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# Study on the Dynamic Change of Regional Water Level and Climate Based on Forecast Equation

###### Accepté: 28 Apr 2022
Détails du magazine
Format
Magazine
eISSN
2444-8656
Première parution
01 Jan 2016
Périodicité
2 fois par an
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Anglais
Regional water level dynamic prediction system architecture

According to the overall structure diagram are shown in figure 1 below the system analysis, to the prediction equation of acquired data in the analysis of the orderly use of analysis, and to use information technology to develop a variety of prediction methods, only in this way can more accurately forecast the dynamic change of groundwater, and the relationship between the period of dynamic change and climate[1].

Selection of prediction factors and prediction equations
Definition 1

The statistical relationship between forecast objects and forecast factors presents diversity, common types include hyperbolic function, trigonometric function, power function and linearity. This paper mainly discusses four types: first, it refers to the type of linear function, and the corresponding formula is shown as follows: $y=x$ {\rm{y}} = {\rm{x}}

The second is for the exponential function type, the corresponding function is as follows: $y=lnx$ {\rm{y}} = \ln {\rm{x}}

Again, the power function type, the corresponding function is as follows: $y=xa$ {\rm{y}} = {\rm{xa}}

Finally refers to the type of exponential function, the corresponding function is as follows: $Y=exp(ax)$ {\rm{Y}} = \exp \left({{\rm{ax}}} \right)

Theorem 1

The corresponding coefficients of the four function relations are calculated respectively for the forecast factor XI and the forecast quantity Yi. The maximum absolute value of the correlation coefficient is the optimal function type of the forecast factor. The prediction equation constructed by this function type belongs to nonlinear prediction equation.

Proposition 2

Since the logarithmic function and power function put forward x>0, range normalization should be implemented for the numerical value of the forecast factor first, and the corresponding formula is shown as follows: $xi′=xi−min{xi}max{xi}−min{xi}Among them i=1,2,…,N$ x_i^{'} = {{{x_i} - \min \left\{{{x_i}} \right\}} \over {\max \left\{{{x_i}} \right\} - \min \left\{{{x_i}} \right\}}}Among\,them\,\,i = 1,2, \ldots,N

In cases where the condition $xi′=0$ x_i^{'} = 0 is met, you can $xi′=0.00001$ x_i^{'} = 0.00001 , which satisfies all types of requirements. The range processing of the prediction factor not only meets the requirement of the function domain, but also prevents the overflow of the calculation due to the large numerical value during the calculation.[2]

The calculation formula of correlation coefficient is as follows: $R=∑(xi′−x¯′)(yi−y¯)∑(xiN−x¯′)2∑(yi−y¯)2$ R = {{\sum {\left({x_i^{'} - {{\bar x}^{'}}} \right)\left({{y_i} - \bar y} \right)}} \over {\sqrt {\sum {{{\left({x_i^N - {{\bar x}^{'}}} \right)}^2}} \sum {{{\left({{y_i} - \bar y} \right)}^2}}}}}

Lemma 3

In the case of calculating linear function types, we can get: $xin=xi, x¯′=∑xi/N$ x_i^n = {x_i},\,{\bar x^{'}} = \sum {{x_i}/N}

In the case of calculating the logarithmic function type, it can be obtained: $xin=linxi′, x¯′=∑lnxi/N$ x_i^n = linx_i^{'},\,{\bar x^{'}} = \sum {\ln {x_i}/N}

In the case of calculating the power function type, we can get: $xin=xi′α, x¯′=∑riα/N$ x_i^n = x_i^{'\alpha},\,{\bar x^{'}} = \sum {r_i^\alpha /N}

In the case of calculating the type of exponential function, it can be obtained: $xin=exp(axi), x¯′=∑exp(axi)$ x_i^n = \exp \left({a{x_i}} \right),\,{\bar x^{'}} = \sum {\exp \left({a{x_i}} \right)}

Conjecture 5 Both linear function and exponential function can be directly obtained | R1 | and | | R2, while the other two functions need to compute parameter alpha, which requires security forecast and the forecast factors of the correlation coefficient reaches maximum on parameter values. In other words, the parameter α needs to be calculated accurately in an appropriate way to ensure that the objective function shown below is maximized: $f(x)=|∑(xi′−x¯′)(yi−y¯)∑(xi′−x¯′)2∑(yi−y)2|$ f\left(x \right) = \left| {{{\sum {\left({x_i^{'} - {{\bar x}^{'}}} \right)\left({{y_i} - \bar y} \right)}} \over {\sqrt {\sum {{{\left({x_i^{'} - {{\bar x}^{'}}} \right)}^2}} \sum {{{\left({{y_i} - y} \right)}^2}}}}}} \right|

The maximum value calculated according to the target function, and clear after the alpha numeric parameters, to compute about | R3, | | R4 |, contrast analysis | R1, | | and | | R2 and R3, | | R4 | the size of the four values, and combining the maximum value choice of function types to build optimization equation.

The parameter α evaluation problem is the nonlinear function optimization problem. At present, there is no unified method for solving nonlinear differential equations, and different types of nonlinear equations need to choose different application algorithms. Due to the complexity of function derivation in the early stage, this paper chooses quadratic difference method for trial calculation[3].

Research on regional water level change and climate dynamics based on forecast equation
Trend test

Example 6. In this paper, mann-Kendall test is used for analysis, also known as M-K test, which can effectively show the trend change characteristics of time series. This method is widely used and recommended by the World Meteorological Organization. It does not need to follow the principle of sample distribution to analyze the variation trend of time series of water quality, air temperature, precipitation and other elements, nor is it disturbed by a few abnormal values. In the test, H0 was originally assumed to be: time series data are n independent samples with identically distributed random variables; Alternative hypothesis H1 is: for all k, j $s=∑k=1n−1∑j=k+1nsgn(xj−xk)$ s = \sum\limits_{k = 1}^{n - 1} {\sum\limits_{j = k + 1}^n {{\mathop{\rm sgn}} \left({{x_j} - {x_k}} \right)}}

In the above formula, xk and xj represent the number of observations in the KTH and JTH years, and the condition that J Thus, it can be obtained: $sgn(xj−xk)={1xj−xk>00xj−xk=0−1xj−xk<0$ {\mathop{\rm sgn}} \left({{x_j} - {x_k}} \right) = \left\{{\matrix{1 \hfill & {{x_j} - {x_k} > 0} \hfill \cr 0 \hfill & {{x_j} - {x_k} = 0} \hfill \cr {- 1} \hfill & {{x_j} - {x_k} < 0} \hfill \cr}} \right.

Open Problem 8. The random sequence approximately follows the normal distribution, so the corresponding mean and variance calculation formula is as follows: $E(S)=0Var(S)=n(n−1)(2n+5)18$ \matrix{{E\left(S \right) = 0} \hfill \cr {Var\left(S \right) = {{n\left({n - 1} \right)\left({2n + 5} \right)} \over {18}}} \hfill \cr}

Thus, when n>10 meets this condition, the standard normal statistical variable Z can be calculated and analyzed using the following formula: $z={S+1Var(s) s>00 s=0S−1Var(s) s<0$ z = \left\{{\matrix{{{{S + 1} \over {\sqrt {Var\left(s \right)}}}\,s > 0} \hfill \cr {0\,s = 0} \hfill \cr {{{S - 1} \over {\sqrt {Var\left(s \right)}}}\,s < 0} \hfill \cr}} \right.

Example 7. In the above formula, Z represents the statistical quantity of the normal distribution. It was originally assumed that there was no trend of change in this sequence. Bilateral trend test was used to find the critical value in the normal distribution table under the condition of clear significance level. When the condition − Z1 − a / 2ZZ1 − a / 2 is met, the original hypothesis can be accepted. On the contrary, Z < − Z1 − a / 2 proves that the sequence has an obvious downward trend, Z > Z1 − a / 2 indicates that the sequence has an obvious upward trend, and α represents the significance level.

Analysis method of groundwater level spatial transformation

In this paper, according to the groundwater level presumption method, GIS technology is used as an analysis tool, kriging method is used to draw the water level change map in the study area, and the spatial difference method is used to explore the hydrometeorological evolution law in the region. Common methods are as follows:

First, inverse distance weighted difference method. This method will predict the values of other regions based on the values of the identified sample points within the prediction region. Generally speaking, the weight of the known sample closer to the prediction point will exceed the weight of the known sample far from the prediction point during the prediction period, and the corresponding formula is shown as follows[4]: $Z^(S0)=∑i=1Nλ1Z(Si)$ \hat Z\left({{S_0}} \right) = \sum\limits_{i = 1}^N {{\lambda _1}Z\left({{S_i}} \right)}

In the above formula, $Z^(S0)$ \hat Z\left({{S_0}} \right) means waiting numerical prediction S0, N for use during the calculation of the forecasting point around the point number, lambda I represents prediction calculation during the use of all kinds of sample weight value, the value will follow sample and increases the distance between the predicted and continues to decline, Z (Si) on behalf of the regional Si acquired measuring value. The calculation formula of clear weight is as follows: $λi=di0−p/∑i=0Ndi0−p∑i=1Nλi=1$ \matrix{{{\lambda _i} = d_{i0}^{- p}/\sum\limits_{i = 0}^N {d_{i0}^{- p}}} \hfill & {\sum\limits_{i = 1}^N {{\lambda _i} = 1}} \hfill \cr}

In the above formula, P represents the parameter, and the optimal value can be determined by taking the minimum value of mean square and prediction error. Di0 represents the distance between the predicted point, S0 and the known sample point Si.

Second, spline difference method. This method Kriger, the application principle of this method is as follows:

Suppose the research area contains N observation points X1, X2... XN, the corresponding variable value is Z (xi), and the estimation of the real value of the waiting prediction point x0 $Z^(X0)$ \hat Z\left({{X_0}} \right) can be expressed by the linear combination of N measured points, as shown below: $z^(x0)=∑i=1Nλiz(xi)$ \hat z\left({{x_0}} \right) = \sum\limits_{i = 1}^N {{\lambda _i}z\left({{x_i}} \right)}

In the above formula, represents the weight value of the ith side point to the waiting detection point, which meets the condition $∑i=1Nλi=1$ \sum\limits_{i = 1}^N {{\lambda _i} = 1} . In the process of weight assignment, the traditional method only needs to consider the distance between the sideslip and the wait-to-measure sample, while the Kriging method not only needs to analyze the distance, but also needs to study the spatial distribution of sideslip and the spatial word correlation of wait-to-measure points.

The above formula proves that the selection and calculation of weights directly affect the estimated value of waiting measurement points. Therefore, the weight calculation ensures that the estimator complies with the following criteria: $E[z^(x0)−z(x0)]=0min Var[z^(x0)−z(x0)]=E[z^(x0)−z(x0)]2$ \matrix{{E\left[ {\hat z\left({{x_0}} \right) - z\left({{x_0}} \right)} \right] = 0} \hfill \cr {\min \,Var\left[ {\hat z\left({{x_0}} \right) - z\left({{x_0}} \right)} \right] = E{{\left[ {\hat z\left({{x_0}} \right) - z\left({{x_0}} \right)} \right]}^2}} \hfill \cr}

At the same time, the Lagrange multiplier method should be used to ensure that the estimator $z^(x0)$ \hat z\left({{x_0}} \right) is the optimal weight value, and the specific formula is as follows: $λ=K−1D$ \lambda = {K^{- 1}}D

In the above formula, $K=[r¯11r¯12…r¯1N1r¯21r¯22…r¯1N1⋮⋮⋮r¯N1r¯N2…r¯NN111…10]$ K = \left[ {\matrix{{{{\bar r}_{11}}} \hfill & {{{\bar r}_{12}}} \hfill & \ldots \hfill & {{{\bar r}_{1N}}} \hfill & 1 \hfill \cr {{{\bar r}_{21}}} \hfill & {{{\bar r}_{22}}} \hfill & \ldots \hfill & {{{\bar r}_{1N}}} \hfill & 1 \hfill \cr \vdots \hfill & \vdots \hfill & {} \hfill & \vdots \hfill & {} \hfill \cr {{{\bar r}_{N1}}} \hfill & {{{\bar r}_{N2}}} \hfill & \ldots \hfill & {{{\bar r}_{NN}}} \hfill & 1 \hfill \cr 1 \hfill & 1 \hfill & \ldots \hfill & 1 \hfill & 0 \hfill \cr}} \right] , $r¯ij=r¯(xi,xj)$ {\bar r_{ij}} = \bar r\left({{x_i},{x_j}} \right) represents the semi-variogram between sample points XI and Xj. After the spatial distance between each other is clear, the semi-variogram model is used to calculate and obtain: $D=[r¯(xi, x0)r¯(x2, x0)…r¯(xN, x0)r]$ D = \left[ {\matrix{{\bar r\left({{x_i},\,{x_0}} \right)} \hfill \cr {\bar r\left({{x_2},\,{x_0}} \right)} \hfill \cr \ldots \hfill \cr {\bar r\left({{x_N},\,{x_0}} \right)} \hfill \cr r \hfill \cr}} \right]

In the above formula, $r¯(xi, x0)$ \bar r\left({{x_i},\,{x_0}} \right) represents the semi-variogram between the sample point XI and the waiting measurement point X0, λ1(i = 1, 2, …, N)=== represents the weight value, and u represents the Lagrange multiplier.

Forecast research

According to the dynamic characteristics of regional water conservancy and meteorology, the discrete data are used to simulate and analyze. The time interval of discrete data affects the closeness of the sequence. All time series models are composed according to trend, period and random noise. It is assumed that H (t) represents the observation sequence of actual groundwater level, and the overall number of selected samples is N, then it can be obtained[5]: $H of t =t of t+P of t + R of t.$ {\rm{H}}\,{\rm{of}}\,{\rm{t}}\, = {\rm{t}}\,{\rm{of}}\,{\rm{t}} + {\rm{P}}\,{\rm{of}}\,{\rm{t}}\, + \,{\rm{R}}\,{\rm{of}}\,{\rm{t}}.

In the above formula, H (t) represents the dynamic observation series of groundwater level, T (t) represents the trend term, P (t) represents the periodic term, and R (t) represents the random term. Specific analysis is as follows:

First, the trend term. In the empirical study, it is found that the trend term is usually regarded as a real-valued function of time T. Since the overall development trend of water level change is definite, polynomial approximation as follows can be used:

At the same time, the undetermined coefficients c0, C1… were determined by multiple regression method., ck and order K. In order to verify the fitting results, correlation coefficient R of trend curve fitting should be studied, and the specific formula is shown as follows: $R2=1−∑t=1n(H(t)−T′(t))2∑t=1n(H(t)−H¯(t))2$ {R^2} = 1 - {{\sum\limits_{t = 1}^n {{{\left({H\left(t \right) - {T^{'}}\left(t \right)} \right)}^2}}} \over {\sum\limits_{t = 1}^n {{{\left({H\left(t \right) - \bar H\left(t \right)} \right)}^2}}}}

In the above formula, n represents the total number of actual measured sequence H (t), and represents the average value of sequence H (t). The closer the calculation result of the above formula is to 1, the closer the linear relation between T' (T) and TK is proved. In the case of definite reliability and degree of freedom, when the value R exceeds the critical value, the regression equation has application value. Second, the period term. This sequence is represented by the superposition of K waves, and the corresponding estimation formula is shown as follows: $P′(t)=a0+∑j=1kajcos(2πnjt)+∑j=1kbj sin(2πnjt)$ {P^{'}}\left(t \right) = {a_0} + \sum\limits_{j = 1}^k {{a_j}\cos \left({{{2\pi} \over n}jt} \right) + \sum\limits_{j = 1}^k {{b_j}\,\sin \left({{{2\pi} \over n}jt} \right)}}

In the above formula, k represents the number of harmonics, usually the integer part of N /2; J is regarded as wave number, and it is generally considered that k partial waves each contain n/1, n/2... N/k cycles, in other words, the frequency of the JTH component wave is J /n; A0, AJ and BJ represent The Fourier coefficients, and the actual calculation formula is as follows: ${a0=1n∑i=1nh(t)aj=2n∑t=1nh(t)cos[2πin(t−1)]j=1,2,…,kbj=2n∑t=1nh(t)sin[2πin(t−1)]$ \left\{{\matrix{{{a_0} = {1 \over n}\sum\limits_{i = 1}^n {h\left(t \right)}} \hfill \cr {{a_j} = {2 \over n}\sum\limits_{t = 1}^n {h\left(t \right)\cos \left[ {{{2\pi i} \over n}\left({t - 1} \right)} \right]j = 1,2, \ldots,k}} \hfill \cr {{b_j} = {2 \over n}\sum\limits_{t = 1}^n {h\left(t \right)\sin \left[ {{{2\pi i} \over n}\left({t - 1} \right)} \right]}} \hfill \cr}} \right.

Third, the random term. Autoregressive model was used for solving and analysis, and the corresponding model formula is shown as follows: $R′(t)=Φ0+Φ1R(t−1)+Φ2R(t−2)+…+ΦPR(t−p)$ {R^{'}}\left(t \right) = {\Phi _0} + {\Phi _1}R\left({t - 1} \right) + {\Phi _2}R\left({t - 2} \right) + \ldots + {\Phi _P}R\left({t - p} \right)

In the above formula, p represents the model order, φ I represents the model autoregression coefficient, and is consistent with I = 1, 2… P. For an autoregressive model of a certain order, the autoregressive coefficient φ I can be obtained by similar multiple regression calculation. This paper uses the AIC criterion to clarify the model order, as shown below: $AIC(p)=nlnσ2+2p$ AIC\left(p \right) = n\ln {\sigma ^2} + 2p

Fourth, accuracy test. By linear superposition of the three components of the above study, the overall prediction model of regional water level change and climate dynamic change can be obtained, as shown below: $H′(t)=T′(t)+P′(t)+R′(t)$ {H^{'}}\left(t \right) = {T^{'}}\left(t \right) + {P^{'}}\left(t \right) + {R^{'}}\left(t \right)

The actual prediction accuracy can be analyzed by posterior error method. Assuming that the dynamic sampling sample of actual measurement is: H1, H2, …, Hn, Hn+ 1, …, Hn + k1, The prediction equation obtained by using the first n sampling values is H' (t), tn+1 ~ tn + k1 the prediction value at the hypothetical time is $H(n+1)′, H(n+2)′,…,H(n+k1)′$ H_{\left({n + 1} \right)}^{'},\,H_{\left({n + 2} \right)}^{'}, \ldots,H_{\left({n + k1} \right)}^{'} , then it can be regarded as the posterior prediction value of the prediction equation, let ɛ I represent the residual of the posterior prediction, then it can be obtained: $εi=Hn+1−Hn+1′$ {\varepsilon _i} = {H_{n + 1}} - H_{n + 1}^{'}

Assuming that the standard deviation of the first n data in the dynamic sample is S1 and the standard deviation of the posterior data residual is S2, the posterior error ratio and small error frequency can be calculated as follows: ${c=s2/s1p={|q(k)−q|<0.6745s1}$ \left\{{\matrix{{c = {s_2}/{s_1}} \hfill \cr {p = \left\{{\left| {q\left(k \right) - q} \right| < 0.6745{s_1}} \right\}} \hfill \cr}} \right.

The evaluation criteria of forecast accuracy are shown in the following table:

Prediction accuracy evaluation standard table

Forecast level good good qualified unqualified
p >0.95 >0.80 >0.70 ≤0.70
c <0.35 <0.50 <0.65 ≥0.65

Assuming that both p and C values are within the specified range, the model can analyze and predict regional water level changes and climate dynamics; otherwise, the model needs to be checked and adjusted. After the model is qualified, it is necessary to simulate the sequence randomly, mainly using the long and short series method to judge the practicality of the model.[6]

In order to construct the time series model, it is necessary to make up some missing data by means of regression analysis to obtain the corresponding observation series, and then analyze the trend term, periodic term and random term, and get the time series model of different stations in the linear superposition, so as to provide mathematical basis for the study of dynamic change.

Result analysis

Combined with the above research methods, the prediction and analysis of groundwater level and climate change in a certain region can be obtained as shown in the following table. Table 2 refers to the calculation results of trend term, Table 3 refers to the calculation results of Fourier coefficient of periodic term, and Table 4 refers to the coefficient analysis results of autoregressive model:

Calculation results of trend items

c0 c1 c2 The correlation coefficient
414.85 −0.2416 0.0014 0.9289

Calculation results of Fourier coefficients of periodic term

k ak bk
10 −0.0616 0.5846

Coefficient analysis results of autoregressive model

Φ0 Φ1 Φ2
−0.428 0.007 0.645

The analysis process of the autoregressive model of water level is shown in Figure 2 below:

According to the overall prediction model constructed by the above research, the variation of groundwater level is calculated, and the fitting analysis is carried out according to the actual measured values. According to the above analysis, it is found that the fitting effect between calculated predicted values and measured values is relatively high. The specific comparison results are shown in Table 5 below[7.8]:

Comparison results between measured and predicted values in the study area

In The measured values Predictive value Absolute error The relative error
In January 407.08 408.0688 −0.99 0.24%
On February 406.95 408.40079 −1.45 0.36%
march 409.92 408.35768 1.56 0.38%
April 409.85 410.09384 −0.24 0.06%
On may 409.82 409.95276 −0.13 0.03%
June 409.93 409.82807 0.10 0.02%
In July 409.97 409.83206 0.14 0.03%
In August 410.00 409.87908 0.12 0.03%
On September 409.95 410.02388 −0.07 0.02%
On October 409.85 410.21619 −0.37 0.09%
In November, 409.80 410.4408 −0.64 0.16%
On December 409.73 410.70923 −0.98 0.24%

According to the analysis in the above table, there is a large gap between the values predicted at the beginning of the year and those predicted at the end of the year, and the trend in the time series prediction shows a downward trend due to the rise of water level. However, the relative error between the two is controlled within the specified range, and the actual test results are good. At the same time, the final results show that the relationship between climate dynamics and regional water level changes is very close. Climate factors themselves change very rapidly and are cyclical in a sense, causing rapid changes in water level dynamics. In the study period of one year, all kinds of climate will follow the seasonal changes in a regular cycle, which will be evident in the dynamic composition of the shallow groundwater. It can be seen that the seasonal and multi-year changes of groundwater dynamics are affected by meteorological factors.

Conclusion

In conclusion, the systematic study of regional water level change and climate dynamic change by using forecast equation can strengthen resource construction and management in China and scientifically deal with the relationship between climate dynamic change and regional water level on the basis of solving the problem of water resource shortage. Therefore, in the context of the new era, facing the increasing demand for water resources, researchers should strengthen their research efforts and use information technology to build a forecast and analysis system, so that they can not only master more accurate and perfect data information, but also have a deep understanding of the relationship between climate dynamics and regional water level changes[9].

#### Calculation results of Fourier coefficients of periodic term

k ak bk
10 −0.0616 0.5846

#### Coefficient analysis results of autoregressive model

Φ0 Φ1 Φ2
−0.428 0.007 0.645

#### Calculation results of trend items

c0 c1 c2 The correlation coefficient
414.85 −0.2416 0.0014 0.9289

#### Prediction accuracy evaluation standard table

Forecast level good good qualified unqualified
p >0.95 >0.80 >0.70 ≤0.70
c <0.35 <0.50 <0.65 ≥0.65

#### Comparison results between measured and predicted values in the study area

In The measured values Predictive value Absolute error The relative error
In January 407.08 408.0688 −0.99 0.24%
On February 406.95 408.40079 −1.45 0.36%
march 409.92 408.35768 1.56 0.38%
April 409.85 410.09384 −0.24 0.06%
On may 409.82 409.95276 −0.13 0.03%
June 409.93 409.82807 0.10 0.02%
In July 409.97 409.83206 0.14 0.03%
In August 410.00 409.87908 0.12 0.03%
On September 409.95 410.02388 −0.07 0.02%
On October 409.85 410.21619 −0.37 0.09%
In November, 409.80 410.4408 −0.64 0.16%
On December 409.73 410.70923 −0.98 0.24%

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