A multi-scale tidal open boundary optimization and prediction method

Abstract

The present application belongs to the technical field of shallow sea environment data reanalysis and prediction, and relates to a multi-scale tidal open boundary optimization and prediction method. The present application optimizes the harmonic constant of the open boundary of the tidal current of the numerical model, calculates the adjustment coefficient factor corresponding to the harmonic constant and the Fourier series expansion thereof based on the observation data, seeks the optimal tidal open boundary condition, reduces the tidal simulation error, and provides higher-precision tidal current prediction data to provide data support for engineering decision-making such as production and life and transportation. The optimization is performed recursively in stages, thereby greatly reducing the required calculation amount and improving the optimization effect.

Description

Technical Field

[0001] This invention belongs to the field of shallow sea environmental data reanalysis and forecasting technology. The invention relates to a multi-scale tidal open boundary optimization and forecasting method. Background Technology

[0002] As one of the most important phenomena in the ocean, tides and currents have a significant impact on human production, daily life, transportation, and trade. Early tide forecasting involved long-term observation of water levels at a single point, followed by harmonic analysis of the observed values ​​to obtain the amplitude and lag angle of the corresponding tidal constituents, and then inverting this to calculate the tidal level. This method is only applicable to nearshore waters with available observation data and cannot forecast tides and currents in open sea areas, presenting a significant limitation. With the development of modern computer technology and numerical models, solving discrete tidal numerical models, such as the barotropic two-dimensional shallow water wave equations, allows for the simulation and forecasting of tides and currents in open sea areas. However, the primary challenge of this method is providing reasonable open boundary conditions for the numerical models. In particular, tidal current data is difficult to observe and cannot be directly obtained from observations in open sea areas. Currently, the open boundary conditions for tides and currents in local sea areas are mostly provided by tidal harmonic constants calculated by global models. In order to apply the harmonic constants in the global model to the regional model and obtain better forecast results, the harmonic constants of each tidal constituent on the open boundary must be corrected.

[0003] Data assimilation methods can extract useful information from observational data and combine this information with numerical models to optimize model parameters, providing a theoretical basis for open-boundary optimization of tidal models. Currently widely used four-dimensional variational data assimilation methods not only require writing adjoint models for the numerical models used, which is a huge workload, but also require storing the integral results of the positive and adjoint models during optimization, consuming significant computational and storage resources. New-generation hybrid data assimilation methods, especially analytical four-dimensional ensemble variational data assimilation methods, combine the advantages of traditional four-dimensional variational methods and ensemble data assimilation methods. They estimate adjoint models from the evolution of ensemble members, effectively avoiding the writing of adjoint models while maintaining the optimization capability of four-dimensional variational methods for nonlinear problems within long time windows, thus reducing computational and storage resource consumption. This approach has great application potential and can be used for tidal open-boundary optimization. Furthermore, it should be noted that in tidal open-boundary optimization, the model open boundary typically consists of hundreds of grid points to be optimized, with eight to ten tidal constituents, resulting in thousands of optimization variables. Due to computational limitations, the set of patterns in typical data assimilation typically contains only around fifty members, far fewer than the number of variables requiring optimization. Therefore, traditional analytical four-dimensional set variational methods are insufficient to simultaneously optimize all variables. Summary of the Invention

[0004] To overcome the shortcomings of existing technologies, this invention provides a multi-scale tidal open boundary optimization and forecasting method. By specially constructing a matching modal function, i.e., a Fourier series, and iteratively optimizing the weighting scheme of the modal function (i.e., the coefficients corresponding to the Fourier series), the required number of set members is reduced, thus improving algorithm efficiency. Therefore, addressing the difficulty of traditional data assimilation methods in optimizing the harmonic constant of tidal open boundaries, this invention proposes an optimization scheme adapted to analytical four-dimensional ensemble variational methods, reducing computational load and improving the simulation and forecasting capabilities of tidal currents.

[0005] The above-mentioned objective of this invention is achieved through the following technical solution: a multi-scale tidal open boundary optimization and forecasting method, characterized by comprising the following steps: S1. Extract the open boundary harmonic constants of the model to be optimized from the global model or the large regional model: Perform conventional harmonic analysis on the water level series output from the existing large regional model, or obtain the harmonic constants of the tidal constituents from the publicly available global tidal data and interpolate them onto the grid of the tidal model as the open boundary water level data of the model to be optimized. S2. Tidal Model Simulation: Based on the provided open-boundary tidal data, a static initial field is used to simulate each tidal constituent one by one to ensure the stability of the numerical model during the simulation process; the simulation duration ensures that the model is stable for at least two corresponding tidal constituent cycles to calculate the harmonic constants. S3. Construct the Fourier function of the adjustment factor: Determine the adjustment scale of the adjustment factor of the open boundary tidal harmonic constant to be optimized, and thus determine the period of the corresponding Fourier function; Perform small perturbations on the Fourier coefficients of the determined background value to determine the form of the adjustment factor of the open boundary tidal current harmonic constant. S4. Optimize the coefficients of the Fourier function with a determined period: In the initial iteration, assuming the parameter is 0 as the initial value, construct the adjustment coefficient vector of the tidal current to be optimized with the perturbed Fourier coefficients, and adjust the tidal current open boundary based on the coefficient vector. Then, bring the obtained open boundary into the A-4DEnVar framework, calculate the corresponding cost function value and its corresponding gradient value, and use A-4DEnVar to optimize the perturbed Fourier coefficients. S5. Recursively optimize the next period scale in the Fourier series: Using the scheme of the previous step, optimize the coefficients of the sine and cosine functions of different periods in the Fourier series in turn until the convergence condition is met, so that the error between the simulation results and the observed values ​​is within an acceptable range. S6. Tidal Open Boundary Condition Reconstruction: Output the optimized tidal and current flow harmonic constants, calculate the open boundary tidal and current flow components from the harmonic constants, and compare them with existing results to determine the effectiveness of the optimization.

[0006] The publicly available global tidal data includes tidal models provided by the National Astronomical Observatory of Japan and TOPEX / Poseidon satellite assimilation tidal model data provided by Oregon State University.

[0007] In the tidal model simulation, the water level and velocity at the open boundary are calculated using harmonic constants. Specifically, for a single point j at the open boundary, if the tidal current velocity V encompasses all tidal components... j Then we have: (1) Here, i is the tidal index, and h... ij Tidal amplitude, σ i Tidal frequency, v 0ij Initial phase, f i g is the intersection factor corresponding to the tidal boundary. i Correcting the angle for the intersection point; in addition to the standard components in the harmonic analysis above, this includes α. ij As the amplitude adjustment factor for this tidal constituent and θ ij As a delay angle adjustment factor; For a single tidal constituent, the above formula can be expressed as: (2) At the initial moment of optimization, the amplitude adjustment factor and the lag angle adjustment factor are set to 0, and different open boundary points have different adjustment factors.

[0008] In the Fourier function that constructs the adjustment factor, the factor is decomposed using a Fourier series, expressed as: (3) Where α is the constant term in the Fourier expansion, k is the index of the Fourier series term, and K is the series cutoff length. and These are the coefficients of the Fourier series, and T is the period length of the adjustment factor, which is directly set to the length of the open boundary in open boundary optimization; expanding the delay angle adjustment factor, we get: (4) Combining equations (1) to (4), it can be seen that to optimize the flow velocity in equation (1), it is only necessary to optimize the coefficients corresponding to the Fourier series for each tidal fraction. Since the corresponding terms of the Fourier series are orthogonal, the optimization process first optimizes α and θ, and then optimizes the coefficients corresponding to the Fourier series for each tidal fraction. Optimize it; for the k-th scale of a single tidal constituent, at most only 4 set members are needed.

[0009] In the optimization of the coefficients of the Fourier function with a determined period, it is assumed that the state variable x in the mode at time t... t satisfy: (5) Where M(*) is the model nonlinear operator, x0 is the model initial field, and λ is the model parameter; in the tidal model to be optimized, parameter λ is the coefficient of the Fourier series corresponding to the scale to be optimized. ; Perturb the parameter λ and construct the parameter perturbation member set. Substituting these parameters into the numerical model and running the numerical model through integration, we obtain a set of state variables for the set members as follows: (6) Let the disturbances of the state variables and parameters be respectively and Then we can obtain: (7) M here t (x 0, λ) is the tangent linear mode corresponding to the pattern, M t (x 0, The transpose of λ is the adjoint mode; by estimating the adjoint mode, we can obtain... (8) The cost function for parameter increments in the A-4DEnVar framework is: (9) Among them, H i For linearizing the observation operator, d i The observation increment is given; the gradient of the cost function is: (10) To obtain the minimum value of the cost function, we only need to start with a suitable initial value, calculate the cost function value and the gradient of the cost function in sequence, and use optimization algorithms such as gradient descent and quasi-Newton methods. Of course, after completing the optimization of the Fourier series corresponding coefficients at the current scale, we retain the current results and optimize the Fourier coefficients at the next scale based on the current scale.

[0010] In the process of optimizing the coefficients of the Fourier function with a defined period, the cost function decreases continuously as the data assimilation process proceeds until it reaches a minimum value. At this point, the tidal simulation reaches its optimal state, and the optimization is complete.

[0011] The advantages of this invention compared to the prior art are: 1. By combining observational data with numerical models using the A-4DEnVar method, the accuracy of open boundary conditions for tidal current models is improved, resulting in more precise tidal current reanalysis products that provide data support for engineering decisions in production, daily life, transportation, and other fields.

[0012] 2. To address the issue of the large number of set members required in the implementation of A-4DEnVar, a Fourier series of adjustment factors is introduced for step-by-step recursive optimization, which greatly reduces the amount of computation required and improves the optimization effect. Attached Figure Description

[0013] Figure 1 This is a flowchart of the method of the present invention.

[0014] Figure 2 This is a flowchart of the data assimilation process based on the A-4DEnVar method of this invention. Detailed Implementation

[0015] The present invention will now be described in detail with reference to the accompanying drawings and specific embodiments, but this does not limit the scope of protection of the present invention. Example

[0016] like Figure 1 This paper presents a multi-scale tidal open-boundary optimization and forecasting method, which mainly includes three modules: observation data, tidal numerical model, and data assimilation. The data assimilation module combines the observation data with the tidal numerical model to improve the simulation effect of tidal currents. The method includes the following steps: S1. Extract the open boundary harmonic constants of the model to be optimized from global or large-area models: Perform routine harmonic analysis on the water level series output from existing large-area models, or obtain the harmonic constants of the tidal constituents from publicly available global tidal data, such as NAO (National Astronomical Observatory of Japan) and TPXO (TOPEX / Poseidon satellite assimilation tidal model provided by Oregon State University), and interpolate them onto the grid of the tidal model as the open boundary water level data of the model to be optimized; calculate the water level and velocity at the open boundary using the harmonic constants. Specifically, for a single point j at the open boundary, if the tidal velocity V includes all tidal constituents... j Then we have: (1) Here, i is the tidal index, and h... ij Tidal amplitude, σ i Tidal frequency, v 0ij Initial phase, f i g is the intersection factor corresponding to the tidal boundary. i Correcting the angle for the intersection point; in addition to the standard components in the harmonic analysis above, this includes α. ij As the amplitude adjustment factor for this tidal constituent and θ ij As a delay angle adjustment factor; For a single tidal constituent, the above formula can be expressed as: (2) At the initial moment of optimization, the amplitude adjustment factor and the lag angle adjustment factor are set to 0, and different open boundary points have different adjustment factors.

[0017] S2. Tidal Model Simulation: Based on the provided open-boundary tidal data, a static initial field is used to simulate each tidal constituent one by one to ensure the stability of the numerical model during the simulation process; the simulation duration ensures that the model is stable for at least two corresponding tidal constituent cycles to calculate the harmonic constants. S3. Constructing the Fourier function of the adjustment factor: Determine the adjustment scale of the adjustment factor for the open-boundary tidal harmonic constant to be optimized, thereby determining the period of the corresponding Fourier function; perform small perturbations on the Fourier coefficients of the determined background value to determine the form of the adjustment factor for the open-boundary tidal harmonic constant; decompose the factor using a Fourier series, expressed as: (3) Where α is the constant term in the Fourier expansion, k is the index of the Fourier series term, and K is the series cutoff length. and These are the coefficients of the Fourier series, and T is the period length of the adjustment factor, which is directly set to the length of the open boundary in open boundary optimization; expanding the delay angle adjustment factor, we get: (4) Combining equations (1) to (4), it can be seen that to optimize the flow velocity in equation (1), it is only necessary to optimize the coefficients corresponding to the Fourier series for each tidal fraction. Since the corresponding terms of the Fourier series are orthogonal, the optimization process first optimizes α and θ, and then optimizes the coefficients corresponding to the Fourier series for each tidal fraction. Optimize it; for the k-th scale of a single tidal constituent, at most only 4 set members are needed.

[0018] S4. Optimize the coefficients of the Fourier function with a defined period: In the initial iteration, assuming the parameter is 0 as the initial value, construct the adjustment coefficient vector of the tidal current to be optimized with the open boundary using the perturbed Fourier coefficients, and adjust the tidal current open boundary based on this coefficient vector. Then, input the obtained open boundary into the A-4DEnVar framework, calculate the corresponding cost function value and its corresponding gradient value, and use A-4DEnVar to optimize the perturbed Fourier coefficients; see attached. Figure 2 As shown, assume that the state variable x in the model at time t is... t satisfy: (5) Where M(*) is the model nonlinear operator, x0 is the model initial field, and λ is the model parameter; in the tidal model to be optimized, parameter λ is the coefficient of the Fourier series corresponding to the scale to be optimized. ; Perturb the parameter λ and construct the parameter perturbation member set. Substituting these parameters into the numerical model and running the numerical model through integration, we obtain a set of state variables for the set members as follows: (6) Let the disturbances of the state variables and parameters be respectively and Then we can obtain: (7) M here t (x 0, λ) is the tangent linear mode corresponding to the pattern, M t (x 0, The transpose of λ is the adjoint mode; by estimating the adjoint mode, we can obtain... (8) Meanwhile, note that the cost function for parameter increments in the A-4DEnVar framework is: (9) Among them, H i For linearizing the observation operator, d i The observation increment is given; the gradient of the cost function is: (10) To obtain the minimum value of the cost function, we only need to start with a suitable initial value, calculate the cost function value and the gradient of the cost function in sequence, and use optimization algorithms such as gradient descent and quasi-Newton methods. Of course, after completing the optimization of the Fourier series corresponding coefficients at the current scale, we retain the current results and optimize the Fourier coefficients at the next scale based on the current scale.

[0019] S5. Recursively optimize for smaller scales: Using the scheme from the previous step, optimize the coefficients of the Fourier function with smaller periods in turn until the convergence condition is met, so that the error between the simulation results and the observed values ​​is within an acceptable range. After optimization, when using TPXO data, the cost function decreases by at least 60%, the tidal amplitude difference is less than 20cm, and the tidal lag angle difference is less than 18°.

[0020] S6. Tidal Open Boundary Condition Reconstruction: Output the optimized tidal and current flow harmonic constants, compare them with existing results, and determine the effectiveness of the optimization.

[0021] The embodiments described above are merely preferred embodiments of the present invention, and not all feasible embodiments of the present invention. Any obvious modifications made by those skilled in the art without departing from the principles and spirit of the present invention should be considered to be included within the scope of protection of the claims of the present invention.

Claims

1. A multi-scale tidal open boundary optimization and forecasting method, characterized in that, Includes the following steps: S1. Extract the open boundary harmonic constants of the model to be optimized from the global model or the large regional model: Perform conventional harmonic analysis on the water level series output from the existing large regional model, or obtain the harmonic constants of the tidal constituents from the publicly available global tidal data and interpolate them onto the grid of the tidal model as the open boundary water level data of the model to be optimized. S2. Tidal Model Simulation: Based on the provided open-boundary tidal data, a static initial field is used to simulate each tidal constituent one by one to ensure the stability of the numerical model during the simulation process; the simulation duration ensures that the model runs for at least two corresponding tidal constituent cycles after stabilization, which are used to calculate the harmonic constants; S3. Construct the Fourier function of the adjustment factor: Determine the adjustment scale of the adjustment factor of the open boundary tidal harmonic constant to be optimized, and thus determine the period of the corresponding Fourier function; Perform small perturbations on the Fourier coefficients of the determined background value to determine the form of the adjustment factor of the open boundary tidal current harmonic constant. S4. Optimize the coefficients of the Fourier function with a determined period: In the initial iteration, assuming the parameter is 0 as the initial value, construct the adjustment coefficient vector of the tidal current to be optimized with the perturbed Fourier coefficients, and adjust the tidal current open boundary based on the coefficient vector. Then, bring the obtained open boundary into the A-4DEnVar framework, calculate the corresponding cost function value and its corresponding gradient value, and use A-4DEnVar to optimize the perturbed Fourier coefficients. S5. Recursively optimize the next period scale in the Fourier series: Using the scheme of the previous step, optimize the coefficients of the sine and cosine functions of different periods in the Fourier series in turn until the convergence condition is met, so that the error between the simulation results and the observed values ​​is within an acceptable range. S6. Tidal Open Boundary Condition Reconstruction: Output the optimized tidal and current flow harmonic constants, calculate the open boundary tidal and current flow components from the harmonic constants, and compare them with existing results to determine the effectiveness of the optimization.

2. The multi-scale tidal open boundary optimization and forecasting method according to claim 1, characterized in that, The publicly available global tidal data includes tidal models provided by the National Astronomical Observatory of Japan and TOPEX / Poseidon satellite assimilation tidal model data provided by Oregon State University.

3. The multi-scale tidal open boundary optimization and forecasting method according to claim 1, characterized in that, In the tidal model simulation, the water level and velocity are calculated using harmonic constants at the open boundary. Specifically, for a single point j at the open boundary, if the tidal current velocity V encompasses all tidal components... j Then there is Here, i is the tidal index, and h... ij Tidal amplitude, σ i Tidal frequency, v 0ij Initial phase, fi is the intersection factor corresponding to the tide split, g i Correcting the angle for the intersection point; in addition to the standard components in the harmonic analysis above, this includes α. ij As the amplitude adjustment factor for this tidal constituent and θ ij As a delay angle adjustment factor; For a single tidal constituent, the above formula can be expressed as: V j =(1+a j )fh j cos[σt+v 0j -g+θ j ] (2) At the initial moment of optimization, the amplitude adjustment factor and the lag angle adjustment factor are set to 0, and different open boundary points have different adjustment factors.

4. The multi-scale tidal open boundary optimization and forecasting method according to claim 1, characterized in that, In the Fourier function that constructs the adjustment factor, the factor is decomposed using a Fourier series, expressed as: Where α is the constant term in the Fourier expansion, k is the index of the Fourier series term, and K is the series cutoff length. and These are the coefficients of the Fourier series, and T is the period length of the adjustment factor, which is directly set as the length of the open boundary in open boundary optimization; expanding the delay angle adjustment factor, we get: Since the corresponding terms of the Fourier series are orthogonal, the optimization process first optimizes α and θ, and then optimizes K = 1, 2, ... in sequence after obtaining the result; for the k-th scale of a single tidal constituent, at most 4 set members are required.

5. The multi-scale tidal open boundary optimization and forecasting method according to claim 1, characterized in that, In the optimization of the coefficients of the Fourier function with a determined period, it is assumed that the state variable x in the mode at time t... t satisfy: x t =M(x0,λ) (5) Where M(*) is the model nonlinear operator, x0 is the model initial field, and λ is the model parameter; in the tidal model to be optimized, the parameter λ is the coefficient of the Fourier series corresponding to the scale to be optimized. Perturb the parameter λ and construct the parameter perturbation member set λ. (1) ,…,λ (N) Substituting these parameters into the numerical model and running the numerical model through integration, we obtain a set of state variables for the set members as follows: Let the disturbances of the state variables and parameters be respectively and Then we can obtain: M here t (x0,λ) is the tangent linear pattern corresponding to the pattern, M t The transpose of (x0,λ) is the adjoint mode; by estimating the adjoint mode, we can obtain... The cost function for parameter increments in the A-4DEnVar framework is: Among them, H i For linearizing the observation operator, d i The observation increment is given; the gradient of the cost function is: To obtain the minimum value of the cost function, we only need to start with a suitable initial value, calculate the cost function value and the gradient of the cost function in sequence, and use optimization algorithms such as gradient descent and quasi-Newton methods.

6. The multi-scale tidal open boundary optimization and forecasting method according to claim 1, characterized in that, In the process of optimizing the coefficients of the Fourier function with a defined period, the cost function decreases continuously as the data assimilation process proceeds until it reaches a minimum value. At this point, the tidal simulation reaches its optimal state, and the optimization is complete.

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