A method for glacier thickness inversion that integrates laminar flow model and stochastic unit interpolation

Abstract

This invention relates to the field of glacier thickness inversion technology, specifically disclosing a glacier thickness inversion method that integrates a laminar flow model and stochastic cell interpolation. This method addresses the problem that traditional laminar flow model-based glacier thickness estimation methods rely too heavily on the glacier's central streamline, which is difficult to accurately obtain in complex regions. By introducing stochastic cell interpolation, it avoids the dependence on the central streamline in glacier thickness estimation, while making fuller use of glacier topography and velocity data. Furthermore, by considering the differences in shape factors of different glaciers in the model and combining them with temperature information, specific dynamic parameters are assigned to the glacier, effectively avoiding arbitrary parameter determination. Experimental results show that the improved method of this invention can obtain a more realistic glacier thickness distribution.

Description

Technical Field

[0001] This invention relates to the field of glacier thickness inversion technology, specifically to a method for glacier thickness inversion that integrates laminar flow models and random unit interpolation. Background Technology

[0002] In glacier research, glacier thickness is a crucial parameter for estimating glacier reserves and developing dynamic models. It also plays a key supporting role in revealing the mechanisms of glacier instability disasters and simulating glacier instability disaster scenarios. Studies have shown that glacier thickness varies significantly across different regions and types. This variation affects the glacier's response to climate change, flow velocity, and meltwater flow. Accurately obtaining glacier thickness information is essential for glacier quality assessment, ablation process prediction, and water resource contribution analysis. While traditional ground drilling and ground-penetrating radar methods can provide relatively accurate location data, their widespread application is limited by cost and efficiency. Combining glacier physical models (e.g., laminar flow models), digital elevation models (DEMs), and ice surface geometric parameters allows for indirect estimation of glacier thickness distribution. Earth observation technologies such as satellite photogrammetry and radar interferometry can systematically acquire key inputs such as glacier surface velocity fields and morphological parameters, significantly improving the efficiency of physical model-based ice thickness estimation and enabling large-scale and even global estimation of glacier thickness and volume. Estimating glacier thickness by combining glacier physical models and remote sensing data has gradually become a new research direction. Existing physical model-based methods for glacier thickness estimation mainly fall into three categories: mass conservation-based methods, shear stress-based methods, and laminar flow model-based methods. However, these existing physical model-based glacier thickness estimation methods largely rely on central streamline extraction, which affects the model's adaptability and accuracy in complex regions. Summary of the Invention

[0003] This invention aims to overcome the shortcomings of existing technologies and provide a method for glacier thickness inversion that integrates laminar flow models and stochastic unit interpolation. This invention combines temperature data to determine the rate factor A, thereby enabling the setting of different rate factor parameters for glaciers in different regions; and determines the shape factor through the glacier cross-sectional width and the thickness at the central streamline. By using random unit interpolation, we can avoid dependence on the streamline at the center of the glacier and more effectively utilize observation data from other parts of the glacier to generate a cross-section with a near-parabolic shape.

[0004] To achieve the above objectives, this invention provides a glacier thickness inversion method that integrates a laminar flow model and random cell interpolation. Based on the laminar flow equation derived from Glen's flow law, ice thickness is calculated, and a random cell selection method is used to generate thickness samples. Then, a spatial interpolation method is used to estimate the glacier thickness over the entire region. Specifically, the laminar flow model is a physical model in glaciology, primarily used to describe the internal deformation and motion of ice masses under gravity. In this model, ice flow is analogous to the laminar flow of a viscous fluid under stress, and its motion is controlled by factors such as shear stress, ice viscosity, and ice thickness. Random cell interpolation is a spatial interpolation method based on randomly selected glacier internal cells, combined with the statistical characteristics of local buffer slopes. It involves repeated thickness estimation and spatial interpolation, followed by averaging of the results to obtain a stable glacier thickness distribution. The glacier thickness inversion method specifically includes the following steps:

[0005] Step 1: Obtain the surface flow velocity of glaciers in the study area using remote sensing. Digital elevation model, glacier boundary vector, glacier surface slope And glacier region mask data, and generate a regular rasterized dataset;

[0006] Step 2: Utilize glacier surface air temperature and empirical temperature difference constant First, the representative ice temperature of the glacier is estimated, and then the flow rate factor of multiple different glacier regions is dynamically calculated by combining the empirical formula of the rate factor. ;

[0007] This invention extrapolates representative ice temperatures based on the annual average air temperature at the glacier mass balance line and dynamically adjusts the mobility factor in the Glen formula using empirical formulas. This will improve adaptability to different glacier regions.

[0008] Step 3: Cross-sectional width of multiple different glacier regions With center streamline thickness Estimate multiple sets of shape factors of the valley And calculate the multiple sets of shape factors. The average value is taken after considering the corresponding glacier thickness values, thereby achieving the effect of shape factor. The invention improves the adaptability of the laminar flow model to different glacier cross-sectional shapes by introducing a shape factor based on the glacier cross-sectional width and center thickness to express the glacier geometry. By averaging the ice thickness results corresponding to multiple values, the uncertainty of the shape factor is dynamically corrected, thereby improving the accuracy and stability of ice thickness inversion.

[0009] Step 4: Install the glacier's internal units according to the set scale. Multiple sample units are randomly selected, and a buffer is constructed around each random sample unit. It is then determined whether the buffer satisfies the slope difference condition. If the requirement of less than 50m is met, proceed to step 5; otherwise, continue to expand the buffer zone until the slope difference limit is met.

[0010] Step 5: Calculate the average slope value for each buffer zone, and calculate the ice thickness of the random sample unit corresponding to each buffer zone based on the laminar flow equation;

[0011] Step 6: Assign fixed or estimated boundary thickness values ​​to the glacier edge unit and adjacent units to ensure the rationality and continuity of the boundary conditions;

[0012] Step 7: For unselected glacier interior units, spatial thickness is completed using the calculated sample unit thickness via inverse distance weighted interpolation to obtain a one-time glacier thickness distribution result. ;

[0013] Step 8: Repeat steps 4 through 7. Each time, the glacier thickness distribution results are recorded, and... The results are averaged to obtain the final estimate of glacier thickness.

[0014] This invention integrates laminar flow models with gridded random cell interpolation techniques. First, ice thickness is accurately inverted on selected random cells based on parameters such as flow velocity, slope, and temperature. Then, these points are used for global interpolation, thereby improving the accuracy and rationality of overall ice thickness estimation.

[0015] Furthermore, in step 2, the ice temperature at the glacier mass equilibrium line is used as the representative temperature of the glacier, assuming the deviation between the annual average air temperature and the ice temperature at the glacier surface. The constant is used to measure the surface temperature of the glacier. Estimate the representative temperature of each glacier; then the flow rate factor. The calculation formula is:

[0016]

[0017] in .

[0018] Furthermore, in step 3, the shape factor The calculation formula is:

[0019] .

[0020] Furthermore, in step 3, the central streamline thickness of the glacier... Through the shear stress equation Please provide a solution.

[0021] Furthermore, in step 5, the laminar flow equation for the glacier thickness is:

[0022]

[0023] In the above formula, It is a constant determined empirically; The base velocity; For glacier thickness, For glacier density, This is the acceleration due to gravity.

[0024] Compared with the prior art, the present invention has the following beneficial effects:

[0025] (1) Rigorous theoretical foundation and scientific parameter setting. The laminar flow model derived from Glen's flow law is based on a mature and reliable theoretical foundation. Unlike traditional methods that rely on manual experience to set parameters, the method of this invention dynamically calculates the flow rate factor and shape factor through physical quantities such as air temperature and slope, realizing the automated and physically driven setting of key parameters, thus enhancing the versatility and accuracy of the method.

[0026] (2) High computational efficiency and applicable to large areas. By dividing the glacier area into regular grids and using random cell selection and buffer construction, the method of the present invention can effectively reduce the amount of computation, avoid point-by-point modeling of the entire area, significantly improve the computational efficiency in large glacier areas, and is highly adaptable and easy to promote.

[0027] (3) Avoid dependence on the central streamline and enhance spatial continuity. Traditional methods based on the central streamline are difficult to apply in complex terrain or multi-flow areas. This invention estimates the ice thickness of unselected units by using a random unit interpolation method, which avoids the extraction and dependence on the central streamline and improves the spatial continuity and robustness of the ice thickness estimation model.

[0028] In addition to the objectives, features, and advantages described above, the present invention has other objectives, features, and advantages. The invention will now be described in further detail with reference to the accompanying drawings. Attached Figure Description

[0029] The accompanying drawings are provided to further illustrate embodiments of the present invention and form part of the specification. They are used together with the following detailed description to explain the embodiments of the present invention, but do not constitute a limitation thereof. In the drawings:

[0030] Figure 1 This is a schematic diagram of the research area model provided by the present invention;

[0031] Figure 2 It is an ideal glacier cross section calculated using three different b-values. Comparison of cross-sectional shapes obtained by inversion with different interpolation parameters;

[0032] Figure 3 This is the result of glacier flow velocity in the main peak area of ​​the West Kunlun Mountains;

[0033] Figure 4 This is a map of glacier thickness in the main peak area of ​​the West Kunlun Mountains obtained by inversion using the method of this invention. Detailed Implementation

[0034] The present invention will now be described in detail with reference to the embodiments shown in the accompanying drawings. However, it should be noted that these embodiments are not intended to limit the present invention. Equivalent transformations or substitutions in function, method, or structure made by those skilled in the art based on these embodiments are all within the scope of protection of the present invention.

[0035] This embodiment provides a glacier thickness inversion method that integrates a laminar flow model and random cell interpolation. Based on the laminar flow equation derived from Glen's flow law, the ice thickness is calculated, and a random cell selection method is used to generate thickness samples. Then, a spatial interpolation method is used to estimate the glacier thickness over the entire glacier area. This method uses the thickness information of randomly selected glacier cells and glacier edge cells to interpolate the thickness over the entire glacier, resulting in a profile of the ice thickness distribution that approximates an ideal shape. The glacier thickness inversion method specifically includes the following steps:

[0036] Step 1: Obtain the surface flow velocity of glaciers in the study area using remote sensing. Digital elevation model, glacier boundary vector, glacier surface slope And glacier region mask data, and generate a regular rasterized dataset.

[0037] Step 2: Utilize glacier surface air temperature and empirical temperature difference constant Estimate the representative ice temperature of the glacier and dynamically calculate the mobility factor using the empirical formula for the rate factor. This led to the acquisition of flow rate factors for multiple different glacier regions. That is, to assign spatially varying mobility factors to different glacial regions. This step reflects the differences in the thermal state within the glacier. The ice temperature at the glacier's mass equilibrium line is used as the representative temperature of the glacier, assuming an empirical temperature difference between the annual average air temperature and the ice temperature at the glacier surface. The constant is used to measure the surface temperature of the glacier. Estimate the representative temperature for each glacier; the glacier surface air temperature data are derived from ERA5 reanalysis data, using empirical temperature difference constants. The setting method is conventional in this field. Liquidity factor. The calculation formula is:

[0038]

[0039] Among them, when the empirical temperature difference constant The mobility factor is at 7℃ (this value is based on statistical analysis of the difference between borehole temperature and surface air temperature in the active layer of Chinese glaciers). .

[0040] Step 3: Cross-sectional width of multiple different glacier regions With center streamline thickness Valley's multiple shape factors and through multiple sets of shape factors After calculating the corresponding thickness values, take the average value. The average thickness at the center streamline is denoted as . As a subsequent calculation of the shape factor The parameters are used to achieve control over the shape factor. Dynamic correction for shape factor. Through calculation formula The thickness value is calculated by selecting all shape factors within the range of 0.6-1 at intervals of 0.01. The average thickness is calculated using the value. The central streamline thickness of the glacier. Through the shear stress equation Please provide a solution.

[0041] Step 4: Install the glacier's internal units according to the set scale. Multiple sample units are randomly selected, and a 3×3 buffer is constructed around each sample unit. It is then determined whether the buffer satisfies the slope difference condition. If the requirement of less than 50m is met, proceed to step 5; otherwise, continue to expand the buffer zone until the slope difference limit is met.

[0042] Step 5: Calculate the average slope for each buffer zone and calculate the ice thickness of the random cell based on the laminar flow equation for glacier thickness; the laminar flow equation for glacier thickness is:

[0043]

[0044] in, It is a constant determined empirically; The base velocity; For glacier thickness, For glacier density, This is the acceleration due to gravity.

[0045] Step 6: Assign fixed or estimated boundary thickness values ​​to the glacier edge unit and adjacent units to ensure the rationality and continuity of the boundary conditions.

[0046] Step 7: For unselected glacier interior units, use the calculated sample unit thickness to perform spatial thickness completion using the inverse distance weighted interpolation method, obtaining a one-time ice thickness distribution result. The sample unit thickness includes the ice thickness of the random unit calculated in step 5 and the boundary thickness values ​​of the glacier edge unit and adjacent unit in step 6.

[0047] Step 8: Repeat steps 4 through 7. Each time, the ice thickness distribution results generated are recorded. The results are averaged to obtain the final ice thickness estimate; among them, According to the ratio set in step 4 The value of is determined by .

[0048] The method of this invention combines a laminar flow model with a random cell selection method to dynamically construct a buffer and calculate the ice thickness of the sample cells under the condition that the slope difference is less than a preset threshold. This takes into account both the physical mechanism of ice thickness inversion and computational efficiency, thereby improving the reliability and spatial representativeness of the ice thickness estimation results.

[0049] Example 1

[0050] like Figure 1 As shown, the study area is divided into four categories: non-glacier units, glacier-adjacent units, glacier-margin units, and glacier-internal units; a certain proportion is randomly selected from the glacier-internal units. The system uses random cells, and constructs a 3×3 buffer around each selected random cell; the buffer size is allowed to grow until the slope difference between the lowest and highest cells within the buffer is reached. Up to 50m; the average slope of all cells within the buffer zone is used as the slope for calculating the thickness of that cell; the thickness of these random cells is obtained using the laminar flow equation. Simultaneously, thickness values ​​are assigned to adjacent glacier cells and glacier edge cells; the thickness of unselected glacier interior cells is interpolated using IDW to obtain the ice thickness distribution within the glacier at one time. Repeat the above steps. Next, get The ice thickness distribution results, for this The final ice thickness distribution result is obtained by averaging the individual ice thickness distribution results.

[0051] Figure 2 It is an ideal glacier cross section calculated using three different b-values. The comparison of cross-sectional shapes obtained by inversion with different interpolation parameters aims to observe the correction ability of different parameter combinations on cross-sectional morphology; the glacier width shown in the figures ranges from 600m to 1500m. While keeping other inversion parameters constant, the increase in width leads to an increase in the number of data points within the cross-section, which increases the ratio of internal cells to edge cells. Therefore, it is necessary to adjust the random cell ratio. The size is used to balance the ratio of these two units. In this invention, the original cross-section without interpolation is called the 'uncorrected cross-section,' and the adjusted cross-section is called the 'corrected cross-section.' Power-law function It is often used to simulate the valley shape of glaciers. When the range is 1.5-2.5, the power-law function curve approximates a parabola, which can represent the bottom of most glacier valley cross-sections with near-ideal shapes. Using... The power-law function describes the idealized cross-sectional morphology of a glacier. The varying data primarily include different surface flow velocities. Different cross-sectional widths and different random unit ratios .from Figure 2 It can be seen that the method of integrating laminar flow model and random element interpolation can generate a profile with an approximately ideal shape without relying on the generation of the center streamline and without fitting the cross-sectional shape of the vertical streamline.

[0052] Figure 3 The results show the glacier flow velocity in the main peak area of ​​the West Kunlun Mountains; Figure 4 The flow velocity results based on the main peak region of the West Kunlun Mountains are presented. Figure 3 The glacier thickness results obtained by inversion using this invention are used. Figure 4 It can be seen that the method of the present invention can make full use of known glacier topography and flow velocity data to generate glacier cross sections that are more consistent with reality, and to a certain extent correct the glacier thickness inverted by the laminar flow model itself, thereby improving the accuracy of glacier storage estimation in large areas.

[0053] This invention addresses the problem that traditional laminar flow model-based glacier thickness estimation methods rely too heavily on the glacier's central streamline, which is difficult to accurately obtain in complex regions. It introduces a stochastic cell interpolation method to avoid this dependence on the central streamline and to make fuller use of glacier topography and velocity data. Furthermore, the model considers the differences in shape factors among different glaciers and incorporates temperature information to assign specific dynamic parameters to the glacier, avoiding arbitrary parameter determination. Experimental results show that the improved method of this invention can obtain a more realistic glacier thickness distribution.

[0054] The above description is merely a preferred embodiment of the present invention and is not intended to limit the invention. Various modifications and variations can be made to the present invention by those skilled in the art. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

Claims

1. A method for glacier thickness inversion that integrates laminar flow model and stochastic unit interpolation, characterized in that, Includes the following steps: Step 1: Obtain the surface flow velocity of glaciers in the study area using remote sensing. Digital elevation model, glacier boundary vector, glacier surface slope And glacier region mask data, and generate a regular rasterized dataset; Step 2: Utilize glacier surface air temperature and empirical temperature difference constant First, the representative ice temperature of the glacier is estimated, and then the flow rate factor of multiple different glacier regions is dynamically calculated by combining the empirical formula of the rate factor. ; Step 3: Cross-sectional width of multiple different glacier regions With center streamline thickness Estimate multiple sets of shape factors of the valley And calculate the multiple sets of shape factors. The average value is taken after considering the corresponding glacier thickness values; Step 4: Install the glacier's internal units according to the set scale. Multiple sample units are randomly selected, and a buffer is constructed around each random sample unit. It is then determined whether the buffer satisfies the slope difference condition. If the requirement of less than 50m is met, proceed to step 5; otherwise, continue to expand the buffer zone until the slope difference limit is met. Step 5: Calculate the average slope value for each buffer zone, and calculate the ice thickness of the random sample unit corresponding to each buffer zone based on the laminar flow equation; Step 6: Assign fixed or estimated boundary thickness values ​​to the glacier edge unit and adjacent units to ensure the rationality and continuity of the boundary conditions; Step 7: For unselected glacier interior units, spatial thickness is completed using the calculated sample unit thickness via inverse distance weighted interpolation to obtain a one-time glacier thickness distribution result. ; Step 8: Repeat steps 4 through 7. Each time, the glacier thickness distribution results are recorded, and... The results are averaged to obtain the final estimate of glacier thickness.

2. The glacier thickness inversion method according to claim 1, characterized in that, In step 2, the ice temperature at the glacier mass equilibrium line is used as the representative temperature of the glacier, assuming the deviation between the annual average air temperature and the ice temperature at the glacier surface. The constant is used to measure the surface temperature of the glacier. Estimate the representative temperature of each glacier; then the flow rate factor. The calculation formula is: in .

3. The glacier thickness inversion method according to claim 1, characterized in that, In step 3, the shape factor The calculation formula is: 。 4. The glacier thickness inversion method according to claim 3, characterized in that, In step 3, the central streamline thickness of the glacier Through the shear stress equation Solve; where, For glacier density, This is the acceleration due to gravity.

5. The glacier thickness inversion method according to claim 1, characterized in that, In step 5, the laminar flow equation for the glacier thickness is: In the above formula, It is a constant determined empirically; The base velocity; For glacier thickness, For glacier density, This is the acceleration due to gravity.

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