A spatiotemporal tracing method for nutrient pollution of pollutants

Abstract

The invention discloses a spatiotemporal tracing method for nutrient pollution of pollutant dischargers, comprising the following steps: S1, constructing a bay nutrient pollution database; S2, rasterizing the watershed and constructing a spatial connection relationship matrix; S3, matching the pollution source with the grid; S4, constructing a profile biogeochemical process model of a river grid and a non-river grid to form a nutrient pollution amount matrix; S5, constructing a nutrient migration cost matrix; S6, using a multi-objective optimization algorithm to calculate a source-path combination with the highest nutrient migration efficiency; S7, matching the river grid with a monitoring section to perform model correction, and using an algorithm to achieve automatic parameter adjustment according to the error; S8, tracing back the main nutrient source-migration combination through a profile biogeochemical process model combined with an optimal source-path combination; S9, performing time series modeling on different nutrient source-migration combinations of different sections to obtain the spatial variation law of the nutrient pollution dischargers at different times.

Description

Technical Field

[0001] The present invention relates to the technical field of pollution source tracing, and in particular to a method for temporal and spatial source tracing of nutrient salt pollution for a pollutant discharger. Background Art

[0002] The existing research on pollution source tracing technology is mainly divided into two categories: experimental method and model method. Among them, the experimental method relies on the isotope method, combined with methods such as Bayesian network, and traces the source and migration process of nitrogen and phosphorus through the application of stable isotopes or radioactive isotopes; however, the experimental method has problems such as strong environmental heterogeneity, short spatiotemporal scale, and high analysis cost. The model method can be further divided into dynamic mechanism model and static equilibrium model. This method overcomes the limitations of the experimental method to a certain extent, and has significant advantages in the promotion and application of different river basins. However, the existing model method also has problems such as unclear pollution subjects, source-migration pathway separation, and future prediction. Therefore, the present invention proposes a spatiotemporal tracing method for nutrient pollution for pollution subjects, which can quickly and accurately analyze the pollution subjects of the excessive sections, and better formulate targeted nutrient pollution reduction policies according to the source-migration combination. Summary of the invention

[0003] The purpose of the present invention is to provide a spatiotemporal tracing method for nutrient pollution for pollutant dischargers. The method is flexible and easy to use, and can analyze the pollutant dischargers of excessive sections relatively quickly and accurately, and better formulate targeted nutrient pollution reduction policies based on source-migration combinations.

[0004] To achieve the above object, the present invention adopts the following technical solutions:

[0005] A method for temporal and spatial tracing of nutrient pollution for pollutant dischargers, comprising the following steps:

[0006] S1. Build a database of nutrient pollution in the bay;

[0007] S2, rasterize the watershed and construct a spatial connection relationship matrix to extract the river grid;

[0008] The specific process of step S2 is as follows: the watershed is divided into 500×500 m grids, and a spatial connection relationship matrix is ​​constructed using a Markov chain based on the topography; the Markov chain method based on topography uses DEM data to calculate the slope and aspect of different grids, and is implemented through a river grid extraction algorithm;

[0009] The slope calculation formula is as follows: , , ,in, Indicates slope; Indicates terrain height exist Directional gradient; Indicates terrain height exist Directional gradient; Represents a grid cell Height; Represents a grid cell Height; Represents a grid cell Height; Represents a grid cell Height; Represents the grid cell in The size of the direction, Represents the grid cell in The magnitude of the direction, i.e. and 500 m;

[0010] The slope aspect calculation formula is as follows: ,In the Markov chain, the transition probability is determined based on the slope and aspect relationships between grid cells;

[0011] The specific calculation formula of the transition probability matrix is ​​as follows: ,in, Represents the grid cell The probability of transferring to grid cell j; Represents the grid cell The slope of the transfer to j; is a smoothing parameter greater than zero, used to control the effect of slope on transition probability; Is a grid cell The neighborhood of includes all possible four-way or eight-way transfer directions; Indicates the first of four or eight directions of transfer grid cells; Represents a grid cell Transfer to The slope of the grid is calculated by the D8 algorithm, and the flow path is determined to extract the river grid.

[0012] S3. Allocate point source nutrient pollution and non-point source nutrient pollution to different grids through Kriging interpolation, model simulation and dynamic coefficient calculation to match the pollution sources with the grids;

[0013] S4. Construct the biogeochemical process model of the river grid and the non-river grid to form the nutrient pollution matrix;

[0014] S5. According to the spatial connection relationship matrix, the connectivity index, soil topographic moisture index and groundwater topographic index are combined, and then the four migration paths of surface runoff, groundwater runoff, river network and pipeline transportation are combined to construct the nutrient migration cost matrix;

[0015] S6. Using a multi-objective optimization algorithm, calculate the source-path combination with the highest nutrient migration efficiency and obtain the best source-path combination;

[0016] S7, matching the river grid with the monitoring section to calibrate the profile biogeochemical process model, and using the gradient descent algorithm to achieve automatic parameter adjustment based on the error;

[0017] S8. By combining the optimal source-path combination with the profile biogeochemical process model, the main nutrient source-migration combination was traced with the cross section as the sink;

[0018] S9. Use the STL model to conduct time series modeling for different nutrient source-migration combinations in different sections, analyze the spatiotemporal variation patterns of nutrient sources in different sections, obtain the spatial variation patterns of nutrient pollution dischargers at different times, and construct different scenarios to analyze the time lag effects of reduction measures for different dischargers.

[0019] Preferably, the bay nutrient pollution database in step S1 is constructed by summarizing river basin meteorological data, hydrological data, water quality data, water ecological data, topographic data, land use data and socio-economic data through statistical yearbooks, literature research, field surveys and historical monitoring data.

[0020] Preferably, the point source nutrient pollution in step S3 includes septic tanks, land and sea aquaculture tail water, urban domestic sewage and industrial point sources; the non-point source nutrient pollution in step S3 includes agricultural fertilizers, pesticides, livestock and poultry manure and atmospheric deposition; the agricultural fertilizers and pesticides are calculated using enhanced regression trees based on land type, soil type, crop type and slope data; the livestock and poultry manure is calculated based on the livestock inventory in the statistical yearbook combined with the pollution census coefficient; the atmospheric deposition is modeled and calculated using the Kriging interpolation toolset; the septic tank is analyzed and determined based on statistical data combined with population distribution, sewage treatment plant location and sewage outlet location; the land and sea aquaculture tail water corresponds to a specific grid; the urban domestic sewage is simulated using the EPA's SWMM model; the industrial point source is calculated based on the pollution source census data and then superimposed on the corresponding grid; non-agricultural fertilizers are visually interpreted and manually added; meteorological data simulation is simulated using the same coupled simple meteorological model as the SWAT model; for vacant data, Kriging interpolation, spline interpolation or autoregressive interpolation is used to fill the space without data.

[0021] Preferably, the profile biogeochemical process model in step S4 is: for nitrogen, five reservoirs of ammonia nitrogen, nitric nitrogen, active organic nitrogen, stable organic nitrogen and organic nitrogen from biological residues are considered, including mineralization, solidification, degradation, nitrification, ammonia volatilization, denitrification and atmospheric deposition processes; for phosphorus, six reservoirs of active inorganic phosphorus, stable inorganic phosphorus, dissolved phosphorus, active organic phosphorus, stable organic phosphorus and organic phosphorus from biological residues are considered, including mineralization, solidification, degradation, inorganic phosphorus adsorption and infiltration processes, and each reservoir is connected through the water balance of each soil layer of the distributed hydrological model, and the pollution corresponding to each grid and each time step is quantified through the profile biogeochemical model simulation to form a pollution amount matrix corresponding to each time step.

[0022] Preferably, the multi-objective optimization algorithm in step S6 takes the highest efficiency and the best robustness as optimization goals, and the constraint condition is that the pollution transfer result conforms to the actual pollution situation, and finally uses a genetic algorithm for solving.

[0023] Preferably, tracing the source to the pollutant discharger at different scales in step S8 should be carried out according to the following steps: first, based on the water quality simulation of the grid where the section is located, reversely determine the source-path combination of pollution; then, based on the source-path combination of pollution, identify the grid where the main pollution source is located, and determine the pollutant discharger corresponding to the section; wherein, the pollutant discharger refers to the specific functional entity in the pollution grid, and the pollutant discharger includes the industry to which the industrial enterprise belongs, the planting types to which agriculture belongs, and the discharge from the sewage treatment plant; finally, analyze the hotspot pollution sources of different pollutant dischargers and their corresponding hotspot migration pathways to evaluate their contribution to the sections with excessive water quality and their pollution reduction potential.

[0024] Preferably, in step S9, the trend term of the STL model is fitted using multiple regression, the period term is fitted using trigonometric functions, and the noise term is fitted using SARIMA combined with an ARCH model, thereby realizing the time analysis of the hotspot source-migration combination; wherein the trend term fitting formula is as follows: ,in, Indicates time Trend items; Indicates time of independent variables; , represents the regression coefficient; is the error term; the fitting formula for the period term is as follows: ,in, Indicates time The periodic term value of ; and are the fitting coefficients; is the cycle length; is the number of harmonics; Indicates The fitting formula of the noise term is as follows: , , ,in, is the backtracking operator; is the seasonal autoregressive polynomial; is a non-seasonal autoregressive polynomial; is the seasonal moving average polynomial; is a non-seasonal moving average polynomial; It is a non-seasonal difference operation; is the seasonal difference operation; is the white noise error term; It's time conditional variance; are independent and identically distributed random variables with mean 0 and variance 1; are the parameters of the ARCH model.

[0025] After adopting the above technical solution, the present invention has the following beneficial effects: First, the method is flexible and easy to use, and can accurately locate specific pollutants, which provides a clear target for targeted pollution control measures. Secondly, the traceability unit of this method is the source-migration combination. When conducting precise pollution control, it is possible to simultaneously consider the reduction of pollution sources and the interception and optimization of migration pathways, and better formulate targeted nutrient pollution reduction policies based on specific source-migration combinations. Finally, this method is also comprehensive in time and space, and can analyze the distribution and change patterns of pollution in time and space, which is crucial for understanding pollution dynamics and formulating long-term governance strategies. BRIEF DESCRIPTION OF THE DRAWINGS

[0026] Figure 1 It is a flow chart of the present invention. DETAILED DESCRIPTION

[0027] In order to make the purpose, technical solution and advantages of the present invention more clearly understood, the present invention is further described in detail below in conjunction with the embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention and are not used to limit the present invention.

[0028] like Figure 1 As shown, a method for temporal and spatial tracing of nutrient pollution for pollutant dischargers includes the following steps:

[0029] S1. Build a database of nutrient pollution in the bay;

[0030] The bay nutrient pollution database in step S1 is constructed by summarizing basin meteorological data, hydrological data, water quality data, water ecological data, topographic data, land use data and socio-economic data through statistical yearbooks, literature research, field surveys and historical monitoring data;

[0031] S2, rasterize the watershed and construct a spatial connection relationship matrix to extract the river grid;

[0032] The specific process of step S2 is as follows: the watershed is divided into 500×500 m grids, and a spatial connection relationship matrix is ​​constructed using a Markov chain based on the topography; the Markov chain method based on topography uses DEM data to calculate the slope and aspect of different grids, and is implemented through a river grid extraction algorithm;

[0033] The slope calculation formula is as follows: , , ,in, Indicates slope; Indicates terrain height exist Directional gradient; Indicates terrain height exist Directional gradient; Represents a grid cell Height; Represents a grid cell Height; Represents a grid cell Height; Represents a grid cell Height; Represents the grid cell in The size of the direction, Represents the grid cell in The magnitude of the direction, i.e. and 500 m;

[0034] The slope aspect calculation formula is as follows: ,In the Markov chain, the transition probability is determined based on the slope and aspect relationships between grid cells;

[0035] The specific calculation formula of the transition probability matrix is ​​as follows: ,in, Represents the grid cell The probability of transferring to grid cell j; Represents the grid cell The slope of the transfer to j; is a smoothing parameter greater than zero, used to control the effect of slope on transition probability; Is a grid cell The neighborhood of includes all possible four-way or eight-way transfer directions; Indicates the first of four or eight directions of transfer grid cells; Represents a grid cell Transfer to The slope of the grid is calculated by the D8 algorithm, and the flow path is determined to extract the river grid.

[0036] S3. Allocate point source nutrient pollution and non-point source nutrient pollution to different grids through Kriging interpolation, model simulation and dynamic coefficient calculation to match the pollution sources with the grids;

[0037] The point source nutrient pollution in step S3 includes septic tanks, land and sea aquaculture tail water, urban domestic sewage and industrial point sources; the non-point source nutrient pollution in step S3 includes agricultural fertilizers, pesticides, livestock and poultry manure and atmospheric deposition; the agricultural fertilizers and pesticides are calculated using enhanced regression trees based on land type, soil type, crop type and slope data; the livestock and poultry manure is calculated based on the livestock inventory in the statistical yearbook combined with the pollution census coefficient; the atmospheric deposition is modeled and calculated using the Kriging interpolation toolset; the septic tank is based on statistical data combined with population distribution, sewage treatment plant location and sewage discharge. The inlet location is analyzed and determined; the land and sea aquaculture tailwater corresponds to a specific grid; the urban domestic sewage is simulated using the EPA's SWMM model; the industrial point source is summarized according to the pollution source census data and superimposed on the corresponding grid for calculation; non-agricultural fertilizers are manually added after visual interpretation; meteorological data simulation is simulated using the same coupled simple meteorological model as the SWAT model; for vacant data, kriging interpolation, spline interpolation or autoregressive interpolation is used to fill the space without data; S4, construct a profile biogeochemical process model for river grids and non-river grids to form a nutrient pollution matrix;

[0038] The profile biogeochemical process model in step S4 is as follows: for nitrogen, five reservoirs of ammonia nitrogen, nitric nitrogen, active organic nitrogen, stable organic nitrogen and organic nitrogen from biological residues are considered, including mineralization, solidification, degradation, nitrification, ammonia volatilization, denitrification and atmospheric deposition processes; for phosphorus, six reservoirs of active inorganic phosphorus, stable inorganic phosphorus, dissolved phosphorus, active organic phosphorus, stable organic phosphorus and organic phosphorus from biological residues are considered, including mineralization, solidification, degradation, inorganic phosphorus adsorption and infiltration processes, and each reservoir is connected through the water balance of each soil layer of the distributed hydrological model, and the pollution corresponding to each grid and each time step is quantified through the profile biogeochemical model simulation to form a pollution amount matrix corresponding to each time step;

[0039] S5. According to the spatial connection relationship matrix, the connectivity index, soil topographic moisture index and groundwater topographic index are combined, and then the four migration paths of surface runoff, groundwater runoff, river network and pipeline transportation are combined to construct the nutrient migration cost matrix;

[0040] S6. Using a multi-objective optimization algorithm, calculate the source-path combination with the highest nutrient migration efficiency and obtain the best source-path combination;

[0041] The multi-objective optimization algorithm in step S6 takes the highest efficiency and the best robustness as the optimization objectives, and the constraint condition is that the pollution transfer result conforms to the actual pollution situation, and finally uses the genetic algorithm to solve;

[0042] S7, matching the river grid with the monitoring section to calibrate the profile biogeochemical process model, and using the gradient descent algorithm to achieve automatic parameter adjustment based on the error;

[0043] S8. By combining the optimal source-path combination with the profile biogeochemical process model, the main nutrient source-migration combination was traced with the cross section as the sink;

[0044] In step S8, tracing the source to the pollutant discharger at different scales should be carried out according to the following steps: first, based on the water quality simulation of the grid where the section is located, reversely determine the source-path combination of the pollution; then, based on the source-path combination of the pollution, identify the grid where the main pollution source is located, and determine the pollutant discharger corresponding to the section; wherein the pollutant discharger refers to the specific functional subject in the pollution grid, and the pollutant discharger includes the industry to which the industrial enterprise belongs, the planting species to which the agriculture belongs, and the discharge of the sewage treatment plant; finally, analyze the hotspot pollution sources of different pollutant dischargers and their corresponding hotspot migration pathways to evaluate their contribution to the section with excessive water quality and their pollution reduction potential;

[0045] S9. Use the STL model to model the time series of different nutrient source-migration combinations in different sections, analyze the temporal and spatial variation of nutrient sources in different sections, obtain the spatial variation of nutrient pollution dischargers at different times, and construct different scenarios to analyze the time lag effect of reduction measures for different dischargers;

[0046] In step S9, the trend term of the STL model is fitted using multiple regression, the period term is fitted using trigonometric functions, and the noise term is fitted using SARIMA combined with an ARCH model, thereby realizing the time analysis of the hotspot source-migration combination; wherein the trend term fitting formula is as follows: ,in, Indicates time Trend items; Indicates time of independent variables; , represents the regression coefficient; is the error term; the fitting formula for the period term is as follows: ,in, Indicates time The periodic term value of ; and are the fitting coefficients; is the cycle length; is the number of harmonics; Indicates The fitting formula of the noise term is as follows: , , ,in, is the backtracking operator; is the seasonal autoregressive polynomial; is a non-seasonal autoregressive polynomial; is the seasonal moving average polynomial; is a non-seasonal moving average polynomial; It is a non-seasonal difference operation; is the seasonal difference operation; is the white noise error term; It's time conditional variance; are independent and identically distributed random variables with mean 0 and variance 1; are the parameters of the ARCH model.

[0047] The above is only a preferred specific embodiment of the present invention, but the protection scope of the present invention is not limited thereto. Any changes or substitutions that can be easily thought of by a person skilled in the art within the technical scope disclosed by the present invention should be included in the protection scope of the present invention. Therefore, the protection scope of the present invention should be based on the protection scope of the claims.

Claims

1. A method for temporal and spatial tracing of nutrient pollution for pollutant dischargers, characterized in that: The following steps are involved: S1. Build a database of nutrient pollution in the bay; S2, rasterize the watershed and construct a spatial connection relationship matrix to extract the river grid; The specific process of step S2 is as follows: the watershed is divided into 500×500 m grids, and a spatial connection relationship matrix is ​​constructed using a Markov chain based on the topography; the Markov chain method based on topography uses DEM data to calculate the slope and aspect of different grids, and is implemented through a river grid extraction algorithm; The slope calculation formula is as follows: , , ,in, Indicates slope; Indicates terrain height exist Directional gradient; Indicates terrain height exist Directional gradient; Represents a grid cell Height; Represents a grid cell Height; Represents a grid cell Height; Represents a grid cell Height; Represents the grid cell in The size of the direction, Represents the grid cell in The magnitude of the direction, i.e. and 500 m; The slope aspect calculation formula is as follows: ,In the Markov chain, the transition probability is determined based on the slope and aspect relationships between grid cells; The specific calculation formula of the transition probability matrix is ​​as follows: ,in, Represents the grid cell Transfer to grid cell probability; Represents the grid cell Transfer to The slope of is a smoothing parameter greater than zero, used to control the effect of slope on transition probability; Is a grid cell The neighborhood of includes all possible four-way or eight-way transfer directions; Indicates the first of four or eight directions of transfer grid cells; Represents a grid cell Transfer to The slope of the grid is calculated by the D8 algorithm, and the flow path is determined to extract the river grid. S3. Allocate point source nutrient pollution and non-point source nutrient pollution to different grids through Kriging interpolation, model simulation and dynamic coefficient calculation to match the pollution sources with the grids; S4. Construct the biogeochemical process model of the river grid and the non-river grid to form the nutrient pollution matrix; S5. According to the spatial connection relationship matrix, the connectivity index, soil topographic moisture index and groundwater topographic index are combined, and then the four migration paths of surface runoff, groundwater runoff, river network and pipeline transportation are combined to construct the nutrient migration cost matrix; S6. Using a multi-objective optimization algorithm, calculate the source-path combination with the highest nutrient migration efficiency and obtain the best source-path combination; S7, matching the river grid with the monitoring section to calibrate the profile biogeochemical process model, and using the gradient descent algorithm to achieve automatic parameter adjustment based on the error; S8. By combining the optimal source-path combination with the profile biogeochemical process model, the main nutrient source-migration combination was traced with the cross section as the sink; S9. Use the STL model to conduct time series modeling for different nutrient source-migration combinations in different sections, analyze the spatiotemporal variation patterns of nutrient sources in different sections, obtain the spatial variation patterns of nutrient pollution dischargers at different times, and construct different scenarios to analyze the time lag effects of reduction measures for different dischargers.

2. A method for temporal and spatial source tracing of nutrient salt pollution for a pollutant discharger as claimed in claim 1, characterized in that: The bay nutrient pollution database in step S1 is constructed by summarizing basin meteorological data, hydrological data, water quality data, water ecological data, topographic data, land use data and socio-economic data through statistical yearbooks, literature research, field surveys and historical monitoring data.

3. A method for temporal and spatial source tracing of nutrient salt pollution for a pollutant discharger as claimed in claim 1, characterized in that: The point source nutrient pollution in step S3 includes septic tanks, land and sea aquaculture tail water, urban domestic sewage and industrial point sources; the non-point source nutrient pollution in step S3 includes agricultural fertilizers, pesticides, livestock and poultry manure and atmospheric deposition; the agricultural fertilizers and pesticides are calculated using enhanced regression trees based on land type, soil type, crop type and slope data; the livestock and poultry manure is calculated based on the livestock inventory in the statistical yearbook combined with the pollution census coefficient; the atmospheric deposition is modeled and calculated using the Kriging interpolation toolset; the septic tank is analyzed and determined based on statistical data combined with population distribution, sewage treatment plant location and sewage outlet location; the land and sea aquaculture tail water corresponds to a specific grid; The urban domestic sewage is simulated using the EPA's SWMM model; the industrial point sources are summarized based on the pollution source census data and then superimposed on the corresponding grid for calculation; non-agricultural fertilizers are manually added after visual interpretation; meteorological data simulation is performed using the same coupled simple meteorological model as the SWAT model; and for missing data, Kriging interpolation, spline interpolation or autoregressive interpolation is used to fill in the space without data.

4. A method for temporal and spatial tracing of nutrient salt pollution for pollutant dischargers as claimed in claim 1, characterized in that: The profile biogeochemical process model described in step S4 is as follows: for nitrogen, five reservoirs of ammonia nitrogen, nitric nitrogen, active organic nitrogen, stable organic nitrogen and organic nitrogen from biological residues are considered, including mineralization, solidification, degradation, nitrification, ammonia volatilization, denitrification and atmospheric deposition processes; for phosphorus, six reservoirs of active inorganic phosphorus, stable inorganic phosphorus, dissolved phosphorus, active organic phosphorus, stable organic phosphorus and organic phosphorus from biological residues are considered, including mineralization, solidification, degradation, inorganic phosphorus adsorption and infiltration processes, and each reservoir is connected through the water balance of each soil layer of the distributed hydrological model, and the pollution corresponding to each grid and each time step is quantified through the profile biogeochemical model simulation to form a pollution amount matrix corresponding to each time step.

5. The method for temporal and spatial source tracing of nutrient salt pollution for a pollutant discharger as claimed in claim 1, characterized in that: The multi-objective optimization algorithm described in step S6 takes the highest efficiency and the best robustness as the optimization objectives, and the constraint condition is that the pollution transfer result conforms to the actual pollution situation, and finally uses the genetic algorithm to solve.

6. A method for temporal and spatial tracing of nutrient salt pollution for pollutant dischargers as claimed in claim 1, characterized in that: In step S8, tracing the source to the pollutant discharger at different scales should be carried out according to the following steps: first, based on the water quality simulation of the grid where the section is located, reversely determine the source-path combination of pollution; then, based on the source-path combination of pollution, identify the grid where the main pollution source is located, and determine the pollutant discharger corresponding to the section; wherein, the pollutant discharger refers to the specific functional entity in the pollution grid, and the pollutant discharger includes the industry to which the industrial enterprise belongs, the planting types to which agriculture belongs, and the discharge from the sewage treatment plant; finally, analyze the hotspot pollution sources of different pollutant dischargers and their corresponding hotspot migration pathways to evaluate their contribution to the sections with excessive water quality and their pollution reduction potential.

7. A method for temporal and spatial source tracing of nutrient salt pollution for pollutant dischargers as claimed in claim 1, characterized in that: In step S9, the trend term of the STL model is fitted using multiple regression, the period term is fitted using trigonometric functions, and the noise term is fitted using SARIMA combined with an ARCH model, thereby realizing the time analysis of the hotspot source-migration combination; wherein the trend term fitting formula is as follows: ,in, Indicates time Trend items; Indicates time of independent variables; represents the regression coefficient; is the error term; the fitting formula for the period term is as follows: ,in, Indicates time The periodic term value of ; and are the fitting coefficients; is the cycle length; is the number of harmonics; Indicates The fitting formula of the noise term is as follows: ; ; ;in, is the backtracking operator; is the seasonal autoregressive polynomial; is a non-seasonal autoregressive polynomial; is the seasonal moving average polynomial; is a non-seasonal moving average polynomial; It is a non-seasonal difference operation; is the seasonal difference operation; is the white noise error term; It's time conditional variance; are independent and identically distributed random variables with mean 0 and variance 1; are the parameters of the ARCH model.

Images