Spatial Interpolation Methods of Temperature Data Based on Geographic Information System—Taking Jiangxi Province as an Example ^†

Abstract

The comfort level of air temperature in a region is one of the influencing factors that affect tourists’ choice of tourism purpose. As a national red cultural mecca, the study of air temperature in Jiangxi Province can provide an important scientific reference for the development of tourism and the dissemination of red culture. Temperature is one of the most important indicators for climate comfort studies. Thus, in this paper, the average air temperature in Jiangxi Province in 2018 was studied. Three interpolation methods of Kriging interpolation, the inverse distance weight method, and the spline function method were used to spatially interpolate the data from 26 weather stations to obtain the spatial distribution map of air temperature for comparative study. At the same time, the method of cross-validation was adopted, and the average error and the root-mean-square error were quoted as the evaluation indexes for accuracy assessment. The conclusions of this paper are as follows: (1) the ME of IDW and spline method can reach 0.02–1.82 °C and the RMSE can reach 1.22–2.72 °C; (2) Kriging interpolation improves the RMSE by 27% and 55% compared to IDW and spline function methods, respectively; (3) considering the relatively sparse distribution of meteorological stations in Jiangxi Province, Kriging interpolation can avoid the extreme value phenomenon due to the influence of distance by reasonably choosing the shape and size associated with the surface space in the process of solving. Moreover, the results of this experimental study show that the accuracy of the kriging interpolation method is higher, so this method is more suitable for the spatial interpolation of the temperature in Jiangxi Province. In conclusion, this study provides a reference for the study of temperature comfort in Jiangxi Province.

1. Introduction

Since the beginning of the 21st century, many issues brought about by climate change have gradually begun to attract the attention of experts and scholars at home and abroad. Climate change alters the environment in which people live and furthermore affects their level of comfort. Thus, it can be seen that the degree of influence of climate in life is increasing, and there is more of a need to study the content of how to more reasonably evaluate the impact of climate comfort level on people’s lives [1]. Climate generally refers to the average conditions in a given area, and the main climatic elements include light, air temperature, precipitation, and wind. Temperature is an important indicator of the distribution and dynamics of atmospheric temperature at the Earth’s surface and is the most fundamental element of climate change research, which also has a very important impact on the structure and function of ecosystems. In this paper, the 2018 average temperature of 26 meteorological stations that can basically cover the whole region of Jiangxi Province is used as the base data. At present, there are many methods used for the spatial interpolation of meteorological elements, while the inverse distance weight method, spline function method, and Kriging interpolation belong to the more accurate interpolation methods among the current interpolation methods, so this paper selects the above three interpolation methods for the spatial interpolation of air temperature to analyze the influence of various interpolation methods on the study of spatial distribution characteristics of the annual mean temperature change and to analyze the spatial variability law of the annual mean air temperature in Jiangxi Province [2].

2. Interpolation Methods and Accuracy Assessment

2.1. Interpolation Method

In the processing of meteorological data, they are generally analyzed using some interpolation tools in ArcGIS10.8 software. The method of utilizing a certain number of sample point data to estimate the value of a raster cell is called interpolation. Interpolation methods can be used to estimate unknown values of any geographic point data, such as elevation, rainfall, chemical pollution levels, noise levels, and so on. Currently, there are many methods most commonly used for the spatial interpolation of meteorological elements, namely, the kriging method, the inverse distance weighting method, the trend surface interpolation method, and the spline function method [3,4,5,6].

2.1.1. Kriging Interpolation

Kriging interpolation analyzes the spatial distribution of variability of the interpolated variable, determines the range of distances that can affect the surrogate interpolation point, and then estimates the surrogate interpolation value using the sample values in that region. The theory underlying the Kriging method is the theory of regional change. This approach shows that the spatial variation in any variable is expressed by the sum of three basic elements. The three basic elements are the structural component related to the mean, i.e., the trend; the component related to local variation; and the random noise term, i.e., the residual error term.

$$Z(x)=m(x)+{\epsilon }^{\prime }(x)+{\epsilon }^{″}$$

(1)

The most important issue in Kriging’s algorithm is the semivariance function that reflects the spatial variation in the variables $\gamma (h)$. Commonly used models include spherical, exponential, and linear models. The spherical model is modeled as follows:

$$\gamma (h)=\left\{\begin{array}{cc}0& h=0\\ C_0+c({\displaystyle \frac{2}{3}}\times {\displaystyle \frac{h}{a}}-{\displaystyle \frac{1}{2}}\times {\displaystyle \frac{h^3}{a^3}})& 0<h<a\\ C_0+C& a<h\end{array}\right.$$

(2)

In this formula, $a$ is the variable range, i.e., impact distance, which indicates the maximum distance of mutual influence. When the distance between two points exceeds this into, the effect is negligible and is basically 0. $h$ is a delay that indicates the distance between two points, $C_0$ is the kernel variance, $C_0+C$ is sorghum and indicates the maximum value of the spatial covariance. The weights are determined using the following formula:

$$\left\{\begin{array}{cc}{\displaystyle {\sum }_{\beta =1}^n{\lambda }_{\beta }\times \overline{\gamma }(V_{\alpha },V_{\beta })+\psi =\overline{\gamma }(V_{\alpha },V)}& \alpha =1,2,\cdots ,n\\ {\displaystyle {\sum }_{\beta =1}^n{\lambda }_{\beta }}=1& \end{array}\right.$$

(3)

In this formula, $\psi$ is the Lagrange multiplier operator needed to compute the minimum variance. The values of the interpolated points can be obtained by substituting the weight coefficients into the following estimation equation:

$$Z_k={\displaystyle \sum _{i=1}^n{\lambda }_i\times z_i}$$

(4)

2.1.2. Inverse Distance Weighting Method Interpolation

The inverse distance weighting method estimates the value of the interpolated points by applying linear weights to the sampled points. Weights are an inverse distance function. The surface on which the interpolation is performed should have a surface with a localized dependent variable. This method assumes that the mapped variable is reduced by being affected by the distance between its sampling location. This is mainly dealt with by using power parameters to control the impact, limiting the points used for interpolation, and using variations in the sum of variable search radius searches and fixed search radii, as well as the use of barriers.

The basic idea of the inverse distance weighting method is the first law of geography, that is, the closer the distance between the two things, the more similar its attributes, and vice versa as it decreases. In the specific interpolation calculations, it is important to interpolate the distance between the point and the sample as the weight, and the closer the interpolation distance the greater the weight given to the sample point, and its contribution is inversely proportional to the distance. The expression is as follows:

$$\widehat{Z}(s_0)={\displaystyle \sum _{i=1}^N{\lambda }_{i}Z(s_i)}$$

(5)

In this formula, $\widehat{Z}\left(S_0\right)$ is the predicted value at $S_0$, $Z\left(S_i\right)$ is the measured value obtained at $S_i$, $N$ is the number of sample points around the predicted points involved in interpolation, and ${\lambda }_i$ is the weight of each sample point used in the prediction calculation process. The formula for determining the weights is as follows:

$${\lambda }_i=d_{i_0}^{-\phi }/{\displaystyle \sum _{i=L}^N{d}_{i_0}^{-\phi }}$$

(6)

2.1.3. Spline Function Interpolation

The spline function method is an interpolation method that uses functions to approximate surfaces. Spline function methods are categorized into regular spline function methods and tensile spline function methods. The regular spline method uses values that may lie outside the range of the sample data to create a gradient smooth surface. The tensile spline function method controls the hardness of the surface based on the characteristics of the modeled phenomenon. It creates less smooth surfaces using values that are more tightly constrained by the range of sample data. The algorithm for the spline function method is surface interpolation $S\left(x,y\right)$ as shown in Equation (7):

$$S(x,y)=T(x,y)+{\displaystyle \sum _{j=1}^N{\lambda }_{i}R(r_j)}\hspace{1em}j=1,2,\cdots ,N$$

(7)

Included among these, $N$ is the number of points, ${\lambda }_i$ are the coefficients obtained by solving the system of linear equations, and different choices of functions $T\left(x,y\right)$ and $R\left(r\right)$ correspond to different methods. For the regular spline function method, $T\left(x,y\right)$ and $R\left(r\right)$ are shown in Equations (8) and (9), respectively:

$$T(x,y)=a_1+a_{2}x+a_{3}y$$

(8)

$$R(r)={\displaystyle \frac{1}{2\pi }}\left\{{\displaystyle \frac{r^2}{4}}\left[\ln ({\displaystyle \frac{r}{2\pi }})+c-1\right]+{\tau }^2\left[K_0({\displaystyle \frac{r}{\tau }})+c+\ln ({\displaystyle \frac{r}{2\pi }})\right]\right\}$$

(9)

Included among these, $a_i$ are the coefficients obtained by solving the system of linear equations, $r$ represents the distance between the point and the sample, ${\tau }^2$ is the weight parameter, $K_0$ is the modified Bessel function, and $c$ is a constant of size equal to 0.577215. For the tensor spline function method, $T\left(x,y\right)$ and $R\left(r\right)$ are shown in Equations (10) and (11), respectively:

$$T(x,y)=a_1$$

(10)

$$R(r)=-{\displaystyle \frac{1}{2\pi {\phi }^2}}\left[\ln \left({\displaystyle \frac{\gamma \phi }{2}}\right)+c+K_0(\gamma \phi )\right]$$

(11)

Included among these, $a_1$ are the coefficients obtained by solving the system of linear equations, $r$ is the distance between the point and the sample, ${\phi }^2$ is the weight parameter, $K_0$ is the modified Bessel function, and $c$ is a constant of size equal to 0.577215.

In spline interpolation, the results obtained by regular spline function interpolation are smoother than those obtained by tension spline function interpolation, and the higher the weight value of regular spline interpolation, the smoother the generated surface is; however, in tension spline interpolation, the higher the weight value, the rougher the surface generated.

2.1.4. Global Polynomial Interpolation

Global polynomial interpolation has the basic principle of using a polynomial to compute predicted values based on the set of sample points in the entire study area and then fit a surface or plane over the entire area. In practice, the fitted surfaces rarely coincide exactly with the known sample points, especially at the edges of the study area, which are more susceptible to extremely high and low sample point values, and therefore the global polynomial interpolation is an inexact value interpolation method [7].

2.2. Spatial Accuracy Assessment Methods

In order to validate the accuracy of the models and the effect of site distribution on the interpolation results, this study used the at-site cross-validation method to determine the optimal interpolation method for annual mean temperature. The advantage of this method is that it maximizes the use of observations and avoids the negative impact on the overall accuracy by reducing the number of observations involved in the interpolation process by reserving observations for accuracy verification. The parameters commonly used to evaluate the accuracy of the cross-validation method are the mean error and root-mean-square error. The mean error reflects the size of the estimation error in general, and the root-mean-square error can reflect the estimation sensitivity and the extreme value effect of utilizing the sample points. Cross-validation can accurately test the relative accuracy between different interpolation methods, and the formula for the accuracy evaluation index is as follows:

$$ME={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n(m_i-z_i)}$$

(12)

$$RMSE=\sqrt{{\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n{(m_i-z_i)}^2}}$$

(13)

Included among these, $m_i$ is the actual observed value of annual mean temperature at the ith station, $z_i$ is the result of spatial interpolation of annual mean temperature at the ith station, and n is the number of stations participating in the validation of spatial interpolation [8,9,10].

3. Data Processing and Analysis

3.1. Data Processing

The data in this paper take Jiangxi Province as an example and take the 2018 average temperature of 26 meteorological stations in Jiangxi Province as the basic data, respectively. They use the Kriging interpolation method, the inverse distance weight method of interpolation, and the global polynomial method of interpolation to carry out spatial interpolation for the 18-year average temperature in the whole territory of Jiangxi Province, Figure 1, Figure 2 and Figure 3 show the spatial distribution of the inverse distance weight method, kriging interpolation and spline function method respectively. The three different spatial interpolation methods can reflect the spatial distribution status of the 2018 mean air temperature in Jiangxi Province to a different extent. The 18-year mean air temperature gradually decreases from south to north, with a maximum of no more than 20.13 °C and a minimum of no less than 12.54 °C. The majority of the southern and central regions are around 18–20 °C, and a certain small portion of the northern region is between 12 and 18 °C.

3.2. Data Analysis

From

Table 1. Error of each interpolation method.

	Inverse Distance Weight Interpolation	Kriging Interpolation	Spline Function Interpolation
ME (°C)	0.02	0.03	1.82
RMSE (°C)	1.67	1.22	2.72

, it can be seen that the number of meteorological stations in the spatial interpolation of the mean air temperature in 2018 by the inverse distance weight method is 26, with a mean error of −0.02 °C and a root-mean-square error of 1.67 °C; the number of meteorological stations in the spatial interpolation of the mean air temperature in 2018 by the kriging interpolation method is 26, with a mean error of 0.03 °C and a root-mean-square error of 1.22 °C; and the number of meteorological stations in the spatial interpolation of the mean air temperature in 2018 by the spline function method is also 26, with a mean error of 1.82 °C and a root-mean-square error of 2.72 °C. The number of meteorological stations in the spatial interpolation of the mean temperature is also 26, with a mean error of 1.82 °C and a root-mean-square error of 2.72 °C. For the 2018 mean temperature, in terms of mean error, inverse distance weighting method < kriging interpolation method < spline function method; in terms of root-mean-square error, kriging interpolation method < inverse distance weighting method < spline function method.

However, in the process of interpolation, only the distance is used as the weight, and the influence of other factors such as azimuthal earning is ignored, so the interpolation results have a large deviation from the measured value, and due to the influence of the extreme point or the uneven distribution of the interpolation area monitoring, the formation of the interpolated point is as the center of the circle loaded with the phenomenon, also known as the “bull’s eye” phenomenon. Therefore, in terms of the comprehensive average error and root-mean-square error, the Kriging interpolation method is more effective and more realistically reflects the spatial distribution of the average temperature in Jiangxi Province in 2018, while the inverse distance weighting method is generally effective, and the spline function law has the worst effect.

4. Conclusions and Discussion

In this paper, the inverse distance weight method, Kriging interpolation method, and spline function method were used, respectively, to spatially interpolate the average temperature of Jiangxi Province in 2018, and the interpolation accuracy and the interpolation effect of the annual average temperature were compared and analyzed, and the following conclusions were drawn. First, the annual mean temperature in Jiangxi Province has a high correlation with latitude, and the correlation with longitude is not obvious, showing a gradual decrease in the mean temperature from south to north. Among several methods of spatial interpolation, compared with the inverse distance weight method and the spline function method, the Kriging interpolation method has higher accuracy and is more suitable for reflecting the spatial distribution of the 2018 average temperature in Jiangxi Province.

Secondly, the spatial interpolation method is the basic method for studying the spatial distribution of regional variables, and various methods have their specific assumptions, scope of application, algorithms, advantages, and disadvantages, so it is necessary to compare multiple methods in the research process and select the best one according to the verification accuracy and the actual situation. Choosing the appropriate interpolation method needs to be combined with the characteristics and spatial properties of its original data itself, and it cannot be assumed that the more complex method will produce better results. In this paper, there are many factors affecting the interpolation results, including topographic factors, spatial distribution, and the number of meteorological stations, and the choice of parameters will have a great impact.

The drawback of this paper is that there is not enough meteorological data to conduct further research at this time. As soon as we have access to the latest data, we will use them as a basis for further research to make this project more worthwhile.