GNSS nonlinear deformation correction method and system based on full-spectrum three-dimensional thermoelastic model

Abstract

The application discloses a GNSS nonlinear deformation correction method based on a full-spectrum three-dimensional thermoelastic model, which comprises the following steps: performing discrete Fourier decomposition on the temperature time sequence of each grid point; performing spatial spherical harmonic expansion and coefficient projection on each frequency after decomposition; based on the spatial spherical harmonic expansion and coefficient projection result, considering the periodicity of temperature, performing positive / cosine representation of the temperature term of each frequency; based on the spatial spherical harmonic expansion and coefficient projection result, corresponding to the periodic temperature, performing spherical harmonic representation and time expansion on the displacement field of deformation; solving the deformation equation by using Love number form in spherical coordinates to obtain the expression of complex Love number; based on the complex spherical harmonic coefficient of the foregoing temperature and the complex Love number, performing full-spectrum synthesis of three-component displacement at the spherical surface; and based on the full-spectrum synthesis result, performing internal consistency and external dispersion verification.

Description

Technical Field

[0001] This invention belongs to the field of satellite geodesy and involves nonlinear deformation modeling and correction of GNSS station coordinate time series, especially a data processing method combining a full-spectrum three-dimensional thermoelastic model. Background Technology

[0002] Long-term continuous observations from Global Navigation Satellite System (GNSS) reference stations provide a high spatiotemporal resolution data foundation for space geodesy and geophysical research. Influenced by multiple sources including crustal tectonic evolution, surface mass load (atmosphere, ocean, and terrestrial hydrology), and environmental temperature, GNSS coordinate time series generally exhibit significant nonlinear variations. Practice shows that non-tidal atmospheric load, non-tidal ocean load, and terrestrial hydrological load can cause considerable surface displacement, which is highly correlated with the vertical component of GNSS and can explain a significant proportion of the vertical annual cycle signal. In addition to mass load, the thermal expansion and contraction (thermoelasticity) effect of bedrock induced by changes in surface and near-surface temperature is also a significant source of GNSS nonlinear variations. Without modeling and correction, this will interfere with the interpretation of geophysical signals such as tectonic deformation, post-glacial isostatics, and long-term water storage changes.

[0003] Theoretically, the basic framework for the thermoelastic response of solid media has long been established. Early models mostly adopted the assumption of a semi-infinite space, indicating that periodic changes in surface temperature would induce corresponding deformations in the subsurface medium, and providing an upper limit estimate for the annual vertical periodic displacement of bedrock. Subsequent applied research based on measured surface temperatures further showed that bedrock thermal expansion can generate millimeter-scale annual vertical amplitudes in some regions, providing considerable explanatory power for the vertical nonlinear components of GNSS. However, most of these works only consider seasonal temperature terms such as annual and semi-annual periods, lacking a systematic characterization of the contribution of non-seasonal high-frequency temperature perturbations.

[0004] To overcome the limitations of "seasonal components only," subsequent studies introduced the finite element method to unify seasonal and non-seasonal components into full-cycle (full-spectrum) thermoelastic displacement, showing that non-seasonal temperature fluctuations in some regions can lead to vertical displacement peaks reaching the millimeter level. On the other hand, half-space and some finite element implementations typically only output vertical displacement and assume that horizontal displacement is negligible, which is inconsistent with the true dynamic response of the three-dimensional Earth medium.

[0005] Considering the Earth's spherical geometry and geocentric constraints, some studies have derived the three-dimensional thermoelastic response driven by the surface temperature field within a spherical framework, and based on this, calculated the global three-dimensional annual periodic displacement. The results show that the vertical amplitude can reach the millimeter scale, and the horizontal component is equally significant, indicating that the three-dimensional thermoelastic effect has significant amplitude and regional differences on a global scale.

[0006] In summary, existing research still has the following shortcomings: 1. It focuses primarily on seasonal temperature changes, neglecting the role of non-seasonal (including intraday to multi-monthly) temperature fluctuations and their frequency-dependent phase lag. 2. Many models only provide vertical results and do not systematically provide horizontal components, making it difficult to meet the needs of reference frame stabilization and joint analysis of the three components. Summary of the Invention

[0007] The purpose of this invention is to provide a GNSS nonlinear deformation correction method based on a full-spectrum three-dimensional thermoelastic model. This method constructs a frequency-dependent response (amplitude and phase) to three-dimensional displacement driven by seasonal and non-seasonal temperature forces within a spherical frame, outputting a station-level three-component (vertical / north-south / east-west) correction sequence. Compared to traditional seasonal thermoelastic models, the proposed model can simultaneously characterize both seasonal and non-seasonal temperature forces, reflecting frequency-dependent phase / amplitude characteristics, and outputs a station-level three-component correction sequence suitable for engineering applications. Furthermore, it provides a systematic evaluation of consistency, effectiveness, and regional variability using long-term multi-station data on a global scale, thereby improving the physical interpretability of GNSS time series and the stability of the reference frame.

[0008] According to one aspect of the present invention, a GNSS nonlinear deformation correction method based on a full-spectrum three-dimensional thermoelastic model is provided, comprising: Perform Discrete Fourier Decomposition on the temperature time series of each grid point; For each frequency after decomposition, perform spatial spherical harmonic expansion and coefficient projection; Based on the spatial spherical harmonic expansion and coefficient projection results, and considering the periodicity of temperature, a frequency-by-frequency temperature term is represented by sine / cosine. Based on the spatial spherical harmonic expansion and coefficient projection results, the displacement field of the deformation is represented spherically harmonically and expanded over time according to the corresponding periodic temperature. Solve the transformed equations in spherical coordinates using the Love number form to obtain the expression for the complex Love number; Based on the aforementioned complex spherical harmonic coefficients and complex Love number at temperature, the full spectrum of the three-component displacement at the spherical surface is synthesized. Based on the full-spectrum synthesis results, internal consistency and external dispersion verification are performed.

[0009] As a further technical solution, the temperature time series of each grid point is subjected to discrete Fourier decomposition, including: set up This is a record of the temperature at the spherical grid points over time, where... These are radial coordinates in a spherical coordinate system; and Let represent the latitude and longitude of the observation station, respectively, and t be the time. Perform a Discrete Fourier Transform on each grid point to decompose it into discrete angular frequencies. The frequency components are used to obtain the amplitude. and And phase information.

[0010] As a further technical solution, spatial spherical harmonic expansion and coefficient projection are performed on each of the decomposed frequencies, including: For each frequency ,Will and According to the fully normalized spherical harmonics Expand; Depend on , and The spherical harmonic coefficients are obtained by projecting the inner product of the sphere. , and give The expression of Legendre polynomials, in which, Here, n and m represent the order and degree of the spherical harmonic function, respectively, and are fully normalized. This indicates the complex conjugate operation.

[0011] As a further technical solution, based on the periodicity of temperature, a frequency-by-frequency sine / cosine representation of the temperature term is performed, including: Considering the periodicity of temperature, the temperature term for each Fourier frequency can be further written in cosine / sine form. , in, and They represent frequencies respectively. The cosine and sine components of the temperature signal.

[0012] As a further technical solution, based on the spatial spherical harmonic expansion and coefficient projection results, and corresponding to the periodic temperature, the displacement field of the deformation is represented spherically harmonicly and expanded over time, including: use Indicates deformation For the corresponding periodic temperature, perform a Fourier expansion of the displacement to give the cosine / sine terms of the radial / tangential components at each frequency. , in, and These represent the radial displacement coefficient and the tangential displacement coefficient, respectively. and These represent the cosine and sine components of the radial displacement at a given frequency, respectively. and These represent the cosine and sine components of the tangential displacement at a given frequency, respectively. The sampling frequency period.

[0013] As a further technical solution, the deformation equation is solved in spherical coordinates using the Love number form to obtain the expression for the complex Love number, including: Solving the transformed equations in spherical coordinates using the Love number form yields the complex Love number. , The expression of depends explicitly on the coefficient of thermal expansion, thermal diffusivity, and Poisson's ratio.

[0014] As a further technical solution, based on the aforementioned complex spherical harmonic coefficients and complex Love number at temperature, a full-spectrum synthesis of the three-component displacement at the spherical surface is performed, including: By pairing and linearly superimposing the complex spherical harmonic coefficients of temperature with the complex Love number at various frequencies, the three-dimensional displacement components at the geocentric sphere coordinate radius are obtained: vertical displacement. North-south displacement East-west displacement .

[0015] As a further technical solution, internal consistency verification is performed based on the full-spectrum synthesis results, including: The temperature range of each station is divided into several intervals; the mean and standard deviation are calculated using the model displacement within each interval, and the empirical boundary is defined by the mean ± 2σ; the accuracy is calculated by the proportion of model displacement falling into each boundary; a consistently low standard deviation and a mean close to 0 indicate that the model is stable and the system bias is small; an accuracy greater than a set threshold indicates high fidelity.

[0016] As a further technical solution, external dispersion verification is performed based on the full-spectrum synthesis results, including: After aligning with the GNSS time series, tectonic trends, step loads, and non-tidal loads are removed, and the three-dimensional thermoelastic response displacement is subtracted. The WRMS reduction rate of GNSS nonlinear changes is statistically analyzed to quantify the explanatory power of the model for temperature-driven deformation.

[0017] According to one aspect of the present invention, a GNSS nonlinear deformation correction system based on a full-spectrum three-dimensional thermoelastic model is provided, comprising: The first main module is used to perform discrete Fourier decomposition on the temperature time series of each grid point; The second main module is used to perform spatial spherical harmonic expansion and coefficient projection on each frequency after decomposition. The third main module is used to perform sine / cosine representation of the temperature term frequency by frequency based on the spatial spherical harmonic expansion and coefficient projection results, taking into account the periodicity of temperature. The fourth main module is used to perform spherical harmonic representation and time expansion of the displacement field of deformation based on the spatial spherical harmonic expansion and coefficient projection results, corresponding to the periodic temperature. The fifth main module is used to solve the deformation equations in spherical coordinates using Love number form to obtain the expression of complex Love numbers; The sixth main module is used to synthesize the full spectrum of the three-component displacement at the sphere based on the complex spherical harmonic coefficients and complex Love number at the aforementioned temperature. The seventh main module is used to verify internal consistency and external dispersion based on the full spectrum synthesis results.

[0018] Compared with the prior art, the beneficial effects of the present invention are as follows: This invention proposes a nonlinear deformation correction method for GNSS based on a full-spectrum three-dimensional thermoelastic model, targeting high-precision GNSS applications under conditions of significant climatic zone differences and strong non-seasonal fluctuations. Theoretically, it employs a full-spectrum representation and complex Love number solution within a spherical frame, uniformly considering both thermodynamic forces and surface thermal loads, and explicitly presents the amplitude-phase lag relationship of the three-component thermoelastic response, offering greater physical consistency and dimensional integrity compared to models with only seasonal terms or half-space components. In application, this invention can directly generate vertical / north-south / east-west three-component station correction sequences and seamlessly integrate them with non-tidal load corrections, significantly reducing the dispersion and residual spectral energy of GNSS time series, minimizing signal aliasing, improving reference frame stability, and reliably separating and interpreting geophysical signals such as tectonic and water storage changes. Furthermore, the method is modularly implemented, parameters are controllable, and it is easy to batch scale and deploy in near real-time, demonstrating strong engineering applicability. Attached Figure Description

[0019] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the accompanying drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the accompanying drawings described below are some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0020] Figure 1 The diagram illustrates a GNSS nonlinear deformation correction method based on a full-spectrum three-dimensional thermoelastic model provided in this embodiment of the invention. The middle diagram is a flowchart of the construction of the full-spectrum three-dimensional thermoelastic model (FSM), the right diagram is a flowchart of the parameter evaluation of the FSM thermoelastic model, and the left diagram is a flowchart of the performance evaluation of the FSM model correction scheme in the GNSS coordinate time series.

[0021] Figure 2The method provided in this embodiment of the invention is compared with the thermoelastic displacement modeling results of traditional methods; the surface temperature time series of low latitude (KAT1), mid latitude (BJFS) and high latitude (AND1) GNSS reference stations (top figure) and the corresponding three-dimensional thermal expansion displacement (bottom figure); the light blue curve shows the displacement predicted by the traditional seasonal model, and the red curve is the output result of the full-spectrum FSM model.

[0022] Figure 3 A schematic diagram of the annual amplitude and non-seasonal temperature variation of global thermoelastic displacement estimated by the method provided in the embodiments of the present invention.

[0023] Figure 4 The reduction rate of root mean square error (WRMS) of 1,698 GNSS displacement time series worldwide (top figure) and the box plot of WRMS reduction rate of each region after modeling and correcting thermoelastic displacement using the method provided in this embodiment of the invention are shown. Detailed Implementation

[0024] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention. In addition, the technical features of the various embodiments or individual embodiments provided by the present invention can be arbitrarily combined to form new technical solutions. Such combinations are not bound by the order of steps and / or structural composition patterns, but must be based on the ability of those skilled in the art to implement them. When the combination of technical solutions is contradictory or cannot be implemented, it should be considered that such a combination of technical solutions does not exist and is not within the scope of protection claimed by the present invention.

[0025] Please see Figure 1 The flowchart illustrates the three-dimensional thermoelastic deformation correction process according to an embodiment of the present invention. This invention provides a GNSS nonlinear deformation correction method based on a full-spectrum three-dimensional thermoelastic model, the specific implementation process of which is as follows: Within the framework of a homogeneous elastic sphere constrained by "geocentric mass," the analytical approach of thermoelastic three-dimensional displacement proposed by Fang et al. (2014) is adopted to decompose the deformation induced by surface temperature changes into two modes: exponential modes (thermal forces) and power-law modes (surface thermal loads). Both contribute to the radial displacement of the sphere with equal order, while the power-law mode dominates the tangential displacement. To overcome the limitation of only considering seasonal terms, this study performs full-spectrum decomposition of the temperature field, introducing seasonal and non-seasonal (including high-frequency) components to jointly drive the three-dimensional thermoelastic response (FSM).

[0026] Step 1: Discrete Fourier Decomposition of Temperature Time Series set up This is a record of the temperature at each grid point on a spherical surface over time; a discrete Fourier transform is performed on each grid point to decompose it into discrete angular frequencies. The frequency components are used to obtain the amplitude. and (Obtained from the folded symmetric spectrum) and phase information; and the sampling frequency is given. Sample length N, spherical coordinates Conventions for symbols, etc.

[0027] , in, The radial coordinates in the spherical coordinate system (unit: meters); and These represent the latitude and longitude of the observation station (unit: radians), respectively. The sampling frequency; The sampling frequency period; Indicates the first Angular frequency of a discrete frequency; and This is the amplitude obtained from the folded symmetric spectrum.

[0028] Step 2: Spatial spherical harmonic expansion and coefficient projection For each frequency ,Will and According to the fully normalized spherical harmonics Expand (maximum number of approximation accuracy); spherical harmonic coefficients Will be by and The inner product of the sphere is obtained by projection; and the following is given. The expression of Legendre polynomials.

[0029] , in, This is a fully normalized spherical harmonic function. n and m represent the order and degree of the spherical harmonic function, respectively. This is the highest order of the spherical harmonics, which determines how many spherical harmonic terms are used in the spherical harmonic expansion to approximate the actual temperature or deformation field. (Astro) This indicates the complex conjugate operation. It is a Legendre polynomial.

[0030] Step 3: Sine / Cosine representation of the frequency-by-frequency temperature term Considering the periodicity of temperature, the temperature term for each Fourier frequency can be further written in cosine / sine form. and These are the cosine and sine components of the frequency, respectively.

[0031] , in and They represent frequencies respectively. The cosine and sine components of the temperature signal.

[0032] Step 4: Spherical harmonic representation and time expansion of the displacement field In a spherically symmetric elastic body, deformation use It is represented by radial and tangential coefficients (including radial unit vector and spherical differential operator); corresponding to the periodic temperature, the displacement is further expanded using Fourier expansion to give the cosine / sine terms of the radial / tangential components at each frequency.

[0033] , in, and These represent the radial displacement coefficient and the tangential displacement coefficient, respectively. It is a radial unit vector; It is a differential operator that correlates tangential displacement with the gradient of the spherical harmonic basis function. It is the elastic modulus; and These represent the cosine and sine components of the radial displacement (deformation perpendicular to the surface) at a given frequency, respectively. and These represent the cosine and sine components of the tangential displacement at a given frequency, respectively.

[0034] Step 5: Love Number Method and Complex Love Number Solving the deformed equations in spherical coordinates using the Love number form yields the complex Love number. The expression of the Love number explicitly depends on the coefficient of thermal expansion, thermal diffusivity, Poisson's ratio, etc.; in the frequency domain, the Love number inherently contains factors that vary with frequency, resulting in phase lag at each frequency.

[0035] , , Wherein, the coefficient of thermal expansion is Thermal diffusivity Poisson's ratio is Regarding the integral constant and For parameter settings, please refer to the study by Fang et al. (2014). Each frequency The deformation response exhibits characteristic phase hysteresis. This lag originates directly from the thermal diffusion operator in the frequency domain.

[0036] Step 6: Full spectrum synthesis of the three-component displacements at the spherical surface By pairing and linearly superimposing the complex spherical harmonic coefficients and complex Love numbers at various frequencies at the above temperatures, the three-dimensional displacement components at the geocentric sphere coordinate radius r are obtained: .

[0037] , , in, Indicates vertical displacement. Indicates north-south displacement. This indicates east-west displacement.

[0038] Step 7: Model Validation (Internal Consistency + External Discreteness) Internal consistency: The temperature range of each station is divided into 5 equal intervals; the mean and standard deviation (STD) are calculated using model displacement within each interval, and the empirical boundary is defined by the mean ± 2σ; the accuracy is calculated as the proportion of model displacements falling within each boundary. A consistently low STD and a mean close to 0 indicate model stability and small systematic bias; an accuracy > 95% reflects high fidelity.

[0039] External performance: After aligning with the GNSS time series, tectonic trends, steps, and non-tidal loads (non-tidal atmospheric load, non-tidal ocean load, and hydrological load) are removed, and the FSM displacement is subtracted. The decrease in WRMS before and after the three-component correction is compared to quantify the model's explanatory power for temperature-driven deformation.

[0040] By comparing the present invention with the traditional seasonal thermoelastic model (only annual / semi-annual items), the full-spectrum three-dimensional thermoelastic model (FSM) of the present invention has better performance in three-dimensional displacement characterization and GNSS correction. Figure 1 The paper presents the construction process of FSM (middle figure), parameter and consistency evaluation (right figure), and the performance evaluation approach of using FSM for GNSS coordinate time series correction (left figure), showing that the method chain is complete and reproducible. Figure 2 The study presents a comparison of surface temperature and corresponding three-dimensional thermal expansion displacement at three stations: low latitude (KAT1), mid latitude (BJFS), and high latitude (AND1). Light blue represents the traditional seasonal model, and red represents the full-spectrum FSM. The results show that the FSM is more consistent with the observations in terms of amplitude and phase. Figure 3 This invention estimates the annual amplitude of global thermoelastic displacement and the spatiotemporal distribution of non-seasonal temperature variations, emphasizing the non-negligible contribution of non-seasonal components to the actual displacement. Figure 4The distribution of WRMS reduction rate after applying FSM to 1,698 GNSS stations worldwide (above figure) and box plots for each region are presented, showing that the present invention can effectively reduce the dispersion of GNSS coordinate time series.

[0041] The implementation of the various embodiments of the present invention is based on programmed processing by a device with processor functionality. Therefore, in practical engineering, the technical solutions and functions of the various embodiments of the present invention are encapsulated into various modules. Based on this reality, and building upon the above embodiments, the embodiments of the present invention provide a GNSS nonlinear deformation correction system based on a full-spectrum three-dimensional thermoelastic model. This system is used to execute the GNSS nonlinear deformation correction method based on a full-spectrum three-dimensional thermoelastic model in the above method embodiments.

[0042] The system includes: The first main module is used to perform discrete Fourier decomposition on the temperature time series of each grid point; The second main module is used to perform spatial spherical harmonic expansion and coefficient projection on each frequency after decomposition. The third main module is used to perform sine / cosine representation of the temperature term frequency by frequency based on the spatial spherical harmonic expansion and coefficient projection results, taking into account the periodicity of temperature. The fourth main module is used to perform spherical harmonic representation and time expansion of the displacement field of deformation based on the spatial spherical harmonic expansion and coefficient projection results, corresponding to the periodic temperature. The fifth main module is used to solve the deformation equations in spherical coordinates using Love number form to obtain the expression of complex Love numbers; The sixth main module is used to synthesize the full spectrum of the three-component displacement at the sphere based on the complex spherical harmonic coefficients and complex Love number at the aforementioned temperature. The seventh main module is used to verify internal consistency and external dispersion based on the full spectrum synthesis results.

[0043] The GNSS nonlinear deformation correction system based on a full-spectrum three-dimensional thermoelastic model provided in this invention addresses the shortcomings of existing research. It employs several modules to construct a system within a spherical frame that uniformly describes the frequency-dependent response (amplitude and phase) of three-dimensional displacement driven by seasonal and non-seasonal temperature forces, outputting a station-level three-component (vertical / north-south / east-west) correction sequence. Compared to traditional seasonal thermoelastic models, the proposed model can simultaneously characterize both seasonal and non-seasonal temperature forces, reflecting frequency-dependent phase / amplitude characteristics. It also outputs a station-level three-component correction sequence suitable for engineering applications. Furthermore, it conducts a systematic evaluation of consistency, effectiveness, and regional variability using long-term multi-station data on a global scale, thereby improving the physical interpretability of GNSS time series and the stability of the reference frame.

[0044] It should be noted that the system embodiments provided by the present invention are used not only to implement the methods in the above method embodiments, but also to implement the methods in other method embodiments provided by the present invention. The only difference is that corresponding functional modules are set. The principle is basically the same as that of the above system embodiments provided by the present invention. As long as those skilled in the art can improve the modules in the above system embodiments by referring to the specific technical solutions in other method embodiments and combining technical features to obtain corresponding technical means and technical solutions composed of these technical means, on the basis of the above system embodiments, and on the premise of ensuring the practicality of the technical solutions, they can obtain corresponding system-like embodiments for implementing the methods in other method-like embodiments.

[0045] In summary, this invention proposes a full-spectrum three-dimensional thermal expansion model: Fourier spectrum analysis is performed on the time series of surface temperature, incorporating both seasonal and non-seasonal temperature variations into the three-dimensional heat conduction and elasticity coupling equations. The three components of thermoelastic displacement are uniformly solved within a spherical framework, and joint corrections and evaluations are performed with GNSS observation sequences. Based on global GNSS reference station data from 2000 to 2024, this invention quantitatively assesses the contribution of thermal expansion effects caused by temperature changes globally, verifying the model's effectiveness in reducing nonlinear divergence, improving reference frame stability, and enhancing geophysical signal separation.

[0046] The terms “comprising” and “having”, and any variations thereof, in the specification, claims, and accompanying drawings of this invention are intended to cover a non-exclusive inclusion, such as a process, method, system, product, or apparatus that includes a series of steps or units, not necessarily limited to those explicitly listed, but may include other steps or units not explicitly listed or inherent to such processes, methods, products, or apparatus.

[0047] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, and not to limit them; although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some or all of the technical features therein; and these modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the technical solutions of the embodiments of the present invention.

Claims

1. A GNSS nonlinear deformation correction method based on a full-spectrum three-dimensional thermoelastic model, characterized in that, include: Discrete Fourier decomposition is performed on the temperature time series of each grid point to obtain multiple discrete angular frequency components; For each frequency after decomposition, perform spatial spherical harmonic expansion and coefficient projection; Based on the spatial spherical harmonic expansion and coefficient projection results, and considering the periodicity of temperature, a frequency-by-frequency temperature term is represented by sine / cosine. Based on the spatial spherical harmonic expansion and coefficient projection results, the displacement field of the deformation is represented spherically harmonically and expanded over time according to the corresponding periodic temperature. Solving the deformation equation in spherical coordinates using the Love number form yields the complex Love number expression, which explicitly depends on the coefficient of thermal expansion, thermal diffusivity, and Poisson's ratio, and the complex Love number at each frequency independently includes a phase lag factor; Based on the aforementioned complex spherical harmonic coefficients and complex Love number at temperature, the full spectrum synthesis of the three-component displacement at the sphere is performed by linear superposition of each frequency. The three components include vertical displacement, north-south displacement, and east-west displacement. Based on the full-spectrum synthesis results, internal consistency and external dispersion verification are performed.

2. The GNSS nonlinear deformation correction method based on a full-spectrum three-dimensional thermoelastic model according to claim 1, characterized in that, Discrete Fourier decomposition is performed on the temperature time series for each grid point, including: set up This is a record of the temperature at the spherical grid points over time, where... These are radial coordinates in a spherical coordinate system. and Let t represent the latitude and longitude of the observation station, respectively, and t be the time. Perform a discrete Fourier transform on each grid point to decompose it into discrete angular frequencies. The frequency components are used to obtain the amplitude. and And phase information.

3. The GNSS nonlinear deformation correction method based on a full-spectrum three-dimensional thermoelastic model according to claim 2, characterized in that, For each frequency after decomposition, perform spatial spherical harmonic expansion and coefficient projection, including: For each frequency ,Will and According to the fully normalized spherical harmonics Expand; Depend on , and The spherical harmonic coefficients are obtained by projecting the inner product of the sphere. , and give The expression of Legendre polynomials, in which, For a fully normalized spherical harmonic function, and Let and represent the order and degree of the spherical harmonic function, respectively. This indicates the complex conjugate operation.

4. The GNSS nonlinear deformation correction method based on a full-spectrum three-dimensional thermoelastic model according to claim 3, characterized in that, Based on the periodicity of temperature, a frequency-by-frequency sine / cosine representation of the temperature term is performed, including: Considering the periodicity of temperature, the temperature term for each Fourier frequency can be written in cosine / sine form. ， in, and They represent frequencies respectively. The cosine and sine components of the temperature signal. The sampling frequency period.

5. The GNSS nonlinear deformation correction method based on a full-spectrum three-dimensional thermoelastic model according to claim 3, characterized in that, Based on the spatial spherical harmonic expansion and coefficient projection results, corresponding to the periodic temperature, the displacement field of the deformation is represented spherically harmonically and expanded over time, including: use Indicates deformation For the corresponding periodic temperature, perform a Fourier expansion of the displacement to give the cosine / sine terms of the radial / tangential components at each frequency. ， ， in, and These represent the radial displacement coefficient and the tangential displacement coefficient, respectively. and These represent the cosine and sine components of the radial displacement at a given frequency, respectively. and These represent the cosine and sine components of the tangential displacement at a given frequency, respectively. The sampling frequency period.

6. The GNSS nonlinear deformation correction method based on a full-spectrum three-dimensional thermoelastic model according to claim 4, characterized in that, Solving the transformed equations in spherical coordinates using the Love number form yields the expression for complex Love numbers, including: Solving the transformed equations in spherical coordinates using the Love number form yields the complex Love number. , The expression of depends explicitly on the coefficient of thermal expansion, thermal diffusivity, and Poisson's ratio.

7. The GNSS nonlinear deformation correction method based on a full-spectrum three-dimensional thermoelastic model according to claim 1, characterized in that, Based on the aforementioned complex spherical harmonic coefficients and complex Love number at temperature, a full-spectrum synthesis of the three-component displacement at the spherical surface is performed, including: By pairing and linearly superimposing the complex spherical harmonic coefficients of temperature with the complex Love number at various frequencies, the geocentric radius can be obtained. Three-dimensional displacement components at the location: vertical displacement North-south displacement East-west displacement .

8. The GNSS nonlinear deformation correction method based on a full-spectrum three-dimensional thermoelastic model according to claim 1, characterized in that, Based on the full-spectrum synthesis results, internal consistency verification is performed, including: The temperature range of each station is divided into several intervals; the mean and standard deviation are calculated using the model displacement within each interval, and the empirical boundary is defined by the mean ± 2σ; the accuracy is calculated by the proportion of model displacement falling into each boundary; a consistently low standard deviation and a mean close to 0 indicate that the model is stable and the system bias is small; an accuracy greater than a set threshold indicates high fidelity.

9. The GNSS nonlinear deformation correction method based on a full-spectrum three-dimensional thermoelastic model according to claim 1, characterized in that, Based on the full-spectrum synthesis results, external dispersion verification is performed, including: After aligning with the GNSS time series, tectonic trends, step loads, and non-tidal loads are removed, and the three-dimensional thermoelastic response displacement is subtracted. The WRMS reduction rate of GNSS nonlinear changes is statistically analyzed to quantify the explanatory power of the model for temperature-driven deformation.

10. A GNSS nonlinear deformation correction system based on a full-spectrum three-dimensional thermoelastic model, characterized in that, include The first main module is used to perform discrete Fourier decomposition on the temperature time series of each grid point to obtain multiple discrete angular frequency components. The second main module is used to perform spatial spherical harmonic expansion and coefficient projection on each frequency after decomposition. The third main module is used to perform sine / cosine representation of the temperature term frequency by frequency based on the spatial spherical harmonic expansion and coefficient projection results, taking into account the periodicity of temperature. The fourth main module is used to perform spherical harmonic representation and time expansion of the displacement field of deformation based on the spatial spherical harmonic expansion and coefficient projection results, corresponding to the periodic temperature. The fifth main module is used to solve the deformation equation in spherical coordinates in the form of Love numbers to obtain the complex Love number expression. The complex Love number explicitly depends on the coefficient of thermal expansion, thermal diffusivity and Poisson's ratio, and the complex Love number at each frequency independently contains a phase lag factor. The sixth main module is used to synthesize the full spectrum of the three-component displacement at the sphere by linear superposition of the frequencies based on the complex spherical harmonic coefficients and complex Love number at the aforementioned temperature. The three components include vertical displacement, north-south displacement and east-west displacement. The seventh main module is used to verify internal consistency and external dispersion based on the full spectrum synthesis results.

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