthermore, proxy data are susceptible to complex errors such as measurement errors, decadal uncertainty, and imperfect relationships between proxy signals and target climate variables. In PDAs, observational error variance is often empirically specified based on the residual variance of a proxy system model calibrated from instrumented data. However, this approach often underestimates true proxy uncertainty because it fails to account for calibration errors (e.g., non-stationarity between past and instrumented periods), structural errors (originating from missing physical, biological, or chemical processes in the PSM), and representativeness errors (due to mismatches between point-scale proxies and gridded state variables). A key factor in the success of EnKF is the use of covariance localization. Early implementations often employed a fixed localization radius. Within this framework, a common approach is to process the sample covariance using a smoothed, distance-dependent function. A standard choice for this function is the Gaussian-like Gaspari-Cohn function. However, as numerous studies have shown, Gaussian-like decay functions are not necessarily optimal. To address the limitations of fixed localization, a series of adaptive localization methods have been developed. These methods typically adjust the localization based on state-related correlation patterns or ensemble estimation errors. Examples include techniques developed by Anderson, Bishop, Hodyss, and Menetrier et al. However, these methods have not yet been applied to the surrogate record assimilation context. In PDAs, due to the sparse distribution of surrogate records, a large covariance localization radius is typically used. While necessary, an excessively large localization radius can introduce spurious long-range correlations, ultimately degrading the quality of the analysis. This limitation suggests that fixed-radius localization is not optimal for PDA applications. Conversely, adaptive localization is more advantageous because it can optimize the influence radius of surrogate records based on the underlying spatial correlation structure, applying wider scales in areas of strong large-scale covariance and stricter constraints in data-rich or locally forcibly driven regions. Currently, research on adaptive localization techniques in PDAs remains very limited. Summary of the Invention

[0004] The technical problem to be solved by this invention is to provide an adaptive localization method based on a ensemble paleoclimate data assimilation framework, construct an observation density field, update the localization radius of model grid points, and combine statistical correlation information within the reconstruction time period to obtain the final mixed weight matrix; while ensuring that each model grid point retains a certain amount of observation influence, the invention utilizes statistical correlation to reduce spurious teleconnections and improve reconstruction results.

[0005] To solve the above-mentioned technical problems, the present invention adopts the following technical solution:

[0006] The adaptive localization method based on the ensemble paleoclimate data assimilation framework includes the following steps:

[0007] Step 1: For the study area, obtain proxy record data for the data assimilation period, divide the data assimilation period into several assimilation windows, and for each assimilation window, obtain the observation density at each climate model grid point under the assimilation window through the Gaussian kernel probability density function, and perform normalization processing to obtain the relative observation density at each climate model grid point.

[0008] Step 2: Set the upper and lower limits of the localization radius, and convert the relative observation density at each climate model grid point into the localization radius of each climate model grid point.

[0009] Step 3: For each climate model grid point, if the proxy record is located within its localization radius, calculate the weight of the proxy record located within the localization radius according to the Gaspari-Cohn fifth-order polynomial.

[0010] Step 4: If the proxy record is located outside its localization radius, calculate the correlation information between the climate model grid and the proxy record, and determine whether the correlation information meets the correlation threshold. If it does, use the correlation information to calculate the weight of the proxy record located outside its localization radius; otherwise, use the Gaspari-Cohn fifth-order polynomial to calculate the weight of the proxy record located outside its localization radius.

[0011] Step 5: Integrate the weights of the proxy records inside and outside the localized radius of the climate model grid to obtain a mixed weight matrix. Use the mixed weight matrix to adjust the covariance matrix in the ensemble Kalman filter method to achieve the final data assimilation.

[0012] Compared with the prior art, the present invention, employing the above technical solution, has the following technical effects:

[0013] 1. This invention combines the classical fixed localization covariance method and statistical correlation in paleoclimate data assimilation methods, which improves the efficiency of using sparse proxy data, reduces the influence of spurious teleconnections, and improves the accuracy of paleoclimate data assimilation methods. At the same time, it maintains the handling of large localization radii in classical localization methods, allowing integration with advanced classical algorithms.

[0014] 2. In the part of obtaining the observation density, this invention uses the Gaussian kernel probability density method and the Scott rule, that is, the bandwidth is determined by the number of observations, to further frame the grid points of the pattern with high observation density to the observations at close range, thereby reducing the influence of spurious correlations.

[0015] 3. In the acquisition of the observation weights outside the localized radius of the model, the present invention uses statistical correlation information. This information only needs to be calculated once during the reconstruction period (or calculated multiple times in different periods) and stored. It requires fewer resources, and compared with the physical correlation of spatial distance, the statistical correlation is more accurate and convincing, while improving the utilization of sparse observations. Attached Figure Description

[0016] Figure 1 This is a flowchart of the adaptive localization method based on the ensemble paleoclimate data assimilation framework of the present invention;

[0017] Figure 2 The diagram shows the distribution of observation density during each assimilation period. (a)-(d) represent the time averages of observation density during 1880, 1940, 1990, and the reconstruction period, respectively.

[0018] Figure 3 The comparison of test results between adaptive localization experiment (E_AL1) and fixed localization radius experiment (E_20000) based on reanalysis data is shown in the figure. (a) is the root mean square error, (b) is the efficiency coefficient, (c) is the EOF mode analysis of temperature reanalysis data, (d) is the EOF mode analysis of temperature reconstruction of the fixed localization radius experiment, and (e) is the EOF mode analysis of temperature reconstruction of the adaptive localization experiment based on reanalysis data.

[0019] Figure 4 The results are the RMSE comparison of the adaptive localization experiment (E_AL2) based on model simulation data, where (a) is E_AL2 and the fixed localization radius experiment (E_20000), and (b) is E_AL2 and the adaptive localization experiment based on reanalysis data (E_AL1). Detailed Implementation

[0020] Embodiments of the present invention are described in detail below, examples of which are illustrated in the accompanying drawings. The embodiments described below with reference to the accompanying drawings are exemplary and are only used to explain the present invention, and should not be construed as limiting the present invention.

[0021] This invention proposes an adaptive localization method based on an ensemble paleoclimate data assimilation framework, the specific process of which is as follows: Figure 1 As shown, it includes the following steps:

[0022] Step 1: Obtain proxy record data and obtain a continuous observation density field using the Gaussian kernel probability density function. Then, normalize the observation density field. Climate model grid. The relative observation density at a location is expressed as Specifically:

[0023] Obtain proxy record data and read the proxy dataset;

[0024] The observation density is estimated for each model grid point using the Gaussian kernel probability density function. For a given grid point, the density calculation expression is:

[0025] ,

[0026] in, This indicates the total number of substitute records. Indicates the first The substitute record for the first The contribution of each grid point in the pattern Indicates bandwidth parameter, The Gaussian kernel function is expressed as follows:

[0027] ,

[0028] Considering that the number of surrogate records will change over time, the bandwidth is determined using the Scott rule, specifically:

[0029] ,

[0030] in, This represents the standard deviation of the substitute records during the assimilation period;

[0031] The density field is normalized to obtain the relative observed density at the model grid points. This is to facilitate the transformation of the localized radius of the pattern grid points.

[0032] Step 2: Convert the observation density at the model grid point into the localized radius of the corresponding model grid point.

[0033] After setting upper and lower limits for the localization radius, for climate model grid points Observe its density field Converted to the corresponding localized radius The expression is:

[0034] ,

[0035] in, This represents the maximum localization radius set, which is reached in sparsely observed regions, and is exponential. Controlling the curvature of the density-to-radius mapping, This is the minimum localization radius set to ensure the impact of proxy records on the grid points of the observation-dense region model.

[0036] Step 3, for climate model grid points If the first Each substitute record is located within its localization radius. Within this system, the weights of the substitute records are calculated using the Gaspari-Cohn fifth-order polynomial:

[0037] ,

[0038] in, Represents a Gaspari-Cohn fifth-order polynomial. , Representing the grid points of the climate model With the The distance between substitute records Represents the density-adjusted grid of the climate model. The localization radius.

[0039] Step 4, for climate model grid points If the first Each substitute record is located within its localization radius. In addition, the correlation information between the pattern grid points and the proxy records is first calculated. Then judge Whether the relevance threshold is met determines the weight of the substitute record. .

[0040] The correlation information between pattern grid points and proxy records is obtained by calculating the Pearson correlation coefficient within the reconstruction period, and its expression is:

[0041] ,

[0042] in, Representing the grid points of the climate model With the Correlation between proxy records This indicates the length of the reconstruction period, which is divided into several assimilation windows. Indicates the first The first assimilation window Values ​​of each climate model grid point Indicates the first The first assimilation window The value of a substitute record and This represents the average value of the climate model grid points and proxy records within the reconstruction period.

[0043] For climate model grids The first one located outside its localization radius The influence weight of each substitute record can be expressed as:

[0044] ,

[0045] Where 0.2 is the set expression threshold.

[0046] Step 5: Integrate the proxy record weights inside and outside the localized radius of the climate model grid to obtain a mixed weight matrix. The covariance matrix is ​​adjusted using this matrix, as expressed by:

[0047] ,

[0048] in, Represents the mixed weight matrix, Represents the covariance function. Indicates the background field. This represents the schema data obtained after the proxy record is processed by the proxy system model.

[0049] The following description uses the reconstruction of surface temperature in the equatorial region as an example. First, based on coral proxy records, a continuous observed density distribution is obtained using the Gaussian kernel probability density function (KDE) and then normalized. Next, the localized radius is assigned to each model grid point using the density field, and the weights of proxy records inside and outside the localized radius are calculated. The weights of proxy records outside the localized radius are adjusted using statistical correlation information between the proxy records and the model grid points. The weights are then integrated to obtain a mixed weight matrix, and the covariance matrix is ​​adjusted. Subsequently, ensemble Kalman filtering is performed to obtain the reconstructed temperature field (i.e., the analysis field). The reconstruction results using adaptive localized radii are compared with those using fixed localized radii to illustrate the advantages and effectiveness of the method presented in this invention.

[0050] In the temperature reconstruction of the low-latitude equatorial region, this invention uses root mean square error (RMSE), efficiency coefficient (CE), and empirical orthogonal function (EOF) to verify the results. The results show that the adaptive localization method based on an ensemble paleoclimate data assimilation framework is superior to the traditional fixed localization method, and the reconstruction results are more consistent with the reanalysis data. The technical approach is as follows: Figure 1 As shown.

[0051] 1) The objective was to reconstruct the temperature anomaly field in the low equatorial latitudes (30°S to 30°N) from 1880 to 2000, using corals as the proxy record. This region was chosen primarily for two reasons: First, the temperature sensitivity of coral proxy records enhances reconstruction reliability; introducing other types of proxy records (such as tree rings and ice cores) on a global scale, due to their generally lower temperature sensitivity, might increase reconstruction uncertainty. Second, the tropics contain key climate systems for paleoclimate research, such as the El Niño-Southern Oscillation (ENSO), allowing for a clear assessment of methodological effectiveness while effectively reducing computational costs. The assimilation method was ensemble Kalman filtering (EnKF), with an assimilation frequency of 1 year and an ensemble size of 100.

[0052] 2) Obtain surrogate record data. Since the number of surrogate records will change, the observation density distribution needs to be re-determined based on the Gaussian kernel probability density function during each assimilation. The bandwidth can be empirically determined based on Scott's rule, such as... Figure 2 Figures (a)-(d) show the observation density distribution and time-averaged observation density distribution for 1880, 1940, and 1990. Normalization was performed to maintain the stability of the calculation.

[0053] 3) Based on the conversion formula between observation density and localization radius, determine the localization radius of the model grid points within the assimilation time period. The upper and lower limits of the localization radius are set to 20,000 km and 5,000 km, respectively. Calculate the distance between the proxy record and the model grid points. If this distance is less than the localization radius of the model grid points, calculate the influence weight of the proxy record. When the distance between the proxy record and the model grid points is greater than the obtained localization radius of the model grid points, the correlation between the two needs to be calculated. The case also calculates and compares the correlation information of two statistics: (1) obtained from the reanalysis dataset (NASA GISTEMP v4) and the proxy record, and (2) obtained from the model prior dataset (the "iCESM1" last millennium and historical simulations) and the proxy record.

[0054] 4) Determine the final mixing weight matrix and adjust the covariance matrix. Complete the subsequent EnKF assimilation algorithm to obtain the analysis field (i.e., the temperature reconstruction field).

[0055] 5) The adaptive localization method proposed in this invention is compared with the traditional fixed localization method. In the traditional fixed localization method, the localization radius of the pattern grid points is set to the same size, the substitute record weights are directly calculated, and the subsequent EnKF assimilation algorithm is performed to obtain the analysis field. For example... Figure 3The adaptive localization experiment (E_AL1), shown in (a)-(e), outperforms the fixed localization experiment (E_20000), producing a smaller root mean square error and a generally higher efficiency coefficient. The degree of improvement is related to data density: moderate improvement in data-rich regions, more significant improvement in medium-density regions such as the equatorial ENSO region, and virtually no change in sparse regions such as South America. Furthermore, the dominant empirical orthogonal function mode of the adaptive localization experiment exhibits a spatial structure similar to that of the observations, and its explained variance is superior to that of the fixed localization experiment.

[0056] Figure 4 Figures (a)-(b) show the RMSE test results for adaptive localization methods using different correlations. The adaptive localization experiment using model prior correlation (E_AL2) shows improvement over the fixed localization radius experiment (E_20000), as evidenced by its lower root mean square error and higher efficiency coefficient. Although its performance is still inferior to the adaptive experiment based on reanalysis data (E_AL1), it demonstrates the intrinsic value of model priors: their physically constrained correlations provide feasible and effective guidance for adaptive localization in the absence of reanalysis data. Overall, applying adaptive localization helps improve reconstruction quality, both in terms of statistical accuracy and fidelity of dominant climate modes.

[0057] The experimental results of this invention demonstrate that the adaptive localization method based on the ensemble paleoclimate data assimilation framework can improve the performance of paleoclimate reconstruction and help to reconstruct past climate variability more accurately, reliably and physically consistent.

[0058] Based on the same inventive concept, embodiments of this application provide a computer device, including a memory, a processor, and a computer program stored in the memory and executable on the processor. When the processor executes the computer program, it implements the steps of the aforementioned adaptive localization method based on a ensemble paleoclimate data assimilation framework.

[0059] Based on the same inventive concept, embodiments of this application provide a computer-readable storage medium storing a computer program that, when executed by a processor, implements the steps of the aforementioned adaptive localization method based on a ensemble paleoclimate data assimilation framework.

[0060] Those skilled in the art will understand that embodiments of the present invention can be provided as methods, systems, or computer program products. Therefore, the present invention can take the form of a completely hardware embodiment, a completely software embodiment, or an embodiment combining software and hardware aspects. Furthermore, the present invention can take the form of a computer program product embodied on one or more computer-usable storage media (including, but not limited to, disk storage, CD-ROM, optical storage, etc.) containing computer-usable program code.

[0061] This invention is described with reference to flowchart illustrations and / or block diagrams of methods, apparatus (systems), and computer program products according to embodiments of the invention. It will be understood that each block of the flowchart illustrations and / or block diagrams, and combinations of blocks in the flowchart illustrations and / or block diagrams, can be implemented by computer program instructions. These computer program instructions can be provided to a processor of a general-purpose computer, special-purpose computer, embedded processor, or other programmable data processing apparatus to produce a machine, such that the instructions, which execute via the processor of the computer or other programmable data processing apparatus, generate instructions for implementing the flowchart illustrations and / or block diagrams. Figure 1 One or more processes and / or boxes Figure 1 A device that provides the functions specified in one or more boxes.

[0062] These computer program instructions may also be stored in a computer-readable storage medium that can direct a computer or other programmable data processing device to function in a particular manner, such that the instructions stored in the computer-readable storage medium produce an article of manufacture including instruction means, which are implemented in a process Figure 1 One or more processes and / or boxes Figure 1 The function specified in one or more boxes.

[0063] These computer program instructions may also be loaded onto a computer or other programmable data processing equipment to cause a series of operational steps to be performed on the computer or other programmable equipment to produce a computer-implemented process, thereby providing instructions that execute on the computer or other programmable equipment for implementing the process. Figure 1 One or more processes and / or boxes Figure 1 The steps of the function specified in one or more boxes.

[0064] The above embodiments are merely illustrative of the technical concept of the present invention and should not be construed as limiting the scope of protection of the present invention. Any modifications made to the technical solutions based on the technical concept proposed in this invention shall fall within the scope of protection of this invention.

Claims

1. An adaptive localization method based on an ensemble paleoclimate data assimilation framework, characterized in that, Includes the following steps: Step 1: For the study area, obtain proxy record data for the data assimilation period, divide the data assimilation period into several assimilation windows, and for each assimilation window, obtain the observation density at each climate model grid point under the assimilation window through the Gaussian kernel probability density function, and perform normalization processing to obtain the relative observation density at each climate model grid point. Step 2: Set the upper and lower limits of the localization radius, and convert the relative observation density at each climate model grid point into the localization radius of each climate model grid point. Step 3: For each climate model grid point, if the proxy record is located within its localization radius, calculate the weight of the proxy record located within the localization radius according to the Gaspari-Cohn fifth-order polynomial. Step 4: If the proxy record is located outside its localization radius, calculate the correlation information between the climate model grid and the proxy record, and determine whether the correlation information meets the correlation threshold. If it does, use the correlation information to calculate the weight of the proxy record located outside its localization radius; otherwise, use the Gaspari-Cohn fifth-order polynomial to calculate the weight of the proxy record located outside its localization radius. Step 5: Integrate the weights of the proxy records inside and outside the localized radius of the climate model grid to obtain a mixed weight matrix. Use the mixed weight matrix to adjust the covariance matrix in the ensemble Kalman filter method to achieve the final data assimilation.

2. The adaptive localization method based on an ensemble paleoclimate data assimilation framework according to claim 1, characterized in that, In step 1, based on the acquired proxy record data, the observation density at each climate model grid point under the assimilation window is estimated using the Gaussian kernel probability density function: ， in, Indicates the first Climate model grid points Observation density at the location This indicates the total number of substitute records within the assimilation window. Indicates the first The substitute record for the first The contribution of each climate model grid point Indicates bandwidth parameter, The Gaussian kernel function is expressed as: ， in, ; Using Scott's rule, the expression is: ， in, This represents the standard deviation of the substitute records within the assimilation window. This yields the observation density field for each assimilation window. The observation density field is then normalized to obtain the relative observation density at each climate model grid point within the assimilation window. .

3. The adaptive localization method based on an ensemble paleoclimate data assimilation framework according to claim 1, characterized in that, In step 2, the relative observation density at the climate model grid points is converted into the localized radius of the climate model grid points, expressed as: ， in, Indicates the first Climate model grid points The localization radius, Indicates the first Relative observation density at each climate model grid point , These represent the preset maximum and minimum localization radii, respectively. This represents the curvature parameter that controls the mapping from density to radius.

4. The adaptive localization method based on an ensemble paleoclimate data assimilation framework according to claim 1, characterized in that, In step 3, the weights of the substitute records within the localized radius of the pattern grid are calculated using the Gaspari-Cohn fifth-order polynomial: ， in, Represents a Gaspari-Cohn fifth-order polynomial. , Indicates the first Climate model grid points With the The distance between substitute records Indicates the first Climate model grid points The localization radius.

5. The adaptive localization method based on an ensemble paleoclimate data assimilation framework according to claim 4, characterized in that, In step 4, the correlation information between climate model grid points and proxy records is obtained by calculating the Pearson correlation coefficient within the data assimilation period, expressed as follows: ， in, Indicates the first The first assimilation window The climate model grid point and the first The relevance of a proxy record. Indicates the length of the time period for data assimilation. Indicates the first The first assimilation window Values ​​of each climate model grid point Indicates the first The first assimilation window The value of a substitute record and These represent the average values ​​of all climate model grid points and proxy records within the data assimilation period, respectively. ， in, Indicates that it is located at the th Climate model grid points The first outside the localization radius The weight of each proxy record.

6. The adaptive localization method based on an ensemble paleoclimate data assimilation framework according to claim 1, characterized in that, In step 5, the covariance matrix in the ensemble Kalman filter method is adjusted using the mixed weight matrix, expressed as follows: ， in, Represents the mixed weight matrix. Indicates matrix multiplication. Represents the covariance function. Indicates the background field of the study area. This represents the pattern prior obtained after the proxy record is processed by the proxy system model.

7. A computer device comprising a memory, a processor, and a computer program stored in the memory and capable of running on the processor, characterized in that, When the processor executes the computer program, it implements the steps of the adaptive localization method based on the ensemble paleoclimate data assimilation framework as described in any one of claims 1 to 6.

8. A computer-readable storage medium storing a computer program, characterized in that, When the computer program is executed by a processor, it implements the steps of the adaptive localization method based on the ensemble paleoclimate data assimilation framework as described in any one of claims 1 to 6.

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