Geothermal Genesis Mechanism of the Yinchuan Basin Based on Thermal Parameter Inversion

Abstract

The Yinchuan Basin harbors significant geothermal resource potential and could be a clean energy source critical for transitioning to a low-carbon economy. However, the current research primarily focuses on the exploration and development of geothermal water in the sedimentary basins, with limited studies on the deep geothermal formation mechanisms and regional geothermal types. Although geophysical methods provide insights into the types and formation mechanisms of deep geothermal resources in the basin, there is still a lack of a connection between quantitative understanding and direct evidence. A series of algorithms based on thermal parameter characteristics can directly extract underground thermal features from raw geophysical signal data, offering a powerful tool for characterizing the structure and aggregation patterns of deep thermal sources. Therefore, this study employed a Bayesian thermal parameter inversion method based on interface information to obtain the spatial distribution of thermal conductivity, surface heat flow, and mantle heat parameters in the Ningxia Basin study area. Additionally, correlation analysis and global sensitivity analysis were conducted to further interpret the predicted results. A comprehensive analysis of the geophysical inversion results showed that the deep thermal anomalies in the basin are primarily controlled by fault activities and the lithospheres’ thermal structure, while shallow high-heat flow anomalies are closely related to convective circulation within faults and heat transfer from deep thermal sources. The established geothermal genesis mechanism and model of the Yinchuan Basin provide crucial support for sustainable regional geothermal development planning and the utilization of deep geothermal resources, contributing to energy security and emission reduction goals.

1. Introduction

Geothermal energy, as a clean source of energy from the Earth’s deep interior, is of significant value for sustainable energy utilization and mitigating the atmospheric pollution caused by fossil fuels [1,2]. The Yinchuan Basin, located in the northwest of China, contains abundant geothermal resources due to its unique geological structure. Geothermal research in the Yinchuan Basin began in the 1980s, when the Ningxia Geological Survey Institute first discovered underground hot water in the Huanghipo Gully of Zhongwei County and successfully drilled the first geothermal well (Y1) with a surface water temperature of 67.5 °C [3]. Subsequent studies have extensively investigated the basin’s geological conditions, geothermal reservoir characteristics, and resource potential, proposing that low- to medium-temperature geothermal resources are the predominant type in the Yinchuan Basin [4]. Zhu et al. (2020) conducted field surveys and drilling verifications, discovering regional faults and other structures in the deep parts of the basin; they concluded that the region has conditions suitable for the formation of banded geothermal reservoirs [5]. Chen et al. (2021) studied the geothermal formation mechanisms and heat transfer modes. They suggested that the region exhibits a “conduction–convection hybrid” geothermal type and categorizing the geothermal type of the eastern uplift zone of the Yinchuan Basin accordingly [6]. These studies have revealed the geo-thermal accumulation pattern of the Yinchuan Basin and laid the foundation for the use of the abundant geothermal resources in the region. However, the current research mainly focuses on the exploration and development of geothermal water in the sedimentary basins, with limited studies on the deep geothermal formation mechanisms and regional geothermal types.

Geophysical methods provide reliable evidence for characterizing the spatial distribution of underground structures and deep geothermal mechanisms. Some geophysical studies have been conducted in the Ningxia Basin, including magnetotelluric, gravity, seismic, and radiometric measurements. Yang et al. (2009) used a 3D seismic array in the northern part of the basin to image the basement (7 km) and upper crust structure, revealing a prominent east-to-west basement depth difference [7]. Liu et al. (2017) used a 135 km long deep seismic reflection profile to reveal the P-wave and S-wave velocity structures beneath the profile, confirming a gradual east-to-west increase in Moho depth (40–48 km crustal thickness) [8]. Wang et al. (2023) obtained the 3D S-wave velocity structure of the crust–mantle system beneath the basin using joint inversion of receiver functions and surface waves, revealing the relationship between the fault zones and dense seismic activity [9]. Zhang (2021) conducted a controlled-source magnetotelluric survey to investigate the distribution characteristics of the Cenozoic sedimentary strata in the shallow and aboveground layers of the basin [10]. Hu et al. (2021) delineated the stepped fault structures and graded characteristics of the basin region using gravity data at a scale of 1:200,000 [11]. Wang (2015) studied the types of thermal reservoirs in the Yinchuan Plain region using major electrical surveys and geological profile analysis [12]. These studies provide strong evidence for understanding the deep geothermal source mechanisms and thermal reservoir patterns in the basin. However, there is a lack of consistency between the results from different geophysical surveys, and the indirect findings require further validation through acquiring direct geothermal resource evidence.

Another method is to indirectly estimate the heat flow distribution through geophysical or petrophysical relationships [13,14]. For example, regional magnetic anomaly data can be used to estimate the Curie depth, and the temperature at the Curie point (580 °C) can be used to estimate the geothermal gradient and the distribution of the terrestrial heat flow [15,16]. The seismic passive noise velocity structure is related to the heat source parameters, with low-velocity anomalies typically associated with elevated temperatures [17,18,19]. Archie’s law relates electrical conductivity to temperature, and electrical methods can be used to obtain the distribution of deep heat sources [20]. Additionally, thermal simulation processes can provide reliable heat source data within the drilling range, relying on prior geological information and geophysical results. Lösing et al. (2020) used a one-dimensional heat equation based on the MCMC method to simulate the temperatures of the underground Curie isotherm, Moho, and LAB interfaces, obtaining heat flux data for Antarctica [21]. Louis et al. (2021) demonstrated a deep 3D temperature model of the UK using finite difference techniques based on given thermal conductivity and heat production rates of the stratigraphic units [22]. McAliley et al. (2019) determined the regional heat-conductivity distribution of the Cooper Basin in Australia using drilling temperature and heat flow data as observational inputs [23]. Bai et al. (2023a) established a geothermal parameter model to understand the underground heat transfer form, conducted numerical simulations and thermal parameter inversion using drilling data and heat flow point information, and analyzed the formation mechanisms of potential geothermal resource reservoirs [24]. Wang (2019) applied Bayesian inversion to the Qinghai Gonghe Basin to obtain geothermal field inversion, improving and refining our understanding of the formation mechanisms of dry hot rock geothermal resources and the evaluation of thermal reservoirs in the region [25].

This study investigated the geothermal resource distribution and genetic mechanisms of the Yinchuan Basin using a Bayesian thermal parameter inversion method constrained by the interface characteristics (Moho, Curie, and LAB depths). First, the regional geological setting and geothermal distribution of the Ningxia Basin are outlined. A one-dimensional thermal model was analytically formulated, incorporating thermal parameters (thermal conductivity and heat production rate) and interface depths as inputs. The inversion employs a Markov Chain Monte Carlo (MCMC) algorithm to estimate posterior distributions of parameters. Two synthetic heat flow column models were designed to validate the method under high-heat anomaly and conventional scenarios. Correlation analysis and global sensitivity analysis were integrated to interpret the predicted thermal characteristics. Finally, combining the inversion results with regional geophysical data, the geothermal genesis mechanism and model of the Yinchuan Basin were established, offering critical insights for regional geothermal development planning and deep resource utilization.

2. Regional Geology and Geothermal Overview of the Ningxia Basin

The Yinchuan Basin is located in the northern part of Ningxia, extending from the eastern edge near Tao Le Town on the east bank of the Yellow River, south to Niushou Mountain, west to the western boundary of the Helan Mountains, and northward, converging in the northeast of Huinong District. The basin is 10–50 km wide from east to west and about 160 km long from north to south, covering an area of approximately 7000 square kilometers (Figure 1. The stratigraphy of the Yinchuan Basin belongs to the North China–Qaidam Basin stratigraphic region, the North China stratigraphic region, and the western edge of the Ordos Basin, with exposed strata from the Paleoproterozoic, Triassic, Jurassic, Neogene, and Quaternary periods [26]. The basin is well developed, with faults that are widely distributed between 106° E and 107° E, mainly in a northeast direction. The major faults include the West Helan Mountain (F3), East Helan Mountain (F4), Luhuatai (F5), Yinchuan (F6), and Yellow River (F7) faults [27,28].

To further study the geothermal resources in the Yinchuan Basin, extensive drilling and temperature measurements have been carried out in the region (Figure 1). Data from three completed deep geothermal wells show that the bottom temperatures of wells SY-01 (1360 m), DRT-03 (1690 m), and HDZ (640 m) are 52.4 °C, 64.03 °C, and 30 °C, respectively [6,29,30]. The logging results indicate that the geothermal field consists of cap rocks, reservoirs, heat sources, and hot water sources. The cap layer is mainly composed of Quaternary loose sediments, with a thickness ranging from 1600 to 2000 m. These sediments include gravel, gravel sandstone, gravel-containing sand, and coarse sand and fine sand mixed with clay. This layer has poor permeability, providing a good seal for preserving the geothermal temperature. The geothermal reservoir mainly consists of crystalline limestone and siliceous dolomite layers from the middle Ordovician hot reservoir section; it has a high porosity, providing excellent conditions for heat storage. The geothermal distribution map shows two high-temperature anomaly regions with high gradient geothermal anomalies: one in the west, along the Helan Mountain fault, and the other in the east, along the strip from Tongfu Township to Linhe Town. Due to the sparse drilling data, an accurate characterization of the regional temperature cannot be achieved. However, the distribution pattern of high temperatures is quite distinct, providing a basis for studying the relationship between geothermal anomalies and the strata and structures.

3. Data and Methods

3.1. Data Description

The Curie depth and Moho depth in the Yinchuan Basin from [31] are shown in Figure 2a,b. The Curie depth in the basin ranges from 15 to 32 km, with a continuous interface uplift in the northern and eastern regions, forming a shallow band distribution oriented in the southeast direction. In the Qingtongxia and Yongning areas, the Curie depth is deeper, corresponding to high-value magnetic anomalies, with the deepest depth reaching 25 km. Additionally, the distribution of the Curie depth shows good spatial continuity, with a distinct gradient belt between shallow and deep layers, trending from northwest to southeast. The Moho depth ranges from 41 to 49 km, with a variation of 8 km, showing a gentler undulation compared to the Curie isotherm. In the central region, the Moho is depressed, ranging from 46 to 49 km, and gradually rises outward, forming a distinct high–low–high depth distribution pattern. In the Moho depression area, there are two strong transition zones with northeastward-trending interface changes, which coincide with the high-gradient gravity anomaly zones. Figure 2c shows the lithosphere–asthenosphere boundary (LAB) distribution for the Ningxia region, sourced from the global LAB database [32]. This study uses kriging interpolation to fill in the LAB interface for the Ningxia region, revealing a depth range of 58 to 113 km, with significant variation and an alternating anomaly distribution.

3.2. Geothermal Parameter Modeling

Assuming that the subsurface thermal state is not affected by convection and only considering the heat transfer process, the one-dimensional temperature conduction equation can be written as [33,34]

$$k_1{\displaystyle \frac{{\partial }^{2}T}{\partial z^2}}=-h(z)$$

(1)

where $T$ represents the temperature, $z$ represents the depth, and $k_1$ and $h\left(z\right)$ represent the average thermal conductivity of the strata and the heat production rate. The radioactive materials in the crust generate heat, but this heat production decreases with depth and eventually disappears. Below the Moho, it is assumed that there is either no or an extremely small amount of radiogenic heat production [35]. The temperature within the lithospheric mantle increases linearly, primarily determined by the thermal conductivity of the deep strata and the heat flux at the Moho:

$$T(z)=T(M)+{\displaystyle \frac{q_D}{k_2}}(z-M)$$

(2)

where $T\left(M\right)$ is the temperature at the Moho, $q_D$ is the heat flux at the Moho, and $k_2$ is the average thermal conductivity within the lithosphere. The heat production above the crust cannot be ignored, and the heat production rate usually depends on the integral over depth:

$${\int }_0^z h\left(z^{\prime }\right)d{z}^{\prime }=H(z)$$

(3)

$${\int }_0^z H\left(z^{\prime }\right)d{z}^{\prime }=\mathcal{H}(z)$$

(4)

By integrating the heat production rate from 0 to $z$ and substituting it into the temperature equation, the expression for heat flux at a specific depth can be obtained:

$$q\left(z\right)=q_0-H\left(z\right)$$

(5)

Equation (5) can be rewritten as

$$T(z)=T(0)+{\displaystyle \frac{q_{0}z-\mathcal{H}(z)}{k}}$$

(6)

where $T\left(0\right)$ and $q_0$ are the surface temperature and surface heat flow, respectively. Assuming the heat production rate decays exponentially with depth,

$$h\left(z\right)=H_{0}exp\left(-z/h_r\right)$$

(7)

where $H_0$ is the heat production rate and $h_r$ is the depth at which $H_0$ decays to 1/e of its initial value. After integrating twice,

$$\left\{\begin{array}{c}H\left(z\right)=H_0{h}_r\left(1-exp\left(-z/h_r\right)\right)\\ \mathcal{H}\left(z\right)=H_0{h}_r\left(z+h_{r}exp\left(-z/h_r\right)-h_r\right)\end{array}\right.$$

(8)

Substituting into Equation (8), we obtain the heat flux and temperature at the Moho depth:

$$q_D=q_0-H_0{h}_r\left(1-exp\left(-M/h_r\right)\right)$$

(9)

$$T(M)=T(0)+{\displaystyle \frac{M{q}_0-H_0{h}_r\left(M+h_{r}exp\left(-M/h_r\right)-h_r\right)}{k_1}}$$

(10)

The temperatures at the Curie depth and the LAB can be expressed as

$$T\left(C\right)=T\left(0\right)+{\displaystyle \frac{q_D\times C+H_0{h_r}^2\left(1-exp\left(-C/h_r\right)\right)}{k_1}}$$

(11)

$$T\left(\mathrm{L}\mathrm{A}\mathrm{B}\right)=T\left(0\right)+{\displaystyle \frac{q_D\times M+H_0{h_r}^2\left(1-exp\left(-C/h_r\right)\right)}{k_1}}+(LAB-M){\displaystyle \frac{q_D}{k_2}}$$

(12)

where $C$ and $LAB$ represent the depths of the Curie depth and the lithosphere–asthenosphere boundary. $T\left(C\right)$ and $T\left(LAB\right)$ are the temperatures at the Curie depth and the LAB. According to previous studies, the Curie temperature of underground magnetite is approximately 600 °C ± 50 °C. This study considered the results of Martos et al. (2015) and assigned a Curie depth temperature of 580 °C to the region [14]. An et al. (2015) proposed that there is an adiabatic layer at the base of the lithosphere, where the temperature at this interface can reach up to 1330 °C and the velocity of the strata below this depth significantly decreases [13]. This study referenced previous results and set the LAB temperature at 1315 °C. It is worth noting that the thermal parameters in actual strata often exhibit a non-uniform distribution with depth. For example, due to compaction and high-temperature melting of underground rock layers, thermal conductivity increases with depth, resulting in a gradual stabilization of the underground temperature and a decrease in deep heat flow [21]. The thermal parameter estimation in this study was obtained using the temperature from lateral interface information. Since the observational data of individual heat flow columns were limited, estimating thermal conductivity at different depths would significantly decrease the accuracy of the inversion. Therefore, this study adopted the concept of average thermal parameters and inverted the thermal conductivity at different depth segments, treating heat flow as a comprehensive response of regional thermal parameters [13,33]. This reasonable simplification improves the computational efficiency and ensures the accuracy of the model.

3.3. MCMC Inversion

In the derivation process mentioned above, there are four unknown parameters that cannot be directly obtained: thermal conductivities $k_1$ and $k_2$, Moho heat flux $q_D$ (or surface heat flux $q_0$), and heat production rate $H_0$. This prevents the use of analytical solutions in the forward process to solve the inverse problem. For high-dimensional problems with multiple unknowns, optimization theory can be used to inversely estimate the best-fitting model parameters. Based on Bayes’ theorem, the calculation of inversion parameters can be achieved [36,37]. Bayes’ theorem can be expressed as

$$P(\alpha \mid d)={\displaystyle \frac{P(d\mid \alpha )P\left(\alpha \right)}{P\left(d\right)}}$$

(13)

where $\alpha$ and $d$ are the parameters to be inverted (i.e., $k_1$, $k_2$, $q_D$ or $q_0$, and $H_0$) and $T\left(C\right)$ and $T\left(LAB\right)$ are the observed data. $P\left(\alpha \right)$ is the prior distribution, representing the knowledge of the model parameters before obtaining the observed data, which was primarily derived from past experience and subjective judgment. $P(d\mid \alpha )$ is the likelihood function, indicating how well the model parameters fit the measurement data—the larger it is, the better the fit. $P(\alpha \mid d)$ is the posterior probability density function, which corrects and updates the prior probability and represents the solution to the inverse problem in a statistical inversion sense. The uniform distribution is used as the prior distribution [38]. Suppose $\alpha$ represents the unknown parameters of the model, and d represents the observed data. There are m unknown parameters in the model, so $\alpha =\left({\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_{\mathrm{m}}\right)$. Assuming each parameter follows a uniform distribution and they are independent of each other, the prior probability density function of the model parameter ${\alpha }_i$ can be defined as

$$P\left({\alpha }_i\right)=\left\{\begin{array}{c}{\displaystyle \frac{1}{b_i-a_i}},a_i\le {\alpha }_i\le b_i\\ 0,\hspace{1em}\hspace{1em}\hspace{1em}Others\hfill \end{array}\right.$$

(14)

where $a_i$ and $b_i$ are the lower and upper limits of the prior distribution, respectively. The total prior distribution $P\left({\alpha }_i\right)$ can be expressed as

$$P(\alpha )={\prod }_{i=1}^m P\left({\alpha }_i\right)$$

(15)

In Bayesian computation, the likelihood function is usually considered in various forms, including Gaussian distribution, Laplace distribution, and Dirac distribution. When observational errors are known, the Gaussian distribution is used as the likelihood function to obtain the maximum likelihood estimate:

$$P(d\mid \alpha )={\displaystyle \frac{1}{{\left(2\pi {\sigma }^2\right)}^{\frac{m}{2}}}}exp\left[-{\displaystyle \frac{\left((d-f(\alpha \left){)}^T\right(d-f\left(\alpha \right))\right.}{2{\sigma }^2}}\right]$$

(16)

The statistics of the posterior distribution were obtained using the statistical inversion theory. This study employed the most common MCMC algorithm as the sampling method. The algorithm steps are as follows [33]:

(1)

Arbitrarily choose an initial value ${\sigma }_i$ and set $i=0$;

(2)

Generate a candidate value $q{\sigma }_{\mathrm{*}}$ from the proposal distribution $q\left({\sigma }_i,\mathit{d}\right)$;

(3)

Calculate the acceptance probability:

$$\alpha \left({\sigma }_0,{\sigma }_{*}\right)=min\left\{1,{\displaystyle \frac{\pi \left({\sigma }_{*}\right)q\left(\mathit{d},{\sigma }_i\right)}{\pi \left({\sigma }_0\right)q\left({\sigma }_i,\mathit{d}\right)}}\right\}$$

(17)

where $\pi \left({\sigma }_{\mathrm{*}}\right)$ is the target probability density function;

(4)

Draw $u$ from $U\left(\mathrm{0,1}\right)$. If $u<\alpha \left({\sigma }_0,{\sigma }_{\mathrm{*}}\right)$, set ${\sigma }_{i+1}={\sigma }_{\mathrm{*}}$; otherwise, set ${\sigma }_{i+1}={\sigma }_0$;

(5)

Increase $i \mathrm{b}\mathrm{y}+1$ and return to step 2.

In the establishment of the Markov chain, the density generated by this candidate is called the proposal distribution, denoted as $q\left({\sigma }_i,\mathrm{d}\right)$. This density can be interpreted as producing a value d from $q\left({\sigma }_i,\mathrm{d}\right)$ when the process is at point ${\sigma }_i$.

3.4. Model Test

In the model testing, we designed two synthetic heat flow column models, considering high-heat anomalies with shallow temperature interfaces and conventional temperature interfaces without geothermal conditions. In Test 1, the Curie depth was 17.1 km, the Moho depth was 45 km, and the LAB depth was 70 km, with a lower shallow thermal conductivity (2.5 W/m∙K) and a higher deep thermal conductivity (4.5 W/m∙K). The mantle heat flow and heat production rate were 30 mW/m² and 1.5 μW/m³, continuously supplying heat to the shallow layers. By substituting the above parameters into the equations, the temperatures at the Curie and LAB interfaces were calculated to be 579.17 °C and 1314.2 °C, and the surface heat flow was 97.5 mW/m², indicating that the uplifted temperature interfaces will cause high heat anomalies at the surface. In Test 2, the Curie depth was 28.6 km, the Moho depth was 50 km, and the LAB depth was 80 km, with a higher shallow thermal conductivity (3.1 W/m∙K) and a lower deep thermal conductivity (3.5 W/m∙K). The mantle heat flow and heat production rate were 45 mW/m² and 0.5 μW/m³. Substituting these parameters into the equations resulted in temperatures of 579.84 °C and 1313.1 °C at the Curie and LAB interfaces, and a surface heat flow of 70 mW/m².

Table 1. Heat flow (HF) column model testing parameter information.

Feature	Interface	Depth	Parameter	True	Prior	Posterior	Variance
Model test 1 High HF	Curie	17.1 [km]	$k_1$ $[\mathrm{W}/\mathrm{m}·$K]	2.5	1.5~3.5	2.7	0.04
	Moho	45 [km]	$k_2$ $[\mathrm{W}/\mathrm{m}·$K]	4.5	3~5	4.3	0.04
	LAB	70 [km]	$q_D$ [mW/m2]	30.0	10~50	30.4	0.03
			$H_0$ [μW/m3]	1.5	1~4	1.6	0.31
Model test 2 Low HF	Curie	28.6 [km]	$k_1$ $[\mathrm{W}/\mathrm{m}·$K]	3.1	2~3.5	2.9	0.31
	Moho	50 [km]	$k_2$ $[\mathrm{W}/\mathrm{m}·$K]	3.5	3~4.5	3.7	0.43
			$q_D$ [mW/m2]	45.0	20~50	44.6	0.32
	LAB	80 [km]	$H_0$ [μW/m3]	0.5	0~5	0.4	0.12

shows the true thermal parameter points and predicted results used in the model testing. The inversion calculation was set to 1 million iterations, with the first 100,000 iterations considered as the preheating process. The last 100,000 iterations were selected as the posterior distribution. During the inversion, the prior settings for the hyperparameters were 1 to 10 to ensure consistent acceptance rates for the calculation results.

The calculation results are shown in Table 1. The four predicted parameters were as follows: crustal thermal conductivity $k_1$ = 2.72/2.96 W/m∙K, mantle thermal conductivity $k_2$ = 4.3/3.5 W/m∙K, mantle heat flow $q_D$ = 30.4/44.6 mW/m², and heat production rate $H_0$ = 1.69/0.5 μW/m³. These parameters exhibited low posterior variances, with the high-heat model showing variances ≤ 0.04 and the low-heat model showing variances ≤ 0.43. The mean posterior closely aligned with prior geological expectations, further validating the inversion’s accuracy. Notably, the slightly higher variance in the low-heat model ($k_2$ = 0.43) reflects uncertainties in the mantle thermal conductivity under reduced heat flow conditions, while the minimal variance in ($q_D=0.32$) underscores the stability of the mantle heat flow estimation. These results collectively confirm the MCMC method’s capability to resolve parameter interactions with statistically robust confidence intervals. Figure 3 displays the intersection diagrams of the predicted parameters. The blue dots represent the posterior distribution from the last 100,000 iterations. From the figure, it is evident that the $k_1$ had a strong correlation with $q_D$ and $H_0$. As the thermal conductivity increased, high heat flow and high heat production characteristics were displayed. Additionally, the $k_2$ had a monotonic relationship with $q_D$, with the thermal conductivity slowly increasing as heat flow rose. There was no correlation between $H_0$, $k_1$, and $q_D$, which can be explained by the independence of the thermal characteristics between deep and shallow layers. Thermal anomalies are mainly influenced by the high heat production in shallow layers and low thermal conductivity in the storage temperature region. The simulation results showed that a shallow Curie depth and high heat production are the causes of high heat anomalies, while deep thermal anomalies originate from deep thermal conductivity, unrelated to heat production or shallow thermal conductivity. Notably, Model 1 and Model 2 tested the northern region of Ningxia, which has a shallower Curie depth, and the southern region, which had a deeper Curie depth. Under the selected thermal parameters, the heat flow in the northern part of Ningxia was greater than 95 mW/m², while the southern heat flow showed an anomalous response below 70 mW/m² within the designated structural range. This is because the two isotherms essentially cannot constrain all the thermal parameters. Based on the distance between the Curie and LAB depths, the possible range of solutions may widen or narrow. For Model 2, due to the shallow LAB, the mantle heat flow was restricted to 40 mW/m², and thus the total heat production in the heat flow column cannot exceed 1.0 μW/m³. For all models, the heat production rate had the highest correlation with the crustal thermal conductivity, indicating that one parameter significantly influences the inversion results. Based on these parameters, the surface heat flow in the study area was determined to be between 70 mW/m² and 95 mW/m².

4. Results

Bayesian inversion was used to estimate the thermal parameters for the Curie, Moho, and LAB interfaces in the Ningxia region, with a total of 714 measurement points. The Curie isotherm and LAB temperatures were set at 580 °C and 1315 °C. The inversion hyperparameters were set between 1 and 10, with a step size of 0.05 to ensure consistent acceptance rates during the iteration process. The prior range for crustal and mantle thermal conductivities was $k_1$ = 1.5–3.5 W/m∙K and $k_2$ = 2.5–5.5 W/m∙K. Mantle heat flow is expected to be influenced by strong fluctuations in the LAB and Moho interfaces, so a wide range was set for $q_D$ (0–200 mW/m²). The heat production rate was unknown and was considered the main controlling factor for heat flow; $H_0$ was set to 0–7 μW/m³. The inversion iteration count for each measurement point was set to 10,000 times, with a total calculation time of about 10 s, and the model was preheated in advance.

Figure 4 shows the Bayesian inversion results based on the interface characteristics for the Ningxia region. The surface heat flow was calculated using Equation (11). The results reflect significant thermal parameter differences between the northeastern and southern parts of Ningxia. The surface heat flow ranged from 56 to 96 mW/m² (Figure 4a), with high heat flow anomalies primarily concentrated in the northern and eastern regions, while geothermal anomalies were observed in the Qingtongxia and Yongning areas. The $k_1$ (Figure 4b) showed a mixed distribution of local high and low anomalies, with low thermal conductivity mainly concentrated in the central and eastern parts of the region. Lower shallow thermal conductivity were found to be better at storing heat from deep sources, forming high heat anomalies, which correlates with the heat flow distribution shown in Figure 4a, i.e., low thermal conductivity areas coincide with high heat flow regions. $H_0$ is one of the main heat sources widely distributed in the crust and typically contributes to regional heat flow anomalies. Figure 4c shows the estimated $H_0$ for the Ningxia region, which ranged from 0.3 to 1.6 μW/m³, with higher anomalies mainly in the eastern and northern regions. Both heat production capacity and surface heat flow exhibited similar spatial distributions in terms of the distance from isotherms, indicating a high correlation between the predicted results. In regions with shallow Curie depths, the heat flow reached up to 90 mW/m², with a heat production rate of 1.5 μW/m³. In contrast, with deeper Curie depths, the heat production rate dropped below 0.5 μW/m³, and the surface heat flow also decreased to around 70 mW/m². The $k_2$(Figure 4d) and $q_D$ (Figure 4e) shared the same spatial distribution, with thermal conductivity ranging from 2.6 to 4.5 W/m∙K and average values slightly higher than those of the crustal thermal conductivity, indicating that thermal conductivity gradually increased with depth. Additionally, lower deep thermal conductivity was observed in the Qingtongxia area, possibly due to regional anomalous geological metamorphism. The mantle heat flow had a lower anomaly range compared to surface heat flow, ranging from 29 to 53 mW/m², suggesting that the geothermal gradient in deep strata tends to flatten, which is the main reason for the increased deep thermal conductivity. The mantle heat flow exhibited lower anomaly responses in the eastern and northern regions, which were controlled not only by the LAB interface but also modulated by the deep thermal conductivity properties and geological structures (fault zones and lithospheric heterogeneity).

Typically, parameters in model calculations are approximated to their true values through updates or perturbations. As the number of parameters and data increases, the model’s nonlinearity and unreliability also increase. The advantage of the Bayesian method is that it can quantify the posterior variance of the calculated results, thereby obtaining a measure of the uncertainty of the results. The posterior variance of the thermal parameters in the Bayesian inversion results for the Ningxia region was calculated and is shown in Figure 5. The results indicate that the overall posterior variance of the thermal parameters was relatively low, less than 10%, suggesting good accuracy. Specifically, the surface heat flow in the northern region had a variance of 6.5 mW/m², which may be related to the uncertainty in the crustal thermal conductivity. Similarly, the mantle heat flow variance (0.3–2.4 mW/m²) was affected by the uncertainty in the mantle thermal conductivity, leading to increased variance in the northern region. In contrast, the heat production rate shows stable standard deviations (0.015–0.16 μW/m³) without a significant spatial correlation, reflecting regional homogeneity in shallow radiogenic heat sources.

5. Discussion

5.1. Correlation and Sensitivity Analysis

There is a relationship between thermal parameters and interface characteristics, which involves the characteristic relationships of the parameters themselves. Analyzing their correlation and sensitivity responses helps us to understand and explain regional thermal anomaly characteristics, providing reliable explanations and bases for geophysical surveys and the formation mechanisms of geothermal anomaly areas.

Figure 6 shows the correlation plot between the interface characteristics and thermal parameters. The lower triangle displays two-dimensional kernel density plots, illustrating the average trend in the data relationships. The values of the Pearson correlation coefficients are shown in the lower triangle. Histograms and one-dimensional kernel density functions for all parameters are displayed along the diagonal. The results showed a strong positive correlation between $q_S$ and both $k_1$ and $H_0$, indicating that surface heat flow is mainly influenced by regional radiogenic heat production and increased thermal conductivity. Additionally, $q_S$ was negatively correlated with the Curie depth; areas with shallower Curie depths tended to have higher heat flow variability. There was no significant correlation between $q_S$ and the Moho, LAB interfaces, and $q_D$. $k_1$ did not exhibit any significant correlation with the other parameters, suggesting that it can be calculated as an independent inversion parameter without being influenced by other factors. There was a negative correlation between $k_2$, Curie depth, and $q_D$, which can be explained by the decrease in $q_D$ as $k_2$ increases, causing high temperatures to propagate towards the shallow layers and affecting the uplift of the Curie depth. Similarly, the LAB interface primarily influenced $q_D$, with the heat flow increasing as the interface was uplifted. Moreover, $H_0$ was related to deep structures, including a negative correlation with $q_D$, indicating that the heat produced by radiogenic elements not only supplies the shallow layers but also contributes to the thermal budget of the deeper strata. It is noteworthy that there was no significant correlation between the Moho depth and the other parameters, suggesting that the methods used to determine this depth and its uncertainties do not significantly impact the heat flow results.

The prediction accuracy largely depends on the types of parameters, and it is necessary to understand the sensitivity of each parameter to the observed values. Sensitivity analysis is used to measure the complex relationships between data. A distance-based global sensitivity method (Distance-based Generalized Sensitivity Analysis (DGSA)) was introduced as a “black box” for estimating uncertainties, to assess the degree of correlation between various parameters and thermal parameters [39]. This method was used to calculate the sensitivity response of the regional parameters, with thermal parameters (including crustal and mantle thermal conductivities, mantle heat flow, and heat production rate) and interface characteristics (Curie isotherm, Moho depth, and LAB interface depth) as the inputs and surface heat flow as the output. The calculation results are shown in Figure 7. The results indicate that heat production rate, magnetic anomalies, and crustal thermal conductivity are the parameters that are the most directly sensitive to surface heat flow, followed by the Curie isotherm, LAB interface depth, and mantle heat flow. The sensitivity of the Moho and gravity anomalies was less than 1, indicating no significant response relationship. This suggests that surface heat flow is primarily influenced by the crust’s radiogenic heat production, thermal conductivity, and the Curie isotherm.

5.2. Geophysical Investigation and Analysis of Yinchuan Basin

Geophysical studies have been conducted around the Ningxia Basin, providing strong data support for interpreting the complex tectonic background and the development of geothermal resources. Magnetotelluric data were collected using the MTU-5A instrument from Phoenix Geophysics. A measurement line was laid from the northwest to the southeast, with a total length of approximately 5.5 km (Figure 8a). The measurement time for each point was over 20 h, and the frequency range was from 0.0005 to 320 Hz. The two-dimensional electrical structure (Figure 8b) shows that the shallow layers predominantly have high resistivity (>1000 Ω·m), and at the horizontal position of 20–30 km, a high-conductivity belt (>100 Ω·m) appears, which trends in the northwest direction and extends to a depth of 15 km. Below 15 km, the resistivity is about 10 Ω·m, which may represent a low-resistivity belt that transports material from the shallow, high-conductivity layers. This is hypothesized to be a response from a fault, possibly associated with magma derived from the mantle. The seismic profile results were taken from [8] (Figure 8c). The results show that the asthenosphere and Moho near the Ordos Block display significant uplift, and from the Alashan Block eastward through the Helan Mountains to the Yinchuan Basin and to the Ordos Block, the Moho gradually rises, reaching its shallowest point near the Yellow River Fault. The asthenosphere and Moho are relatively shallow, with deep, high-conductivity, high-temperature molten material intruding upward into the crust, thus increasing the local temperature of the crust. The Yellow River Fault (F6) extends downward into the lower crust and cuts through the Moho, entering the lithospheric mantle. Along the profile, the crust thickness is 40–48 km, and the Moho shows a characteristic higher-to-the-east and lower-to-the-west pattern. On the side of the Ordos Block, the deep Moho in the study area shows minimal undulation, and the Yellow River Fault, as a major deep fault, provides a pathway for deep heat transfer to the shallow thermal reservoirs, which aligns with the local high heat anomalies shown in the predicted heat flow distribution.

5.3. Comprehensive Interpretation of Yinchuan Basin

Based on the seismic, magnetotelluric, and thermal parameter inversion results, we established a geothermal mechanism model for the western margin of the Ordos Basin [40] (Figure 9). The model reveals that at the Yellow River Fault, the Moho was uplifted, rising to approximately 40 km toward the east. Similarly, the Curie depth was uplifted to a depth of 15 km, indicating an overall uplift of the basement. The remote stress field from the Tibetan Plateau in later stages placed the Ordos Plateau’s periphery in an extensional environment. This extension facilitated the upward migration of subsurface thermal materials, accelerating the extensional process and forming multiple Cenozoic faulted basins around the Ordos Basin during the Himalayan period. These basins experienced basement uplift and were provided with pathways for thermal material ascent through normal faults induced by the extensional environment.

The regional increase in surface heat flow is primarily attributed to two heat transfer mechanisms: groundwater convection induced by atmospheric precipitation in the west and thermal conduction along deep faults. Precipitation transports surface water along the eastern Helan Mountain Fault to deeper layers, where it is heated and cycled by heat sources at fracture intersections within the stable block. The heated groundwater then ascends along the Yellow River Fault, forming convective heat. The regional heat source likely results from the combined effects of mantle heat flow and partially molten material near the intersection of the Yellow River Fault and Helan Mountain Fault. The deep heat source primarily uses major fault zones as conduits to transfer heat to shallow layers, heating groundwater and warming the shallow cap layers, ultimately forming high heat anomalies in the eastern region. According to the thermal parameter inversion results, the surface heat flow in the eastern shallow region is generally high, exceeding 90 mW/m², while heat flow in the western region is relatively low. In deeper layers, the heat flow is higher in the west and lower in the east. The heat source mainly ascends along southwest-dipping fault zones, which act as thermal conduits. This is accompanied by surface water in the western region supplying the faults, forming a convective circulation and medium heat transfer. As the heat flow migrates eastward, it results in high heat anomalies in the eastern shallow layers.

6. Conclusions

This study investigated the distribution and genetic mechanisms of geothermal resources in the Yinchuan Basin by establishing a Bayesian thermal parameter inversion method constrained by interface-derived information. The results revealed distinct spatial heterogeneity in the thermal anomalies: high-heat flow zones (>90 mW/m²) are predominantly concentrated in the eastern basin along the Yellow River Fault, with a maximum surface heat flow of 96 mW/m², while the western regions exhibit moderate values (70–85 mW/m²). These anomalies are quantitatively linked to crustal thermal conductivity (2.5–3.5 W/m·K) and radiogenic heat production (0.3–1.6 μW/m³), with posterior uncertainties of <10% across the key parameters. The integration of geophysical data further demonstrated that deep mantle heat flow (29–53 mW/m²) and fault-controlled convective processes jointly regulate the geothermal system. By resolving the dual controls of lithospheric thermal structure and shallow hydrothermal activity, this work provides a quantitative framework for delineating prospective geothermal zones and optimizing resource exploration strategies in sedimentary basins.