Block-level air quality dynamic traceability method and system based on domestic credit and innovation platform

Abstract

The invention discloses a block-level air quality dynamic traceability method and system based on a domestic credit and creativity platform, belongs to the technical field of block-level air quality dynamic traceability, integrates 12 types of data sources of an air quality monitoring station, a traffic checkpoint and an industrial internet of things, supports minute-level data updating, carries a three-level nested grid parallel computing architecture, and provides a dynamic traceability system for the block-level air quality. The method supports bidirectional data coupling and distributed calculation of 10km, 3km and 250m grids, comprises a three-dimensional value surface rendering component and a dynamic particle tracking component, realizes pollution cloud picture rendering delay less than or equal to 500ms and particle tracking precision less than or equal to 50m, sets a threshold trigger type early warning mechanism, and automatically generates management and control suggestions in combination with a pollution contribution decoupling result.

Description

Technical Field

[0001] The present invention relates to a dynamic tracing method for block-level air quality, and in particular to a dynamic tracing method and system for block-level air quality based on a domestic information and communication technology platform, belonging to the technical field of dynamic tracing technology for block-level air quality. Background Art

[0002] In the existing technology, there is an imbalance between the demand for 50-meter-level modeling and the limited computing power supply of imported hardware, the conflict between the minute-level weight update requirement and model convergence, and the gap between the theoretical computing power and actual utilization of domestic chips. These contradictions are particularly prominent in the Pearl River Delta and Yangtze River Delta urban agglomeration areas. Traditional imported GPU solutions have high daily computing energy consumption and are difficult to guarantee data security. This urgently requires the development of a new generation of independent and controllable high-precision traceability technology system. The proposal of this invention is based on this major technical demand. Summary of the Invention

[0003] The main purpose of this invention is to provide a block-level air quality dynamic tracing method and system based on a domestic information and communication technology platform.

[0004] The purpose of the present invention can be achieved by adopting the following technical solutions:

[0005] A block-level air quality dynamic traceability system based on a domestically produced information and communication platform. The multi-source data acquisition module integrates 12 data sources, including air quality monitoring stations, traffic checkpoints, and the Industrial Internet of Things, and supports minute-level data updates.

[0006] Grid computing engine: equipped with a three-level nested grid parallel computing architecture, supporting bidirectional data coupling and distributed computing for 10km, 3km, and 250m grids;

[0007] Dynamic visualization terminal: includes 3D value surface rendering components and dynamic particle tracking components, achieving pollution cloud rendering delay ≤ 500ms and particle tracking accuracy ≤ 50m;

[0008] Early warning and disposal module: Set up a threshold-triggered early warning mechanism and automatically generate control suggestions based on the pollution contribution decoupling results.

[0009] Preferably, the grid computing engine adopts quadrilateral grid partitioning technology, sets rotatable subgrids in the 250m grid at the block layer, and supports a subgrid cutting allocation algorithm.

[0010] Preferably, the dynamic visualization terminal further includes:

[0011] 3D value surface rendering component, using MarchingCubes algorithm to achieve 3D modeling of pollution cloud map;

[0012] Dynamic particle tracking component, tracking pollutant transmission paths based on the Runge-Kutta 4th-order method;

[0013] The heat map generation component uses Kriging spatial interpolation technology to visualize the spatial distribution of pollution concentration.

[0014] Preferably, the data sources integrated by the multi-source data acquisition module include: topographic data, urban canopy data, meteorological driving data, emission data, chemical boundary data, road network data, population data, and shipping data.

[0015] Preferably, the early warning and disposal module generates control suggestions by constructing a pollution contribution decoupling optimization equation, and the optimization equation is: min||WX-Y|| 2 ;

[0016] The constraints are:

[0017] α+β+γ=1;

[0018] 0≤α≤1;

[0019] 0≤β≤1;

[0020] 0≤γ≤1;

[0021] 0≤X;

[0022] Where W is the dynamic weight matrix of local contribution, regional transmission, and background concentration;

[0023] β is the impact weight of regional transmission;

[0024] α is the influence weight of local contribution;

[0025] γ is the influence weight of background concentration;

[0026] WX-Y is the difference between the model-predicted contribution concentration and the observed concentration;

[0027] min represents the minimum difference of the objective function, that is, the optimal solution;

[0028] α, β, and γ must be greater than 0 and their sum must be 1;

[0029] X is the pollution contribution concentration vector, which must be greater than 0, and Y is the observed concentration vector.

[0030] Preferably, the dynamic weight matrix W is calculated by the following formula:

[0031]

[0032] Where α is the comprehensive weight of local emissions and meteorological conditions, 0.6 is the empirical parameter of emissions, ranging from 0.5 to 0.7, 0.4 is the empirical parameter of wind field vector, ranging from 0.3 to 0.5, and E industry is the emission from industrial sources in the region, E total is the total emission in the area, V is the wind speed, and t is the time;

[0033]

[0034] Where β is the comprehensive weight of the boundary layer height and regional transport process, H PBL is the boundary layer height, 2000 is the normalized reference value of the boundary layer height, 0.5 is the empirical parameter for the weight distribution of the boundary layer height and regional transmission, θ is the angle between the wind direction and the target area, and t is the time;

[0035] γ(t)=1-α(t)-β(t);

[0036] Through weight normalization, we ensure that α+β+γ≡1 and satisfy the weight constraint.

[0037] Preferably, it also includes a GPU accelerated computing cluster, which achieves a full-domain computing time of ≤2 hours based on domestically produced trusted computing servers.

[0038] A block-level air quality dynamic tracing method based on a domestically produced information technology platform includes the following steps:

[0039] Grid division: Construct a three-level nested grid system of 10km, 3km, and 250m, and establish a bidirectional data coupling channel between grids;

[0040] Data collection: Integrate 12 types of data sources to obtain minute-by-minute updated air quality monitoring data, meteorological data, and emission data;

[0041] Dynamic weight calculation: Based on the emission-meteorological coupling equation, the local contribution, regional transmission, and background concentration weight matrices are generated in real time and updated at the minute level;

[0042] Pollution decoupling: Analyze the contribution of each pollution source to the target area through an optimization algorithm with constraints;

[0043] Visualization and early warning: Dynamically display pollution distribution and transmission paths through 3D rendering and particle tracking technology, trigger early warnings based on thresholds, and generate control recommendations.

[0044] Preferably, in the dynamic weight calculation step, a dynamic correction term for mobile source emission factors is introduced:

[0045]

[0046] Where t is time, fweek is the weekly emission factor for mobile sources.

[0047] Preferably, the pollution decoupling step adopts a sequential quadratic programming algorithm or an alternating direction multiplier method to solve the optimization equation, and verifies the uniqueness of the solution through Monte Carlo simulation.

[0048] Beneficial technical effects of the present invention:

[0049] The present invention provides a block-level air quality dynamic tracing method and system based on a domestic information and innovation platform, which adopts a three-level nested grid of 10km (regional layer)-3km (city layer)-250m (block layer) to realize the dynamic modeling of block-level (250m) air quality for the first time. Compared with the traditional kilometer-level grid, the morphological representation accuracy of complex terrains such as industrial park boundaries and urban streets is improved by 4-16 times, solving the problem of fuzzy positioning of micro-scale pollution sources (such as single factories and traffic intersections) in traditional models.

[0050] Bidirectional data coupling technology enables refined interaction between grids at different levels. For example, regional meteorological fields drive chemical transmission at the urban level, and block-level pollution sources infer regional contribution weights, ensuring consistency and accuracy of cross-scale simulations.

[0051] Distributed parallel computing improves computing efficiency. Based on the domestically produced GPU acceleration cluster and combined with the sub-grid cutting allocation algorithm, it achieves a global computing time of ≤2 hours. Compared with traditional imported GPU solutions, the computing efficiency is increased by 30%, and the daily energy consumption is reduced by 40%, meeting the high-concurrency computing needs of urban agglomerations such as the Pearl River Delta and the Yangtze River Delta.

[0052] The distributed computing architecture supports minute-level data updates and responds in real time to dynamic source intensity changes such as industrial emissions and traffic flows. It solves the problem of a surge in tracing errors in fixed-weight models under complex conditions such as the periphery of typhoons and calm weather (the error is reduced to ≤15%). BRIEF DESCRIPTION OF THE DRAWINGS

[0053] Figure 1 A schematic diagram of three-level grid coupling calculation according to a preferred embodiment of a block-level air quality dynamic tracing method and system based on a domestically produced information innovation platform of the present invention;

[0054] Figure 2 A flow chart of the dynamic weight calculation steps of a preferred embodiment of a block-level air quality dynamic tracing method and system based on a domestically produced information innovation platform according to the present invention;

[0055] Figure 3 A pollution contribution decoupling interface for a preferred embodiment of a block-level air quality dynamic tracing method and system based on a domestically produced information innovation platform according to the present invention;

[0056] Figure 4 A dynamic tracing diagram of the transmission path of a preferred embodiment of a block-level air quality dynamic tracing method and system based on a domestically produced information innovation platform according to the present invention;

[0057] Figure 5 A contribution pie chart of a preferred embodiment of a block-level air quality dynamic tracing method and system based on a domestically produced information innovation platform according to the present invention;

[0058] Figure 6 This is a diagram of a key source warning list according to a preferred embodiment of a block-level air quality dynamic tracing method and system based on a domestically produced information innovation platform of the present invention;

[0059] Figure 7 This is a system hardware architecture diagram of a preferred embodiment of a block-level air quality dynamic tracing method and system based on a domestic information and innovation platform according to the present invention. DETAILED DESCRIPTION

[0060] In order to make the technical solution of the present invention more clear and specific to those skilled in the art, the present invention is further described in detail below with reference to embodiments and drawings, but the embodiments of the present invention are not limited thereto.

[0061] This system includes the following core modules:

[0062] Multi-level grid coupler: achieves three-level nesting of 10km→3km→250m

[0063] Dynamic weight calculation engine: update weight parameters in minutes

[0064] Pollution fingerprint separator: parsing local / regional / background contributions

[0065] GPU accelerated computing cluster: domestic information innovation server cluster

[0066] The present invention uses a sub-grid cutting and allocation algorithm and a quadrilateral grid partitioning technique to establish a dynamic weight coefficient matrix:

[0067]

[0068] Where α is the comprehensive weight of local emissions and meteorological conditions, 0.6 is the empirical parameter of emissions, ranging from 0.5 to 0.7, 0.4 is the empirical parameter of wind field vector, ranging from 0.3 to 0.5, and E industry is the emission from industrial sources in the region, E total is the total emission in the area, V is the wind speed, and t is the time.

[0069]

[0070] Where β is the comprehensive weight of the boundary layer height and regional transport process, H PBL is the boundary layer height, 2000 is the normalized reference value of the boundary layer height, 0.5 is the empirical parameter for the weight distribution of the boundary layer height and regional transmission, θ is the angle between the wind direction and the target area, and t is the time.

[0071] γ(t)=1-α(t)-β(t)

[0072] Through weight normalization, we ensure that α+β+γ≡1 and satisfy the weight constraint.

[0073]

[0074] Where W is the dynamic weight matrix of local contribution, regional transmission, and background concentration, β is the influence weight of regional transmission, α is the influence weight of local contribution, γ is the influence weight of background concentration, t is time, and T represents the transpose.

[0075] Pollution contribution decoupling method, constructing an optimization equation with constraints:

[0076] min||WX-Y|| 2

[0077] The constraints are:

[0078] α+β+γ=1

[0079] 0≤α≤1

[0080] 0≤β≤1

[0081] 0≤γ≤1

[0082] 0≤X

[0083] Where W is the dynamic weight matrix of local contribution, regional transmission, and background concentration, β is the impact weight of regional transmission, α is the impact weight of local contribution, γ is the impact weight of background concentration, WX-Y is the difference between the model predicted contribution concentration and the observed concentration, min represents the minimum difference of the objective function, that is, the optimal solution, α, β and γ must be greater than 0 and their sum is 1, X is the pollution contribution concentration vector, X must be greater than 0, and Y is the observed concentration vector.

[0084] The sequential quadratic programming (SQP) algorithm is used to solve the problem, and the uniqueness of the solution is verified by Monte Carlo simulation.

[0085] Real-time traceability visualization system

[0086] Develop a 3D pollution cloud rendering engine. The key technical parameters are shown in Table 1 below:

[0087] Module Technical indicators Implementation Dynamic Path Tracing Particle tracking accuracy ≤50m Parallel computing architecture Heatmap generation Rendering delay ≤ 500ms WebGL2.0 accelerated rendering Contribution Dashboard Data refresh frequency 1Hz Websocket real-time data push

[0088] Example 1 (Industrial Park Traceability)

[0089] Step 1: Mesh division;

[0090] Generate a 5km radius analysis domain with Foshan Shunde Industrial Park as the center

[0091] Establish 852 250m quadrilateral subgrids

[0092] Step 2: Data Collection

[0093] Call relevant data interfaces and model output results to obtain the required basic data:

[0094] Industrial emission source data: including emission quantity, emission species, and emission location information of all industrial sources in the industrial park;

[0095] Traffic activity data: Access historical traffic flow data provided by the AutoNavi Map API, including traffic volume, speed, and road-level information, to estimate traffic source emissions;

[0096] Model result data: Load the output result file of the model corresponding to the traceability area to obtain the simulated meteorological element data, including wind speed, wind direction, temperature, and boundary layer height variables.

[0097] Step 3: Dynamic weight calculation

[0098] Based on the meteorological results simulated by the model and the factors affecting the intensity of emission source activity, the contribution weight coefficients of different pollution sources are dynamically calculated. The weights change over time, reflecting the differences in the dominant factors at different times. For example, when the weight coefficient α = 0.58 reflects that industrial emissions are at their peak during this period and have the greatest contribution to pollution, the weight coefficient β = 0.29 reflects the influence of the dominant wind direction (southeast wind) on the regional transport of pollutants during this period. The weight coefficient γ = 0.13 represents the combined weight of other influencing factors (background concentration).

[0099] Step 4: Pollution Decoupling

[0100] Based on data collection, model simulation and dynamic weight calculation, qualitative and quantitative analysis of the source composition of pollutants in the industrial park:

[0101] Local contribution: 63.2% (including 31.7% from the injection molding industry)

[0102] Regional transmission: 28.1% (Guangzhou Huangpu Port Area)

[0103] Background concentration: 8.7%

[0104] Step 5: Visualize the Output

[0105] The results of pollutant source analysis are presented in a visual form, including:

[0106] Generate 3D dynamic tracing animation: Dynamically display the impact of pollutant emission, transmission, diffusion and accumulation processes on industrial parks in 3D space;

[0107] Identify key controlled enterprises: Identify key controlled enterprises in the industrial park that contribute most to local pollution, and visually display their locations and the spatiotemporal impact of their emissions on surrounding concentrations.

[0108] Example 2 (Analysis of Traffic Pollution);

[0109] Special treatment:

[0110] Introducing real-time road network data with an update frequency of 5 minutes

[0111] Add dynamic correction items to mobile source emission factors:

[0112]

[0113] Where λ is the correction coefficient of mobile source emission factor, 1 is the base value of the correction coefficient of mobile source emission factor, 0.3 is the daily fluctuation range of mobile source emission, t is time, f week is the weekly emission factor of mobile sources, and 0.2 is the influence coefficient of the weekly emission factor of mobile sources.

[0114] Construct a three-level nested grid system at the regional level (10km), city level (3km), and block level (250m), and establish a bidirectional data coupling channel between grids;

[0115] At the block level, quadrilateral subgrid division technology is used, and each subgrid unit contains 12 rotatable subgrids;

[0116] Develop a dynamic weight calculation engine that integrates emission intensity data, three-dimensional meteorological data, and urban morphology parameters in real time to generate a weight matrix that is updated minute by minute;

[0117] A sequential quadratic programming algorithm with constraints is used to decouple the pollution contribution.

[0118] Deploy GPU-accelerated computing clusters to achieve full-domain computing time of ≤2 hours.

[0119] The working method of the dynamic weight calculation engine includes:

[0120] Establish an emission-meteorological coupling equation to dynamically adjust the proportion of industrial source emissions and the influence of wind speed direction, reflecting the pollution diffusion characteristics under different meteorological conditions in real time. For example, in calm weather, industrial emissions dominate; in strong winds, wind speed direction significantly affects the transmission path of pollutants. The specific calculation formula is as follows:

[0121]

[0122] Where α is the emission-meteorological coupling correction factor, K1 is the emission empirical parameter with a range of 0.5-0.7, K2 is the wind field vector empirical parameter with a range of 0.3-0.5, Qindustry is the industrial source emission in the region, Q total is the total emission in the area, V is the wind speed, and θ is the angle between the wind direction and the grid boundary.

[0123] The boundary layer height correction factor is introduced to quantify the impact of vertical diffusion capacity on pollution transmission. For example, when the boundary layer height is low, the correction factor increases the weight of regional transmission and reflects the pollutant retention effect. The specific calculation formula is as follows:

[0124]

[0125] Where β is the boundary layer height correction factor, PBLH is the boundary layer height, i and j are grid indices, and W i,j is the weight of the area transfer within the grid, E i,j is the emission intensity within the grid.

[0126] The pollution contribution decoupling calculation specifically includes:

[0127] Construct the optimization objective function, which consists of two parts: data fitting term |WX-Y| 2 and the total variation regularization term TV(X).

[0128] The data fitting term ensures that the predicted pollution contribution is as close as possible to the observed concentration, and the total variation regularization term constrains the continuity of the spatial distribution of the pollution contribution. At the same time, the trade-off parameter λ is considered to control the balance between the data fitting term and the total variation regularization term. The specific calculation formula is as follows:

[0129] min||WX-Y|| 2 +λ·TV(X)

[0130] Where W is the dynamic weight matrix of local, regional transmission, and background concentrations, X is the contribution concentration of each pollution source, Y is the actual observed concentration, WX-Y is the difference between the model-predicted contribution concentration and the observed concentration, min represents the minimum difference of the objective function, that is, the optimal solution, TV(X) is the constraint term, and λ is the trade-off parameter.

[0131]

[0132] Where TV(X) is the total variation regularization term, i.e. the constraint term, X i+1,j -X i,j and X i,j+1 -X i,jis the difference between the contribution concentrations of adjacent grids. The total variation regularization term makes the model-predicted contribution concentrations remain continuous in space.

[0133] The alternating direction method of multipliers (ADMM) is used for iterative solution. Through the decomposition-coordination iterative mechanism, the complex problem is decomposed into multiple sub-problems, and these sub-problems are optimized alternately to finally approach the global optimal solution.

[0134] Set convergence conditions to ensure that the algorithm efficiently and reliably approaches the optimal solution by quantifying constraint satisfaction and variable stability. The specific conditions are as follows:

[0135] ||X (K+1) -X K ||2≤10 -4 Where K is the number of iterative solutions.

[0136] An air quality dynamic tracing system, characterized by comprising:

[0137] Multi-source data acquisition module: integrates 12 types of data sources including air quality monitoring stations, traffic checkpoints, and industrial IoT;

[0138] Grid computing engine: supports parallel computing of three-level nested grids;

[0139] Dynamic visualization terminal: with pollution cloud map rendering, source path tracing, and contribution analysis dashboard functions;

[0140] Early warning and disposal module: Set threshold-triggered early warning and disposal suggestion generation functions.

[0141] The dynamic visualization terminal includes: a three-dimensional value surface rendering component: using the MarchingCubes algorithm;

[0142] Dynamic particle tracking component: based on Runge-Kutta 4th order method;

[0143] Heat map generation component: Apply Kriging spatial interpolation technology;

[0144] Cross-platform display component: supports WebGL and Unity3D multi-engine output.

[0145] The above is only a further embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any technician familiar with the technical field can replace or change the technical solution and concept of the present invention within the scope disclosed by the present invention, which falls within the scope of protection of the present invention.

Claims

1. A block-level air quality dynamic tracing system based on a domestically produced information technology platform, characterized by: Multi-source data acquisition module: Integrates 12 types of data sources including air quality monitoring stations, traffic checkpoints, and industrial IoT, and supports minute-level data updates; Grid computing engine: equipped with a three-level nested grid parallel computing architecture, supporting bidirectional data coupling and distributed computing for 10km, 3km, and 250m grids; Dynamic visualization terminal: includes 3D value surface rendering components and dynamic particle tracking components, achieving pollution cloud rendering delay ≤ 500ms and particle tracking accuracy ≤ 50m; Early warning and disposal module: Set up a threshold-triggered early warning mechanism and automatically generate control suggestions based on the pollution contribution decoupling results.

2. A block-level air quality dynamic tracing method based on a domestically produced information technology platform according to claim 1, characterized in that: The grid computing engine adopts quadrilateral grid partitioning technology, sets rotatable subgrids in the 250m grid at the block layer, and supports subgrid cutting and allocation algorithm.

3. The block-level air quality dynamic tracing system based on the domestically-produced information and innovation platform according to claim 1 is characterized by: The dynamic visualization terminal also includes: 3D value surface rendering component, using MarchingCubes algorithm to achieve 3D modeling of pollution cloud map; Dynamic particle tracking component, tracking pollutant transmission paths based on the Runge-Kutta 4th-order method; The heat map generation component uses Kriging spatial interpolation technology to visualize the spatial distribution of pollution concentration.

4. The block-level air quality dynamic tracing system based on the domestically produced information technology platform according to claim 1 is characterized by: The data sources integrated by the multi-source data acquisition module include: topographic data, urban canopy data, meteorological driving data, emission data, chemical boundary data, road network data, population data, and shipping data.

5. The block-level air quality dynamic tracing system based on the domestically-produced information and innovation platform according to claim 1 is characterized by: The early warning and disposal module generates control suggestions by constructing a pollution contribution decoupling optimization equation, which is: min|WX-Y| 2 ; The constraints are: α+β+γ=1; 0≤α≤1； 0≤β≤1； 0≤γ≤1; 0≤X； Where W is the dynamic weight matrix of local contribution, regional transmission, and background concentration; β is the impact weight of regional transmission; α is the influence weight of local contribution; γ is the influence weight of background concentration; WX-Y is the difference between the model-predicted contribution concentration and the observed concentration; min represents the minimum difference of the objective function, that is, the optimal solution; α, β, and γ must be greater than 0 and their sum must be 1; X is the pollution contribution concentration vector, which must be greater than 0, and Y is the observed concentration vector.

6. The block-level air quality dynamic tracing system based on the domestically-produced information and innovation platform according to claim 1 is characterized by: The dynamic weight matrix W is calculated by the following formula: Where α is the comprehensive weight of local emissions and meteorological conditions, 0.6 is the empirical parameter of emissions, ranging from 0.5 to 0.7, 0.4 is the empirical parameter of wind field vector, ranging from 0.3 to 0.5, and E industry is the emission from industrial sources in the region, E total is the total emission in the area, V is the wind speed, and t is the time; Where β is the comprehensive weight of the boundary layer height and regional transport process, H PBL is the boundary layer height, 2000 is the normalized reference value of the boundary layer height, 0.5 is the empirical parameter for the weight distribution of the boundary layer height and regional transmission, θ is the angle between the wind direction and the target area, and t is the time; γ(t)=1-α(t)-β(t); Through weight normalization, we ensure that α+β+γ≡1 and satisfy the weight constraint.

7. The method for dynamic tracing of block-level air quality based on a domestically-developed information and innovation platform according to claim 1 is characterized by: It also includes a GPU-accelerated computing cluster, which uses domestically produced trusted computing servers to achieve a full-domain computing time of ≤2 hours.

8. A block-level air quality dynamic tracing method based on a domestically produced trust-based innovation platform, based on the block-level air quality dynamic tracing system based on a domestically produced trust-based innovation platform described in claims 1-7, characterized in that: The following steps are involved: Grid division: Construct a three-level nested grid system of 10km, 3km, and 250m, and establish a bidirectional data coupling channel between grids; Data collection: Integrate 12 types of data sources to obtain minute-by-minute updated air quality monitoring data, meteorological data, and emission data; Dynamic weight calculation: Based on the emission-meteorological coupling equation, the local contribution, regional transmission, and background concentration weight matrices are generated in real time and updated at the minute level; Pollution decoupling: Analyze the contribution of each pollution source to the target area through an optimization algorithm with constraints; Visualization and early warning: Dynamically display pollution distribution and transmission paths through 3D rendering and particle tracking technology, trigger early warnings based on thresholds, and generate control recommendations.

9. The method for dynamic tracing of block-level air quality based on a domestically-developed information and innovation platform according to claim 1 is characterized by: In the dynamic weight calculation step, a dynamic correction term for mobile source emission factors is introduced: Where t is time, f week is the weekly emission factor for mobile sources.

10. The method for dynamic tracing of block-level air quality based on a domestically-produced information and innovation platform according to claim 1 is characterized by: The pollution decoupling step adopts a sequential quadratic programming algorithm or an alternating direction multiplier method to solve the optimization equation, and the uniqueness of the solution is verified by Monte Carlo simulation.

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