Fractional Uncertain Forecasting of the Impact of Groundwater Over-Exploitation on Temperature in the Largest Groundwater Depression Cone

Abstract

China currently faces critical climatic conditions, with persistent global warming trends and extreme heat waves across the northern hemisphere. To explore the predictive trajectory of regional extreme high temperature influenced by groundwater over-exploitation, the SGMC(1,N) was established. Additionally, the SGMC(1,N) model was validated using 2019–2023 observational data from the world’s largest groundwater depression cone. The results demonstrate superior performance, with the model achieving a MAPE of 1.97% compared to benchmark models. Scenario simulations with annual groundwater reduction rates (−15%, −20%, −25%) successfully project extreme heat evolution for 2024–2028. When the decline rate of annual groundwater over-exploitation is set at −20%, a 6.66 °C temperature reduction from baseline by 2028 is projected. Stable decline trends emerge when GOE reduction exceeds 20%. To mitigate regional extreme heat, implementing phased groundwater extraction quotas and total extraction cap regulations is recommended.

1. Introduction

Persistent high temperature has affected the economy in various fields, and extreme heat and drought have led to a reduction in food production. In addition, they have led to shortages in the supply of electricity and water resources in many countries. Vautard et al. used a new set of climate models and statistical extreme value models, and studied two extreme heat waves in Western Europe [1]. Deng assessed some new development concepts and measures proposed by typical cities around the world in response to extreme climate disasters [2]. Beillouin et al. found that extreme weather can increase the likelihood of widespread crop loss based on multiple yield data for nine crops from 17 European countries; analyzing historical data since 1901, with a focus on 2018; and evaluating how climate extremes affect yields [3]. It is also very significant to analyze the reasons for yield losses to use regionally specific single and composite extreme climates. In Europe, the likelihood of a combination of prolonged drought and high temperature events in summer has increased [4]. Hassan and Nile predicted future climate change trends in Iraq [5]. Li et al. considered the extreme of the hot summer in East Asia in 2022; for this component, they examined the relevant atmospheric circulation and evaluated a multi-model U.S. ensemble of real-time projections from the north [6]. Loria and F. studied the impact of ground deformation caused by underground heat islands on civil infrastructure. Underground climate change poses a silent threat to civilian infrastructure in the Chicago Loop and other urban areas around the world [7]. From a mechanistic perspective, the impacts of rising temperatures on groundwater levels can be primarily categorized through two interrelated pathways. Firstly, elevated temperatures directly impact groundwater recharge systems by enhancing surface evaporation, thereby increasing surface runoff, ultimately leading to a reduction in groundwater levels [8]. Secondly, the increased agricultural water demand induced by higher temperatures intensifies groundwater extraction, which disrupts the dynamic equilibrium of aquifer systems [9]. This dual-effect mechanism underscores the complex interplay between climatic forcing and human activities in contemporary groundwater resource management challenges.

However, the above studies did not take into account the impact of groundwater over-exploitation (GOE) on extremely high temperature (EHT). The current situation of GOE in many regions is still severe, and GOE will lead to the intensification of drought problems to a certain extent. Baniasadi et al. evaluated the negative externality of GOE on the economic environment in the plain of Kerman province, Iran, and adopted methods and measures to reduce groundwater consumption [10]. Li et al. proposed a GOE discrimination model based on groundwater dynamic information and multi-factor analysis. By analyzing the groundwater flow field, groundwater depression funnel and water storage change in the Songhua River–Naoli River area in the plain, the groundwater mining output in these areas was determined [11]. Chen et al. studied the features of natural supply in shallow aquifers in the GOE area of North China. The links between groundwater level depth, rainfall, and shallow aquifer recharge were determined [12]. Golian et al.’s study found that by managing groundwater overdraft, it can maintain aquifers and prevent groundwater decline, which, in turn, minimizes or stops ground subsidence [13]. The above study focuses on the negative impacts of GOE on the environment, predicts the depth of the future groundwater tables, etc., and lays a solid foundation for subsequent studies.

In summary, research on both temperature and groundwater has been conducted with increasingly diversified perspectives. Most empirical studies focus on the significant negative impacts of temperature on groundwater systems, indirectly indicating temperature–groundwater correlations. However, research on the influence of GOE on future temperature trends remains nascent. Current temperature prediction models predominantly rely on natural climatic variables (such as CO₂ emissions, land use, and historical temperatures), yet the role of human-driven GOE is often neglected. Although groundwater control policies exist globally, their EHT mitigation potential has not been quantified, hindering water–climate strategy integration. GOE is introduced as an independent predictor, simulating EHT trends under different GOE scenarios. By predicting EHT, this work aims to quantify the mitigation effects of different over-extraction control schemes on extreme heat events. These findings also provide important scientific value for transitioning sustainable water resource management toward climate-adaptive models.

Traditional statistical methods require accurate data and a sufficiently large sample to obtain more accurate results, and they derive correlations, rather than causation, by analyzing and extrapolating existing data. Conventional hydrological models require high-resolution geospatial data and long-term observations, whereas this study only obtained limited GOE data (2019–2023). Such data inadequacy fails to meet the calibration demands of physical models (e.g., SWAT requiring >10-year daily meteorological inputs) or machine learning training thresholds (ANN needing >100 samples). In contrast, gray prediction models extract latent patterns from small samples via data generation techniques, achieving reliable accuracy with ≥4 temporal points [14]. This demonstrates their applicability in GOE scenarios with sparse monitoring. Particle Swarm Optimization (PSO) coefficients can adapt to non-stationary interactions between GOE and EHT processes. This article uses data from 2019 to 2023, which cannot be processed using traditional statistical methods due to the limited data available for GOE. The gray prediction model does not require a large amount of historical data and can predict the future development trends, so this paper adopts the gray prediction model. Professor Deng proposed the gray system model [15], which could forecast future development trends using limited data. The core concept involves latent patterns from limited data through accumulation generation operators, establishing gray differential equations to simulate system dynamics. Compared with other time-series forecasting models, gray prediction models require fewer variables and minimal information, demanding only limited time-series data for modeling. Therefore, it is receiving increasing attention in many studies. For example, Duman et al. established the NBGMC(1,n) model and estimated electronic waste based on the data from Washington [16]. Zhang et al. used the GMCN (1, N) model with similar climatic conditions to forecast PM2.5 and PM10 concentrations in Shijiazhuang [17]. Wu et al. used gray convex relationship analysis and a gray multivariable model considering the total population; they predicted electricity consumption in nine provinces of Shandong [18]. Rehman et al. used the NDGM model to predict GHG emissions from the main sectors in Pukistan [19]. Xie et al. proposed a new reverse-integration fractional-order gray model to forecast Chinese annual electricity consumption [20]. Cai and Wu predicted the growth figures and the exact time of carbon peak in sixteen provinces with increasing proportions of carbon emissions by proposing a new gray model [21]. All of the above studies were carried out by using gray prediction models, which can obtain better prediction results by using limited data. We can use a gray prediction model to study the impact of GOE on high temperatures. Different growth rates of GOE are considered and the SGMC(1,N) [22] is used to predict the extremely high temperature. The SGMC(1,N) model represents another extended form of the fractional-order gray model. Wu et al. proposed a gray model with fractional-order accumulation, which can flexibly capture the dynamic characteristics of data through variable-order mechanisms [23]. Similarly, the SGMC(1,N) model employs the PSO algorithm to determine the optimal non-integer order, which facilitates smoothing of the intrinsic volatility of original data sequences.

The organizational structure of this paper consists of the following sections: The second section expounds the study zone and data. In the third part, modeling and prediction methods are introduced. Based on the third part, the fourth section analyzes the relationship between the average groundwater over-exploitation (AGOE) and AEHT in the world’s largest groundwater depression cone. Next, the fifth part discusses the results of the prediction of GOE for temperature. Finally, the conclusions are presented in the sixth part.

2. Materials and Methodology

2.1. Research Zone and Data

The world’s largest groundwater depression cone is situated in the North China Plain, forming the core of China’s Capital Economic Circle [24,25]. The North China Plain is geographically positioned between 114°30′–119°30′ E longitude and 37°20′–42°40′ N latitude. It is bounded by the Bohai Sea coast to the southeast, the Inner Mongolia Autonomous Region to the north, Liaoning Province to the northeast, Shanxi Province to the northwest, Shaanxi Province to the west, and Henan Province to the southwest.

The groundwater resources of the North China Plain are predominantly hosted within Quaternary aquifers, which are characterized by abundant sedimentary deposits including sands, gravels, cobbles, and soil particles of varying grain sizes. These aquifers exhibit complex hydrogeological architectures due to interbedded clay layers and heterogeneous lithological sequences. This geological complexity creates favorable conditions for groundwater recharge and long-term storage. The Quaternary aquifer system comprises two distinct units: shallow phreatic aquifers and deep confined aquifers. Both units have developed cones of depression resulting from intensive groundwater extraction. The shallow phreatic aquifers, situated at depths ranging from several meters to tens of meters below the ground surface, primarily receive vertical recharge through atmospheric precipitation and stream channel infiltration. Their discharge mechanisms are dominated by anthropogenic extraction and natural evapotranspiration. In contrast, the deep groundwater systems are hosted within unconsolidated porous aquifers at significantly greater depths, exhibiting semi-confined to confined hydraulic properties. Recharge in these deep aquifers originates from delayed precipitation infiltration, surface water leakage, and lateral groundwater inflow. Discharge occurs predominantly through intensive anthropogenic extraction, accounting for over 90% of total outflow. The spatial configurations of groundwater cones of depression exhibit significant variability, attributable to differential extraction intensities, recharge capacities, and hydrogeological characteristics. Shallow depression cones are mainly distributed in the mountainous plain area, while deep funnels are mainly distributed in the eastern plain [26]. The shallow groundwater drawdown funnels mainly include Tianzhu Tongzhou, Gaolisu, Baoding, Shijiazhuang, Ningbailong, Feixiang, and Bazhou, while the deep groundwater depression cone drawdown funnels mainly include Tianjin, Ninghe, Tanghai, Langfang, Qingxian Dacheng, Cangzhou, Jizaoheng, etc. [27].

It has been demonstrated that human GOE is the main cause of groundwater depression cones [28]. The deep groundwater system in the North China Plain has experienced severe disruption of its natural flow regime due to prolonged intensive exploitation. The original hydraulic continuity has been fragmented into multiple intermediate and localized flow systems centered on cones of depression, which cannot effectively restore storage capacity within natural recharge cycles. This situation directly leads to a continuous decline in the groundwater level in the region, forming a unique funnel-shaped water level distribution. A series of ecological problems can be caused by groundwater depression cones [29], such as the depletion of water resources and scraping of wells, as well as ground subsidence, cracking, and collapse. It also exacerbates the degradation of the geological environment and affects the stability of key infrastructure.

Since 2002, the Gravity Recovery and Climate Experiment (GRACE) satellite mission has revealed a groundwater depletion rate of 6–8 billion metric tons per year. Concurrently, monitored groundwater levels have exhibited a sharp decline, with concomitant changes observed in the surrounding groundwater flow field dynamics. Feng et al. conducted in-depth research on groundwater dynamics in North China using GRACE satellite gravity. Their study achieved the first quantitative estimation of groundwater storage changes in this region, systematically delineating spatio-temporal distribution patterns and long-term trends in North China over the past decade [30]. The changes in groundwater are shown in Figure 1.

Figure 1 illustrates that all GRACE data sources (a–c) consistently indicate that the shallow groundwater depression cones in North China are predominantly located in the alluvial fans at the piedmont of the Taihang Mountains, while the deep groundwater depression cones are primarily situated in the central Hebei Plain. Additionally, Figure 1d demonstrates that the forward modeling results are in agreement with the GRACE observations. In Figure 1d, the red circles denote the deep groundwater depression cones and the black circles represent the shallow groundwater depression cones. The rate of groundwater depletion in North China based on GRACE was 2.2 ± 0.3 cm/yr from 2003 to 2010, equivalent to an annual loss of 8.3 ± 1.1 km³ of water.

The North China Plain has cumulatively over-exploited more than 180 billion cubic meters of groundwater. The cumulative over-exploitation in the Beijing–Tianjin–Hebei Plain area is more than 150 billion cubic meters, which accounts for approximately five-sixths of the GOE in North China. The water resources in the three provinces and cities account for only 0.7% of the national total water resources, but carry 8% of the national population and 10% of the total national economic volume. The area and volume of over-exploitation account for about 37% of the country, indicating a kind of predatory exploitation [31]. Cao et al. found that the number of groundwater depression funnels formed in the Beijing–Tianjin–Hebei region has reached 31, forming a large group of groundwater cones of depression [32]. The Beijing–Tianjin–Hebei region serves as China’s capital economic hub and a globally critical groundwater crisis convergence zone. Deep confined aquifers exhibit millennial-scale recharge cycles, where water table decline caused by over-exploitation has permanently disrupted natural flow fields. Shallow cones of depression trigger soil desiccation and land subsidence [33], while deep groundwater depression cones induce bedrock stress imbalances. The above analyses indicate that the Beijing–Tianjin–Hebei region is the most serious GOE area in the North China Plain. The surrounding provinces of Shanxi and Shandong exhibit less severe over-exploitation compared to the three provinces and cities; moreover, reliable datasets from these areas are also difficult to acquire. Therefore, this paper selects the Beijing–Tianjin–Hebei region as the subject if research to represent the world’s largest groundwater depression cone. For clarity, the geographic location of the study area is shown in Figure 2.

The Beijing–Tianjin–Hebei region is located in the northern North China Plain, bordered by the Yanshan Mountains to the north and Taihang Mountains to the west. Its semi-enclosed topography restricts external water replenishment, making groundwater the primary sustainable resource supporting megapolitan socioeconomic systems [34]. According to provincial/municipal statistical yearbooks, Beijing’s domestic water consumption reached 4.07 billion m³ in 2023, marking a 1.75% increase from the previous year. This growth may be attributed to urban expansion and population growth, with additional supply from the ongoing South-to-North Water Diversion Project. Hebei Province recorded total water usage of 18.652 billion m³, reflecting a 2.2% year-on-year rise, potentially linked to industrial expansion and agricultural irrigation demands. Tianjin reported reduced domestic water consumption at 3.272 billion m³ (−2.4% growth rate), demonstrating improved water use efficiency across sectors through effective conservation measures.

This metropolitan cluster encompasses Beijing Municipality, Tianjin Municipality, and 11 prefecture-level cities within Hebei Province. The terrain is predominantly plains interspersed with mountainous and hilly areas. Characterized by a temperate monsoon climate, the region exhibits distinct seasonal variations: hot and rainy summers contrast sharply with cold, dry winters. The Beijing–Tianjin–Hebei region exhibited a significant upward trend in summer mean maximum temperatures from 1961 to 2018, with substantial increases in extreme heat and heatwave frequency [35]. Compared to the reference period (2005–2014), extreme heat intensity is projected to rise by 0.71 °C and 2.12 °C by the mid and late 21st century, respectively, exceeding those under global warming alone by 0.23 °C and 0.58 °C [36]. Additionally, this paper investigated and organized the EHT in Shandong and Shanxi as well as the AEHT in the world’s largest groundwater depression cone from 2015 to 2023. The temperatures in Shanxi, Shandong, and the world’s largest groundwater depression cone for the past few years are shown in

Table 1. Temperatures in Shanxi, Shandong, and the world’s largest groundwater depression cone (°C).

Year	Shanxi	Shandong	The World’s Largest Groundwater Depression Cone
2015	38	39	40.4
2016	38	37	38.2
2017	38	37	40.3
2018	38	40	40.9
2019	39	39	39.63
2020	39	38	39.47
2021	40.4	38	38.23
2022	40	40	41.3
2023	40	40	41.6

.

To make the data more convenient for analysis, we compared them in the form of a bar chart. The bar chart of EHT over the years is shown in Figure 3.

From this figure, we can see that the EHT in the world’s largest groundwater depression cone is mostly higher than the EHT in Shandong and Shanxi. And the GOE in the world’s largest groundwater depression cone is far greater than that in Shandong and Shanxi. Indirectly, this shows that the greater the GOE, the higher the EHT. Therefore, it is of great significance to study the relationship between GOE and EHT in the world’s largest groundwater depression cone, as well as predict future EHT for the ecological environment.

The data on EHT and GOE used in this paper, ranging from 2019 to 2023, were sourced from official websites such as the Water Administration Bureau [37], People’s Political Consultative Conference Network [38], Yanzhao Water Conservancy [39], and Statistical Yearbook [40,41,42].

2.2. SGMC(1,N) Modeling Process

In gray system theory, the smoothed-accumulation-generated algorithm is usually employed. The reason why we use this method is that it has the characteristics of new information priority to some extent and improves the accuracy of model prediction. The SGMC(1,N) model applies weighted smoothing to historical data, effectively reducing volatility in time series. Concurrently, it captures potential persistent trends in the datasets. The specific flowchart of the SGMC(1,N) is presented in Figure 4.

The model construction process of SGMC(1,N) is as follows:

Firstly, the original data are input into the SGMC(1,N) model. Assuming that the original non-negative sequence is recorded as

$${{\mathrm{Q}}_i}^{\left(0\right)}=\left\{{q_i}^{\left(0\right)}(1),{q_i}^{\left(0\right)}(2),\cdots {q_i}^{\left(0\right)}(m)\right\},i=1,2,\cdots ,n.$$

(1)

When $i=1,$ ${{\mathrm{Q}}_1}^{\left(0\right)}=\left\{{q_1}^{\left(0\right)}(1),{q_1}^{\left(0\right)}(2),\cdots {q_1}^{\left(0\right)}(m)\right\}$ is a system characteristic sequence. When $i=2,3,\cdots ,n,$ ${{\mathrm{Q}}_i}^{\left(0\right)}=\left\{{q_i}^{\left(0\right)}(1),{q_i}^{\left(0\right)}(2),\cdots {q_i}^{\left(0\right)}(m)\right\}$ is the related factors sequence.

Then, ${{\mathrm{Q}}_i}^{\left(1\right)}$ is obtained by the first-order smoothed-accumulation-generated operator of ${{\mathrm{Q}}_i}^{\left(0\right)}$.

$$Q_i^{(1)}=\left\{q_i^{(1)}\left(1\right),q_i^{(1)}\left(2\right),\cdots q_i^{(1)}\left(m\right)\right\},i=1,2,\cdots ,n.$$

(2)

Let $Q_i^{(\delta )}=\left.\left\{q_i^{(\delta )}\left(1\right),q_i^{(\delta )}\left(2\right),\cdots ,q_i^{(\delta )}\left(m\right)\right.\right\},i=1,2,\cdots n$, which is the $\delta$-th-order smoothed-accumulation-generated sequence of Equation (1), where the $\delta$-th-order smoothed-accumulation-generated operator is defined as

$$q_i^{(\delta )}(t)=(1-\delta )q_i^{(1)}\left(t-1\right)+\delta q_i^{(0)}\left(t\right),i=1,2,\cdots ,n,t=1,2,\cdots ,m.$$

(3)

In the sequence generation process, $\delta$ is calculated by PSO. $\delta$ can adjust the weight between current and historical information to minimize the mean absolute percentage error (MAPE) of the model, where the parameter $\delta \in (0,1]$.

The whitening equation expression of SGMC(1,N) can be obtained.

$${\displaystyle \frac{d{q_1}^{\left(\delta \right)}(t)}{dt}}+k_1{q_1}^{\left(\delta \right)}\left(t\right)=k_2{q_2}^{\left(\delta \right)}\left(t\right)+k_3{q_3}^{\left(\delta \right)}\left(t\right)+\cdots +k_n{q_n}^{\left(\delta \right)}\left(t\right)+c(t=1,2,\cdots m).$$

(4)

where ${\displaystyle \frac{d{q_1}^{\left(\delta \right)}\left(t\right)}{dt}}={q_1}^{\left(\delta \right)}\left(t+1\right)-{q_1}^{\left(\delta \right)}\left(t\right)$.

So, Equation (5) can be changed to

$$q_1^{(0)}\left(t\right)+k_1{z_1}^{\left(\delta \right)}\left(t\right)=k_2{z_2}^{\left(\delta \right)}\left(t\right)+k_3{z_3}^{\left(\delta \right)}\left(t\right)+\cdots +k_n{z_n}^{\left(\delta \right)}\left(t\right)+c(t=1,2,\cdots m).$$

(5)

where $k_1,k_2,\cdots ,k_n,c$ are parameters to be estimated for the SGMC(1,N) model. $z_1^{(1)}(1),z_2^{(1)}\left(2\right),\cdots ,z_n^{(1)}\left(t\right)$ is the background value sequence of $q_i^{(\delta )}\left(t\right)$ in the interval (t − 1, t), which can be represented by the mean formula,  $z_i^{(\delta )}(t)={\displaystyle \frac{q_i^{(\delta )}(t-1)+q_i^{(\delta )}(t)}{2}},t=2\cdots m$. The parameters are estimated using the least squares method and denoted as

$$\left[\begin{array}{c}{k}_1\\ k_2\\ ⋮\\ k_n\\ c\end{array}\right]={\left(B^{T}B\right)}^{-1}{B}^{T}D.$$

(6)

where $D=\left[\begin{array}{c}{q_1}^{\left(\delta \right)}\left(2\right)-{q_1}^{\left(\delta \right)}\left(1\right)\\ {q_1}^{\left(\delta \right)}\left(3\right)-{q_1}^{\left(\delta \right)}\left(2\right)\\ ⋮\\ {q_1}^{\left(\delta \right)}\left(m\right)-{q_1}^{\left(\delta \right)}\left(m-1\right)\end{array}\right]$, $B=\left[\begin{array}{ccccc}-{\displaystyle \frac{q_1^{\left(\delta \right)}\left(1\right)+q_1^{\left(\delta \right)}\left(2\right)}{2}}& {\displaystyle \frac{q_2^{\left(\delta \right)}\left(1\right)+q_2^{\left(\delta \right)}\left(2\right)}{2}}& \cdots & {\displaystyle \frac{q_n^{\left(\delta \right)}\left(1\right)+q_n^{\left(\delta \right)}\left(2\right)}{2}}& 1\\ -{\displaystyle \frac{q_1^{\left(\delta \right)}\left(2\right)+q_1^{\left(\delta \right)}\left(3\right)}{2}}& {\displaystyle \frac{q_2^{\left(\delta \right)}\left(1\right)+q_2^{\left(\delta \right)}\left(2\right)}{2}}& \cdots & {\displaystyle \frac{q_n^{\left(\delta \right)}\left(1\right)+q_n^{\left(\delta \right)}\left(2\right)}{2}}& 1\\ ⋮& ⋮& ⋮& ⋮& ⋮\\ -{\displaystyle \frac{q_1^{\left(\delta \right)}\left(m-1\right)+q_1^{\left(\delta \right)}\left(m\right)}{2}}& {\displaystyle \frac{q_2^{\left(\delta \right)}\left(m-1\right)+q_2^{\left(\delta \right)}\left(m\right)}{2}}& \cdots & {\displaystyle \frac{q_n^{\left(\delta \right)}\left(m-1\right)+q_n^{\left(\delta \right)}\left(m\right)}{2}}& 1\end{array}\right]$.

$B$ is the data matrix, and $D$ is the predicted sequence.

According to the Gaussian formula, the approximate time response function of SGMC(1,N) is

$${\widehat{q}}_1^{\left(\delta \right)}(t)=q_1^{\left(0\right)}(1)e^{-k_1\left(t-1\right)}+{\displaystyle \sum _{\tau =2}^t\left(e^{-k_1\left(t-\tau +0.5\right)}\times \frac{g\left(\tau \right)+g\left(\tau -1\right)}{2}\right)}.$$

(7)

where $g\left(\tau \right)=k_2{q_2}^{\left(\delta \right)}\left(\tau \right)+k_3{q_3}^{\left(\delta \right)}\left(\tau \right)+\cdots +k_n{q_n}^{\left(\delta \right)}\left(\tau \right)+c$.

Let ${\widehat{q}}_1^{\left(\delta \right)}(1)=q_1^{\left(0\right)}(1)$; the predictive sequence of the original can be obtained by accumulated subtraction.

$${\widehat{q}}_1^{\left(0\right)}(t+1)={\displaystyle \frac{{\widehat{q}}_1^{\left(\delta \right)}(t+1)-\left(1-\delta \right){\displaystyle \sum _{i=1}^k{q}_1^{(0)}(t)}}{\delta }}.$$

(8)

MAPE is a widely used metric to evaluate model accuracy. It quantifies prediction deviations by calculating absolute percentage errors between predicted and actual values, then averaging them across all data points. When the MAPE value is less than 10%, it is generally considered to indicate a relatively good forecasting model with high prediction accuracy. The expression for MAPE is as follows.

$$\mathrm{MAPE}={\displaystyle \frac{1}{m}}{\displaystyle \sum _{k=1}^m\left|\frac{{\widehat{q}}_1^{\left(0\right)}(t)-q_1^{\left(0\right)}(t)}{q_1^{\left(0\right)}(t)}\right|}\times 100\%.$$

(9)

3. Results and Discussion

3.1. Results of Fitting Average Groundwater Over-Exploitation to AEHT

In the world’s largest groundwater depression cone, the EHT has been influenced by the GOE over the years. Based on the link between the two, this article establishes the SGMC(1,2) model to forecast the AEHT trend at different growth rates of GOE over the years. For the convenience of research, the AGOE and AEHT in the world’s largest groundwater depression cone from 2019 to 2023 are presented in

Table 2. AGOE and AEHT in the world’s largest groundwater depression cone.

Year	2019	2020	2021	2022	2023
AGOE (100 million cubic meters)	5.45	4.1	4.6	4.1	2.23
AEHT (°C)	39.63	39.47	38.43	41.30	40.67

.

Table 2 reveals that the AGOE in the world’s largest groundwater depression cone has great fluctuations, and the AEHT shows an increasing trend over the years. It is also found that the GOE and the temperature show a non-linear relationship. Since the SGMC(1,2) model is suitable for studying the non-linear relationship and poverty information data, using the SGMC(1,2) model to analyze this set of data is very reasonable.

The original data sequence is ${\mathrm{Y}}_1^{\left(0\right)}=\left\{5.45,4.1,4.6,4.1,2.23\right\},$ ${\mathrm{Y}}_2^{\left(0\right)}=\left\{39.63,39.47,38.43,41.30,40.67\right\}$.

The optimal value $\delta =0.8612$ is obtained through the PSO algorithm. PSO utilizes swarm intelligence to explore parameter space and identify the globally optimal solution. In this model, $\delta$ is adjusted to maximize the consistency between the predicted data and original data, thereby minimizing the MAPE. When $\delta$ deviates from this value, MAPE increases significantly and reduces the prediction accuracy. Any deviation from the $\delta$ value results in a significant increase in MAPE and a consequent reduction in prediction accuracy. The parameter $\delta =0.8612$ represents the fractional order of the SGMC(1,N) model. This non-integer configuration significantly enhances the fitting capability for non-stationary data through dynamic order adaptation. This model serves as a non-linear extension of the fractional-order accumulation gray model by transcending traditional first-order accumulation constraints. This enhancement significantly improves the adaptability of the gray system in complex scenarios.

When $\delta =0.8612$,

$$\mathrm{D}=\left[\begin{array}{c}-0.1435\\ 4.5888\\ 7.8039\\ 5.1878\end{array}\right],\mathrm{B}=\left[\begin{array}{ccc}-39.5616& 4.8687& 1\\ -41.7842& 4.7873& 1\\ -47.9806& 5.3911& 1\\ -54.4764& 4.9745& 1\end{array}\right].$$

By using the above matrix, the parameters of the SGMC(1,2) model can be found:

$$\left[\begin{array}{c}{k}_1\\ k_2\\ c\end{array}\right]=\left[\begin{array}{c}-0.1988\\ 6.5324\\ -37.472\end{array}\right].$$

Therefore, the whitening differential equation of SGMC(1,2) is

$${\displaystyle \frac{d{y}_1(t)}{dt}}-0.1988{y_1}^{\left(1\right)}\left(t\right)=6.5324{y_2}^{\left(1\right)}\left(t\right)-37.472.$$

We can obtain a time response equation,

$$y_1^{\left(1\right)}(t)=y_1^{\left(0\right)}(1)e^{0.1988\left(t-1\right)}+{\displaystyle \sum _{\tau =2}^t\left(e^{0.1988\left(t-\tau +0.5\right)}\times \frac{g\left(\tau \right)+g\left(\tau -1\right)}{2}\right)},$$

where $g\left(t\right)=6.5324{y_2}^{\left(1\right)}\left(t\right)-37.472$.

Therefore, ${\widehat{y}}_1^{(1)}=37.6,{\widehat{y}}_2^{(1)}=38.63,{\widehat{y}}_3^{(1)}=39.61,{\widehat{y}}_4^{(1)}=40.84,{\widehat{y}}_5^{(1)}=41.55$,

$${\hat{\mathrm{Y}}}_1^{(0)}=\left\{39.63,42.49,38.43,40.71,40.99\right\}$$

3.2. Comparison of Fitting Accuracy Between Different Models of GOE and EHT

The SGMC(1,2), GMCN(1,2), GM(0,2), and GM(1,2) models are used for comparison, and the fitting accuracy of AGOE and AEHT is obtained. Figure 5 and

Table 3. Fitting values of the four models.

Year	Actual Value	SGMC(1,2)	GMCN(1,2)	GM(0,2)	GM(1,2)
2019	39.63	39.63	39.63	36.28	39.63
2020	39.47	42.49	37.37	41.70	47.28
2021	38.43	38.43	38.34	46.78	56.88
2022	41.30	40.71	38.7	41.70	47.24
2023	40.67	40.99	38.96	22.68	24.91
MAPE (%)		1.97	3.21	16.2	24

, respectively, present the fitting curves and value results of the four models.

To systematically validate the effectiveness of the proposed SGMC(1,N) model, this study employed a comparative analytical framework incorporating representative gray system theoretical models—GM(1,N), GM(0,N), and GMCN(1,N)—as benchmark references. Through rigorous experimental evaluation using the mean absolute percentage error (MAPE) as the primary performance metric, the predictive capabilities of each model were quantitatively assessed. According to established criteria in gray system theory, models achieving MAPE values below 10% demonstrate functional validity for research applications.

The performance metrics across the test dataset are summarized in Table 3. The SGMC(1,N) model exhibits superior predictive precision, with a MAPE of 1.97%. This represents a substantial improvement over conventional approaches, particularly when compared to the GM(1,2) (24%), GM(0,2) (16.2%), and GMCN(1,2) (3.21%) models. The enhanced fitting capability of the SGMC(1,2) variant is further visually substantiated in Figure 5, where its trajectory closely aligns with observational data, while competing models exhibit significant deviations from empirical values. Consequently, this model demonstrates strong applicability while maintaining prediction reliability. It performs particularly well in hydrological applications requiring temperature prediction within large-scale groundwater depression cones.

3.3. Prediction and Discussion of AEHT

The smaller the MAPE, the higher the correlation between high temperature and GOE. In this article, the SGMC(1,2) model has higher accuracy. The AEHT of the world’s groundwater depression cones can be predicted by using this model at different growth rates of GOE.

The huge voids beneath the surface layers in North China have created groundwater depression cones in the past few years, leading to a range of ecological and environmental problems and affecting the sustainable development of society. Nowadays, through improvement, a reduction in GOE, and artificial intervention, the groundwater has been a significant improvement in North China. This paper combines the data related to the average amount of GOE in the world’s largest groundwater depression cone and assumes that the growth rate of GOE is −15%, −20%, and −25%. The percentage growth rate is determined for three reasons. Firstly, in recent years, over-exploitation has led to severe water shortages in the world’s largest groundwater depression cone. We aim to predict the impact on AEHT at different rates. If GOE decreases, will the climate improve in the future? Secondly, based on historical data of GOE, the annual growth rate is approximately −17.26%. The three selected growth rates encompass the core distribution range of historical variations, thereby enhancing the scientific rigor and thoroughness of the study. Thirdly, the selection of these three growth rates is intended to reflect future trends in temperature changes under varying intensities of regulatory policies. Modeling the SGMC(1,2) can predict the AEHT in the world’s largest groundwater depression cone, where the calculation process is identical to that in the third section.

This article takes the GOE from 2019 to 2023 as a sequence of relevant factors. Then, it predicts the AEHT in the world’s largest groundwater depression cone from 2024 to 2028 by establishing an SGMC(1,2) model. The prediction curves and value results are shown in Figure 6 and

Table 4. Temperature prediction results of GOE at different growth rates (°C).

Influencing Factor	Growth Rate	2024	2025	2026	2027	2028
GOE	−15%	38.46	37.40	37.78	39.63	43.03
	−20%	38.05	35.83	34.38	33.75	34.01
	−25%	37.65	34.33	31.20	28.40	26.02

.

When the average growth rate of GOE is −15%, the trend of high temperatures in the world’s largest groundwater depression cone shows a decrease first, followed by an increase. This trend of temperature change is unstable. When the growth rate is −20% or −25%, the temperature curve as a whole shows a downward trend, and the lower the GOE is, the lower the temperature is. Therefore, when the GOE decreases by more than 15%, the improvement in temperature is more obvious. And the larger the reduction in GOE is, the faster the rate of temperature decrease will be. This is mainly due to the reduction in and control of GOE, which mitigates the risks associated with surface drought, collapse, and low precipitation. Fu et al. revealed that severe shortages in groundwater resources can lead to an increase in regional temperatures [43]. This finding is, to some extent, consistent with the results presented in this paper. Therefore, reducing the extraction of groundwater would help alleviate the stress on regional hydrological systems and contribute to the stabilization of local climatic conditions.

The population in North China accounts for only 6% of the total water resources in the country, but industrial and agricultural development has a great demand for water resources, exacerbating the supply–demand contradiction of water resources. The Ministry of Agriculture of the northern provinces that engage in water-saving agriculture has also conducted multi-frequency technological promotion in recent years. With the implementation of the South-to-North Water Diversion Project, GOE in North China has been alleviated, which could also contribute to reducing the rate of temperature rise.

4. Conclusions and Future Perspectives

In this paper, firstly, by fitting the data, it is found that the SGMC(1,2) model can be employed to study the relationship between GOE and EHT and make predictions by using fewer data. This reflects the effectiveness of the model. Then, the AEHT in the world’s largest groundwater depression cone is predicted under different growth rates of GOE. It is found that when the growth rate of GOE decreases, the temperature decreases, and when it increases, the temperature increases.

Therefore, the following recommendations are made based on the predicted results. Firstly, to ensure that urban water use is not affected, it is necessary to reduce GOE, extract groundwater under pressure, and switch to surface water. The extraction of water resources should be carried out according to the actual situation, and scientific methods should be utilized to invoke water resources.

Secondly, it is recommended that relevant departments actively enforce the national policy regarding water-saving technology. This action should be carried out in strict accordance with the principle of balanced space, governance systems, and dual efforts. For the necessary industries, agricultural and industrial water should be subjected to further industrial and domestic water compression and replacement.

In addition, water resources should be utilized wisely in daily life to effectively raise people’s awareness of water conservation. Agricultural and industrial water use should also be formulated to comply with water standards, and water conservation measures, phased groundwater extraction quotas, and total extraction cap regulations should be established.

Finally, it is essential to continue to optimize the South-to-North Water Diversion Project system and implement normalized water replenishment for rivers and lakes in North China. If the demand for water increases in the future, other ways to rationalize the utilization of water resources should also be applied.

This paper employs the SGMC(1,N) model to identify GOE as a critical driving factor, enabling EHT trend prediction under small-sample conditions. This study exclusively incorporates GOE as the driving factor, while the synergistic or antagonistic interactions between GOE and other critical hydrological and meteorological elements remain a research gap. Under the constraints of observational data, this research could not employ physical models requiring multi-parameter coupling. Additionally, potential time-lag effects between GOE and the EHT might obscure the detection of their causal relationships.

Other influencing factors on high temperature should be taken into account in future studies. The cumulative effects of greenhouse gas concentrations exert a significant forcing effect on long-term temperature change trends. Urban green space coverage, agricultural irrigation practices, and soil moisture may further modulate the spatio-temporal distribution characteristics of extreme heatwave events through alterations in surface heat redistribution. Future research should integrate these multi-scale driving factors to enhance the model’s predictive capability for extreme heatwave events. Furthermore, the gray prediction model can be further improved to solve other similar problems of prediction in other cities. Future studies could integrate gray prediction models with mechanistic hydrological models. This integration would balance the complementary strengths of data-driven and mechanism-driven approaches under sufficient datasets. Machine learning or intelligent optimization algorithms could synchronously calibrate the parameters of both models. Such calibration strategies may enhance predictive performance for extreme high-temperature events.