Optimizing Interpolation Methods and Point Distances for Accurate Earthquake Hazard Mapping

Abstract

Earthquake hazard mapping assesses and visualizes seismic hazards in a region using data from specific points. Conducting a seismic hazard analysis for each point is essential, while continuous assessment for all points is impractical. The practical approach involves identifying hazards at specific points and utilizing interpolation for the rest. This method considers grid point spacing and chooses the right interpolation technique for estimating hazards at other points. This article examines different point distances and interpolation methods through a case study. To gauge accuracy, it tests 15 point distances and employs two interpolation methods, inverse distance weighted and ordinary kriging. Point distances are chosen as a percentage of longitude and latitude, ranging from 0.02 to 0.3. A baseline distance of 0.02 is set, and other distances and interpolation methods are compared with it. Five statistical indicators assess the methods. Ordinary kriging interpolation shows greater accuracy. With error rates and hazard map similarities in mind, a distance of 0.14 points seems optimal, balancing computational time and accuracy needs. Based on the research findings, this approach offers a cost-effective method for creating seismic hazard maps. It enables informed risk assessments for structures spanning various geographic areas, like linear infrastructures.

1. Introduction

Natural hazards can lead to unpredictable and severe consequences in affected areas. Understanding the potential of natural hazards and the factors contributing to their occurrence is essential for effective preparedness and damage mitigation [1]. Natural hazard analysis is essential for systematically assessing potential threats from natural events [2]. This analysis entails the identification, mapping, and evaluation of natural hazards. Earthquakes are natural hazards and result from a sudden energy release in the ground, causing ground shaking. Earthquakes have the potential to cause secondary consequences such as liquefaction, landslides, fires, and tsunamis [3]. These events pose significant risks to human life, infrastructure, and environment, with the potential for widespread destruction and loss [4]. Consequently, assessing seismic risk is essential for understanding and mitigating the potential impacts of such seismic events.

A seismic risk assessment consists of three main steps: seismic hazard analysis (SHA), seismic vulnerability analysis, and an exposure assessment. SHA evaluates potential earthquake consequences by examining tectonic and seismic conditions. Exposure assesses susceptibility to hazards through observational insights. Vulnerability quantifies asset damage under different hazard intensities [5,6,7,8]. SHA is a key step in seismic risk assessments, assessing earthquake hazards and their impact using geological and seismological data to estimate ground motion parameters [9]. Employing advanced methods and technologies, SHA quantifies earthquake hazards by assessing ground shaking intensity at specific locations [10]. The two common methods for SHA are probabilistic and deterministic [11]. Despite recent advancements in predicting earthquake location, time, and magnitude, uncertainties remain [12]. Many studies have been conducted in the field of SHA [13,14,15,16]. Outcomes often have high uncertainty and may lack definitive conclusions. However, utilizing methods like decision trees and integrating influential factors in a weighted manner can noticeably enhance the accuracy of hazard intensity prediction [17,18].

SHA results can be applied to single or multiple sites within an area. When considering multiple sites, a solution is a must to address potential hazards across the region. Given the variability in ground motion parameter intensities across different areas, the necessity for a seismic hazard map is evident. These maps inform seismic macrozonation [19,20], pipeline route design, land-use planning, building codes, and disaster preparedness efforts [21], necessitating hazard analyses across multiple points [22]. In a pipeline route design project, a macro assessment of seismic hazards is essential to evaluate the seismic hazard and consequence seismic risk of each candidate route. One upcoming challenge is determining local geological conditions, a process that is often impractical everywhere due to high costs and complexity. In this scenario, an effective approach is to collect sufficient soil characteristic samples along the routes. Using these samples, local site conditions of the remaining length of the route can be estimated. A seismic hazard map is indeed a continuous process, and SHA methods offer valuable insights into ground motion parameters at specific locations. To address this limitation and obtain a more comprehensive view of seismic hazards across an entire region, seismic hazard values from various locations can be used to create seismic hazard maps. To achieve this goal, spatial interpolation methods, commonly applied in fields like engineering and environmental science, are employed. These techniques enable the generation of estimations for locations where seismic hazard data have not been directly measured for any reason. They rely on sampled data points to create a comprehensive view of seismic hazards across the entire region by producing seismic hazard maps [23]. The selection of the spatial interpolation method and the number of data utilized are crucial factors in creating a seismic hazard map. The choice of locations for spatial interpolation can have a considerable impact on the accuracy of the interpolations. In the process of the location selection, it is important to strike a balance to avoid choosing locations that are either too far or too close together. Choosing widely spaced locations can underestimate hazard areas, compromising accuracy while saving time and computational resources. On the other hand, choosing locations that are spaced closely increases computational workload.

Spatial interpolation methods play a remarkable role in geospatial analysis, providing valuable insights into the distribution of phenomena across a given area. Interpolation techniques can be categorized, based on the concept of the prediction model, into two main groups, deterministic and stochastic. Deterministic interpolation techniques create surfaces based on similarity, whereas stochastic methods consider spatial autocorrelation among measured points [24]. These methods are essential for estimating values at unobserved locations using known values from sampled locations [25]. The predictions of these methods are influenced by various factors, making it challenging to select the most appropriate input data method [26]. Various methods and techniques are employed to assess engineering issues in fields like botany, geology, agriculture, hydrology, and climatology using geographic information system (GIS) interpolation tools [27]. Researchers utilize these techniques to bridge gaps between distinct data points, creating comprehensive and continuous representations of phenomena across diverse landscapes. The identification of soil types [28,29], groundwater levels [30,31,32], and precipitation levels [33,34] exemplifies the use of spatial interpolation methods in the field of civil engineering.

To assess earthquake hazards, various spatial interpolation methods have been employed over the past several years, based on point measurements using the scenario method. Keshavarzi and Mansoori Moghadam [35] prepared a zoning map of earthquake-induced acceleration in the Bushehr province of Iran by selecting study sites based on a grid size of 0.02° and utilizing deterministic spatial interpolation along the grid. Farahani et al. [36] performed a SHA in Iran using the HAZUS methodology [37] and utilized spatial interpolation methods to generate a hazard map for their study. Ahmad et al. [38] generated a zoning map for seismic hazards in Pakistan, using a grid size of 0.1° for selecting study sites and employing geostatistical interpolation techniques. Alavi et al. [39] investigated seismic hazards and risk maps associated with strong ground motion and soil liquefaction in southern Iran utilizing the ordinary kriging interpolation (OKI) method within ArcMap. They chose the study sites on the basis of a grid size of 0.08° to conduct an SHA and generate result maps. Numerous studies have been conducted in this field, for example [40,41,42].

Properly selecting locations and choosing suitable interpolation methods are crucial in reducing both time and computational requirements, directly impacting the outcomes of spatial interpolation methods [43]. Each spatial interpolation method has unique advantages, applicability across various contexts, and varying prediction errors depending on the algorithm used [44]. Choosing the appropriate spatial interpolation technique is crucial for achieving the most accurate results, as different interpolation methods can yield varying outcomes. On a large scale with a broad study radius, location selection can be based on a function that incorporates latitude and longitude. Generally, the differences in seismic hazard map results are minor when there are sufficient data, but when data are lacking, errors become significant. On the other hand, the spatial interpolation method can noticeably influence the effectiveness of the seismic hazard map. Using statistical indicators, optimal interpolation methods and ideal number of data points can be determined by assessing the errors between predicted and actual values.

The aim of this study is to identify a suitable spatial interpolation method, taking into account the efficiency of the available data. To achieve this aim, a method for the creation of seismic hazard maps is presented, considering varying quantities of data. The study area is divided into smaller cells, and the center of each cell is considered as a site for SHA. Subsequently, based on the results of site-specific hazard analysis (peak ground acceleration (PGA) with a return period of 975 years), the spatial distribution of seismic hazard in the region is performed using two spatial interpolation methods, OKI and inverse distance weighted (IDW). Then, the predicted values are assessed utilizing a combination of five statistical indicators with an appropriateness index (A.I). Through the repetition of this process with the division of cells into other sizes, the statistical indicator values are compared, and the interpolation method and required amount of data are determined.

2. SHA

SHA considers factors such as the location, magnitude, frequency, and type of earthquake, geology and topography of the area, and soil conditions. The analysis of earthquake hazards involves deterministic and probabilistic approaches [45]. In the deterministic approach, a specific earthquake scenario is assumed, and the earthquake hazard is evaluated based on this scenario. Initially, the locations and characteristics of all potential sources contributing to a significant earthquake are determined, including the specification of a maximum credible earthquake. The ground motion intensity decreases as one moves away from the earthquake location, requiring the selection of attenuation relationships. These relationships dictate the reduction in ground motion parameters concerning distance from the earthquake source and certain magnitudes [46]. In general, peak ground motion parameters decrease with increasing distance from the earthquake site and decreasing earthquake magnitude [45]. The final step involves calculating the earthquake intensity at the desired location using the specific scenario and the chosen attenuation relationship. The probabilistic approach is considered the most appropriate for addressing the inherent uncertainties in natural hazard-related phenomena [47]. This method incorporates uncertainties such as earthquake magnitude, location, and time [48]. In the probabilistic approach, the probability distributions for earthquake location and magnitude, and attenuation relationship, are considered. Thus, all potential earthquake scenarios are covered in SHA, incorporating the effect of all seismic sources with their specified activity rates [45,49]. The probabilistic method comprises three main parts: (1) identifying the geometry of seismic sources, (2) seismicity of the sources, and (3) attenuation relationship. Using a logical tree, different scenarios are weighted based on expert opinions [50]. The geometry of seismic sources is determined based on fault activity and earthquake distribution as point, line, or area sources. The seismic parameters of the sources indicate the earthquake occurrence rate and seismic activity [51]. The result of the probabilistic analysis is the annual exceedance probability of ground motion parameters at any desired seismic level. The mathematical calculation of the probabilistic SHA and the impact of the aforementioned steps are according to Equation (1) [45].

$$P(IM>x)={\displaystyle \sum _{i=1}^{n_{sources}}\lambda (M_i>m_{\min })}{\displaystyle \underset{m_{\min }}{\overset{m_{\max }}{\int }}{\displaystyle \underset{0}{\overset{r_{\max }}{\int }}P(IM>x|m,r)*f_{M_i}(m)*f_{R_i}(r)}}drdm$$

(1)

where λ(M_i > m_min) is the annual occurrence rate of earthquakes greater than m_min (lower bound value of earthquake magnitude) for the i-th source, P(IM > x|m,r) denotes the conditional probability of the IM exceeding x given m and r. ƒ_Ri(r) and ƒ_Mi (m) signify the probability density distributions for distance and magnitude, respectively, for the i-th seismic source. P(IM > x) is the annual exceedance probability of IM (ground motion parameter) from x. The distribution of IM is commonly represented by a lognormal model (ln(IM) is assumed to follow a normal distribution). Therefore, the exceedance probability can be determined utilizing the cumulative distribution function of a standard normal distribution, denoted as Φ(x), based on Equation (2) [16].

$$P(IM>x|m,r)=1-\Phi (\frac{\ln (IM)-{\mu }_{IM}}{{\sigma }_{IM}})$$

(2)

where µ_IM and σ_IM are the median and standard deviation of the distribution of IM, respectively, and are determined based on the seismic source and site [52]. The probability density distribution for earthquake magnitudes should be calculated using the frequency–magnitude relationship, which represents the background seismicity of the study region and is most commonly determined by Equation (3) [45].

$$f_{M_i}(m)=\frac{b*\ln (10)*{10}^{-b(m-m_{\min })}}{1-{10}^{-b(m_{\max }-m_{\min })}} m_{\min }<m<m_{\max }$$

(3)

where m_max is the upper bound value of magnitude used to demonstrate that the maximum magnitude may be generated. The coefficient b describes the specific relationship between the magnitude and total number of earthquakes, typically close to 1.0 in seismically active regions, and is represented by the Gutenberg–Richter coefficient [53]. This coefficient is calculated based on the annual number of seismic events within a specified magnitude range (N_m) and the earthquake catalog period in years unit (T), as per Equation (4).

$${\log }_{10}(N_m/T)=a-bm$$

(4)

The probability density distribution of the distance in seismic source geometry (linear or area) is derived by considering the proportion of the rupture length or area to the total length or area. For instance, in the case of area seismic sources, Equation (5) utilizes both the rupture length (L) and radius of the area source (R) [54].

$$f_{R_i}(r)=\left\{\begin{array}{c}\frac{2r}{R^2}+\frac{L}{2R^2}\\ \frac{1}{R}\\ 0\end{array}\right.\begin{array}{c}\\ \\ \end{array}\begin{array}{c}r\le R-\frac{L}{2}\\ R-\frac{L}{2}\le r\le R\\ r\ge R\end{array}$$

(5)

where r is the distance from rupture to site.

3. Spatial Analysis Interpolation

In recent years, GIS spatial interpolation capabilities have advanced remarkably for the modeling, analysis, and visualization of continuous fields. The importance of the spatial interpolation method in sample selection lies in its ability to provide accurate, precise, and reliable estimates while considering the spatial characteristics of the data [55]. The choice should be tailored to the specific features and distribution of the sample points, ensuring that the interpolation method aligns with the analysis goals. Interpolation techniques within GIS are broadly categorized into two main groups based on assumptions, strengths, and limitations: local neighborhood approach and geostatistical approach [56]. The local neighborhood approach predicts the value of a variable at an unknown location by calculating the weighted average of values from nearby locations. This method employs a subset of input points located within a specified distance or quantity to estimate the value at each output location [57]. On the other hand, the geostatistical approach estimates the value of a variable at an unknown location by considering the statistical properties of measured values in nearby locations. This method utilizes statistical models to address the spatial correlation and variation of input points, offering measures of uncertainty for the estimated values [43]. IDW and OKI are prominent and widely used examples of these approaches.

3.1. IDW Interpolation

The IDW method assumes that the value of a parameter at observation points closer to the prediction point is more similar to it than at more distant points [43]. However, the IDW interpolation is sensitive to the choice of distance coefficient and number of neighboring points used for the interpolation [58]. The IDW interpolation estimates the unknown value of Z(x₀) at point x₀ utilizing the values of a given number of observation points x_i (or by a given radius of the search neighborhood), weighted by an inverse function of the distance between the unknown point and observation, as presented in Equation (6).

$$Z(x_0)=\frac{{\displaystyle \sum _{i=1}^n\frac{Z(x_i)}{h_{ij}^{\beta }}}}{{\displaystyle \sum _{i=1}^n\frac{1}{h_{ij}^{\beta }}}}$$

(6)

where n denotes the total number of sample values utilized in the interpolation process of x₀. Z(x_i) represents the i-th sample data value, while h_ij signifies the separation distance between the interpolated value and corresponding sample data point. The parameter β serves as the power of the inverse distance ratio, acting as a weight for predictions at unsampled locations, conventionally assumed to be a value of 2 [59]. By defining higher power values, the nearest point can be further emphasized, making the fitted surface more meticulous. Another important factor in the efficiency of the IDW method is the search radius, which can be used in both variable and fixed forms [60]. The variable and fixed search radii are two methods used to select input sample points for interpolation. With the variable search radius, a specified number of input sample points are chosen for interpolation, while the fixed search radius selects points within a predetermined distance for interpolation.

3.2. OKI

OKI is a geostatistical interpolation method utilized to estimate values at unobserved locations within a study domain. Unlike IDW, kriging considers both the distance between observed locations and spatial correlation or variability in the data. OKI is based on the principles of spatial autocorrelation and semivariogram modeling, aiming to provide optimal predictions by accounting for spatial dependence in the data. This method involves calculating a semivariogram to understand spatial variability, establishing a system of linear equations based on the semivariogram model, and determining kriging weights [61]. These weights are then applied to observed values to predict values at unobserved locations [62]. OKI is valued for its ability to provide accurate spatial predictions while quantifying uncertainty. The interpolated value at each point is calculated based on the kriging method according to Equation (7).

$$Z^{*}(x_0)={\displaystyle \sum _{i=1}^n{\lambda }_{i}Z(x_i)}$$

(7)

where λ_i is the kriging weight assigned to the observed value at location x_i determined by the semivariogram. Higher weights show a greater influence on the observed data [63]. The OKI method involves a two-step process for prediction: first, fitting a model, and then making the prediction. In the model fitting step, a semivariogram is constructed based on spatial autocorrelation in data to estimate the statistical dependence, referred to as spatial autocorrelation, as outlined in Equation (8) based on the regionalized variable theory [64].

$$\gamma (h)=\frac{1}{2N_h}{\displaystyle \sum _{i=1}^{N_h}{[Z(x_i)-Z(x_i+h)]}^2}$$

(8)

where γ(h) represents the semivariogram, with h denoting the separation distance, and N_h representing the number of sample pairs used to calculate each class of h. Z(x_i) designates the observed value at location x_i, while Z(x_i + h) represents the observed value at a point located at a distance h from Z(x_i). While various models can be applied based on the form of the semivariogram, the spherical model is frequently utilized [65]. A spherical model describes a scenario in which spatial autocorrelation decreases gradually with distance until it diminishes to zero once the distance surpasses a specific threshold value. As the kriging predictor is both unbiased and minimizes variance, in the prediction phase, one can determine unknown weights utilizing Equation (9).

$${\sigma }_k^2(x_0)=\mu +{\displaystyle \sum _{i=1}^n{\lambda }_i}\gamma (x_0-x_i),{\displaystyle \sum _{i=1}^n{\lambda }_i}=1$$

(9)

In this context, µ is the Lagrange constant, and γ(x₀ − x_i) represents the semivariogram value corresponding to the distance between x₀ and x_i [66]. Ensuring that the sum of λ_i equals one guarantees an unbiased prediction [67].

4. Assessment Methods

In the realm of data analysis and predictive modeling, the evaluation of the model performance is paramount to ensure the reliability and accuracy of generated results. Given the extensive use of spatial interpolation methods, there is a growing concern regarding their accuracy and precision. The efficacy of spatial interpolation methods is assessed through a cross validation approach with statistical criteria [68,69]. In this study, five key statistical criteria were employed, including the mean error (ME), mean absolute error (MAE), root mean square error (RMSE), coefficient of determination (Pearson R²), and maximum error (MaxE), which serve as metrics for error measurement. These fundamental tools play a crucial role in quantifying the errors associated with predictive models, shedding light on the extent to which the model’s outputs deviate from the observed data. In the following, the statistical criteria are presented:

4.1. ME

ME is used to evaluate the extent of bias in the estimates, providing a relative measure of the error’s size and calculated with Equation (10). It provides information about the direction and magnitude of the errors. Substantial discrepancies between predicted and actual values are evident in large ME values.

$$\mathrm{ME}=\frac{{\displaystyle \sum _{i=1}^n(\widehat{z}(x_i)-z(x_i))}}{n}$$

(10)

where, z(x_i) and $\widehat{z}\left(x_i\right)$ represent the actual value and predicted value at point x_i, respectively, and n is the number of data samples. If ME is close to zero, it displays that, on average, the predictions are accurate. Positive ME values suggest overestimation, while negative values suggest underestimation.

4.2. MAE

MAE is a statistical metric utilized to quantify the average absolute errors between the predicted and actual values. It serves as a measure of the accuracy of predictions [70], providing a straightforward assessment of the absolute differences between the predicted and observed values without considering the direction of errors. MAE is computed using Equation (11).

$$\mathrm{MAE}=\frac{{\displaystyle \sum _{i=1}^n\left|\widehat{z}(x_i)-z(x_i)\right|}}{n}$$

(11)

where, z(x_i) and $\widehat{z}\left(x_i\right)$ designate the actual value and predicted value at point x_i, respectively, and n is the number of data samples. A lower MAE value demonstrates better accuracy in predictions, as it means that, on average, the model’s predictions are closer to the actual values.

4.3. RMSE

RMSE serves as a crucial parameter for assessing the accuracy of spatial analysis in GIS. It is a statistical measure that calculates the average magnitude of the differences between the predicted and actual values. It is, particularly useful when dealing with numerical data where the goal is to minimize the error between the predicted and actual outcomes. RMSE is computed for each model prediction employing Equation (12).

$$\mathrm{RMSE}=\sqrt{\frac{{\displaystyle \sum _{i=1}^n{(\widehat{z}(x_i)-z(x_i))}^2}}{n}}$$

(12)

where z(x_i) and $\widehat{z}\left(x_i\right)$ represent the actual value and predicted value at point x_i, respectively, and n is the number of data samples. RMSE serves as a metric for assessing the magnitude of errors, yet it exhibits sensitivity to outliers owing to its emphasis on significant errors. Similar to MAE, a lower RMSE value presents better accuracy in predictions, suggesting that the model’s predictions are, on average, closer to the actual values.

4.4. Pearson R²

Pearson R² (P.R²), which measures the correlation (linear) between the actual and predicted covariances, is a criterion for evaluating the reliability and effectiveness of interpolation models. The resultant values range between −1 and 1, where a value closer to −1 illustrates a strong negative correlation and a value closer to 1 depicts a strong positive correlation. P.R² is widely used to measure the goodness of fit for different interpolation methods. This coefficient also serves to quantify the overall accuracy of the spatial interpolation model by measuring the proportion of variability in the actual data that is replicated in the interpolated values and represented by Equation (13) [71].

$$\mathrm{P}{.\mathrm{R}}^2=\frac{{[{\displaystyle {\sum }_{i=1}^n(\widehat{z}(x_i)-{\widehat{z}}_{ave})\ast (z(x_i)-z_{ave})}]}^2}{{\displaystyle {\sum }_{i=1}^n{(\widehat{z}(x_i)-{\widehat{z}}_{ave})}^2\ast {\displaystyle {\sum }_{i=1}^n{(z(x_i)-z_{ave})}^2}}}$$

(13)

where ${\widehat{z}}_{ave}$ and $z_{ave}$ denote the average of the predicted values and average of actual values, respectively. $z\left(x_i\right)$ and $\widehat{z}\left(x_i\right)$ represent the actual value and predicted value at point x_i, respectively, and n is the number of data samples.

4.5. MaxE

MaxE refers to the largest possible absolute difference between the predicted and actual values. Determining MaxE is essential for assessing the reliability of the measurement and prediction models. MaxE allows for a clear understanding of the level of uncertainty associated with the predictions and is calculated by Equation (14).

$$\mathrm{MaxE}=\max (\left|\frac{\widehat{z}(x_i)-z(x_i)}{z(x_i)}\right|)$$

(14)

where $z\left(x_i\right)$ and $\widehat{z}\left(x_i\right)$ designate the actual value and predicted value at point x_i, respectively, and n is the number of data samples.

4.6. Quantified Assessment

Each of the aforementioned statistical criteria has a distinct concept and cannot be directly compared with the others. Although the ME, MAE, RMSE, and MaxE optimal values are closer to zero and P.R² is closer to one, relying solely on individual criteria for performance decisions can be complex. To simplify the evaluation within a decision-making framework for comparing interpolation methods, A.I is used. A.I is a composite measure designed to integrate multiple statistical criteria that consider various aspects or dimensions to provide a comprehensive assessment of a model’s effectiveness. In this method, a positive impact is observed when a decrease in the criterion value leads to the model (such as spatial interpolation methods) being deemed more suitable than others, and vice versa. Among these parameters, a decrease in the parameter value (for all parameters except P.R²) suggests that the model is suitable. To compute A.I, all parameter values are normalized to a non-dimensional scale utilizing the minimum–maximum normalization method. For the normalization of ME, MAE, RMSE, and MaxE, Equation (15) is employed.

$$R_i=\frac{[\max (C_j)-C_j]}{[\max (C_j)-\min (C_j)]}$$

(15)

For a larger value of P.R² that is a positive contribution to appropriateness, Equation (16) is used.

$$R_i=\frac{[C_j-\min (C_j)]}{[\max (C_j)-\min (C_j)]}$$

(16)

Here, R_i is the normalized value of parameter i and C_j represents the parameter value for the statistical criterion j, where max(C_j) and min(C_j) respectively denote the maximum and minimum values of the statistical criterion j across all models. This method normalizes parameter values to a range of 0 to 1, making them non-dimensional. A value closer to 1 indicates greater suitability of the technique, while lower values suggest otherwise. Using these standardized values, the A.I model is computed utilizing Equation (17) [72].

$$\mathrm{A}.\mathrm{I}=\frac{{\displaystyle \prod _{i=1}^n{R}_i}}{n}$$

(17)

where n is the total number of parameters considered for the computation of A.I.

5. Methodology

In this section, a proposed method for identifying optimal sites for seismic hazard maps along with a suitable spatial interpolation method is presented. Specifically, the goal was to find locations that result in seismic hazard maps with the lowest errors in the process of predicting values. To achieve the research objectives, the proposed methodology consists of five steps:

(1)

Geospatial partitioning: dividing the study area into smaller rectangular cells with varied longitudes and latitudes based on the geospatial zoning coefficient (GZC). GZC is a scalar value ranging smaller than one used to scale the longitude and latitude dimensions, resulting in the creation of rectangular cells with GZC size within the study area.

(2)

SHA: identifying and predicting seismic hazard intensity parameters value across each cell.

(3)

Geospatial hazard classification: seismic hazard map of the study area using interpolation among data points within each cell and spatial distribution of seismic hazard.

(4)

Cross validation analysis: evaluating predicted values by comparing them with actual values.

(5)

Outcome: optimal site selection for SHA and spatial interpolation methods.

The methodology begins by dividing the study area into smaller cells utilizing latitude and longitude percentages based on GZC. Subsequently, the center of each cell is designated as a site and represents the focal point for conducting SHA within that cell.

The second step is to perform an SHA. Seismic sources modeling with a specific radius is considered for each site. Employing a probabilistic SHA, the annual exceedance intensities of strong ground motion parameters such as PGA are calculated for each site.

By performing an SHA, in the third step, a seismic hazard map for the study area is prepared using spatial interpolation methods. This step uses deterministic and geostatistical interpolation methods, specifically IDW and OKI.

In the fourth step, new sites are considered based on the repetition of the first step with different GZCs. Subsequently, the second and third steps are repeated for these locations. Then, a set of sites where SHA is conducted is regarded as the baseline. The predicted values for each of these sites are determined for each GZC, and statistical indicators are compared among them.

In the final step, the fourth phase of the methodology assesses error values utilizing two interpolation methods with A.I. The outcomes of this section contribute to determining the optimal number of sites and the most suitable interpolation method. Figure 1 shows a detailed description of the proposed method’s framework.

6. Case Study

The proposed method was employed to identify suitable sites for SHA and to create a seismic hazard map within a specific region of Iran. The study area spanned a radius of 100 km, was centered around the city of Shiraz, and extended between longitudes 51° and 54° east and latitudes 28° and 31° north, covering a total area of 70,685 km² (Figure 2). This region, situated within the active seismic zone of the Zagros region, is a consequence of the dynamic interaction between the Eurasian and Arabian tectonic plates [73]. Given its geographical location, the study area is susceptible to multiple active faults.

6.1. Partitioning Study Area

GZC is a quantitative parameter used in this study to spatially partition the study area into specific cells based on geographical coordinates. GZC, by multiplying in the units of longitude and latitude, subdivided the study area into distinct and smaller sections. Dividing the region into smaller grids enhanced accuracy by capturing variations in seismic activity and soil conditions, allowing for detailed and precise spatial analysis. This organized approach simplified data analysis, aiding in identifying patterns and correlations related to seismic hazards in specific regions. After dividing the study area based on GZC, the centers of each cell were considered as individual site locations, where the subsequent step involved applying the SHA process. Gridding maintained consistent grid cell sizes for uniform seismic hazard evaluation across the area. In the current research work, various GZC values (

Table 1. Site number for each GZC classification.

GZC	Site Number	GZC	Site Number	GZC	Site Number
0.02	7317	0.12	205	0.22	60
0.04	1828	0.14	151	0.24	50
0.06	810	0.16	113	0.26	41
0.08	455	0.18	91	0.28	36
0.1	291	0.2	74	0.3	30

) were utilized to partition the study area into smaller sections. Applying this methodology resulted in diverse classifications across the study area, leading to varied numbers and locations of sites for each distinct GZC value. Figure 3 illustrates the number and locations of sites subdivided based on the GZC values ranging from 0.02 to 0.3 in the study area. These ranges of grid sizes were chosen on the basis of them being approximately equal to the standard error in the earthquake epicenter determination in the study area. For the GZC values exceeding 0.3, no locations within the study area were used for conducting SHA. As evident in Figure 3, an increase in the GZC value resulted in a decrease in the number of sites, and each site became representative of a larger cell with a greater area. Furthermore, recommended sites based on each GZC varied in geology, such as soil properties and regional characteristics from one another.

6.2. SHA

After identifying the location of sites based on the previous step using GZC, SHA was conducted for each site to estimate the strong ground motion parameters. In this study, Open Quake was utilized for a probabilistic SHA [74]. Instrumental earthquake records obtained from the International Institute of Earthquake Engineering and Seismology [75] website spanning the years 1926 to 2020 were used for SHA. In Figure 4, the regional earthquake distribution is depicted based on the moment magnitude. The method proposed in the Iranian guidelines for SHA (No. 626) was employed to eliminate foreshocks and aftershocks, and to enhance the magnitude of the main earthquakes. Based on Figure 5, seismic sources are classified into linear and area sources. In the linear models, active faults served as seismic sources, while in area models, seismic sources were identified by examining earthquake distribution and comparing their mechanisms with faults [76]. The seismic parameters for linear sources were selected from the Earthquake Model of Middle East project data [77], and Gutenberg–Richter coefficients were utilized to determine the seismic parameters for the area sources. The minimum moment magnitude for all seismic sources was set at 4.5. In SHA, the logical tree allocated 35% weight to the linear model and 65% to the area model. To compute PGA, four distinct attenuation equations of strong ground motions were used [78,79,80,81].

Figure 6 demonstrates the results of the probabilistic SHA for sites (PGA estimate based on Figure 5) corresponding to each GZC, represented by graded increasing symbols. In Figure 6, a regular pattern in the probabilistic SHA values is observed, and areas with different hazards are noticeable. The pattern became more extensive as GZC decreases. Soil properties (based on shear wave velocity proposed by [82]) and distance from seismic sources affected seismic hazard attenuation. Given the variations in site locations and numbers across each GZC value, the values of the mentioned factors differed, resulting in diverse hazard levels. This variability considerably affected the accuracy of localized seismic hazard predictions and mapping.

6.3. Classification of Geospatial Hazard

By utilizing the results of SHA for each site corresponding to each GZC, this section examines the spatial distribution of seismic hazards across the entire region based on the mentioned spatial interpolation methods. Figure 7 and Figure 8 represent the spatial distribution of seismic hazards (PGA of a 975-year return period) using the IDW and OKI methods, respectively.

7. Results

The spatial distribution of seismic hazard resulting from sites corresponding to each GZC varies with each spatial interpolation method, exhibiting different levels of accuracy. In this section, the task of identifying the optimal interpolation method from among the two employed methods, along with determining the optimal GZC, is addressed. The selection of an optimal spatial interpolation method, coupled with an optimized GZC, is fundamental for obtaining higher precision and efficacy in an SHA. While a visual interpretation of Figure 7 and Figure 8 can provide a preliminary indication of the most suitable interpolation method for the case study and dataset, it is not conclusive. Therefore, the need for statistical analysis becomes obvious. To achieve this goal, it is necessary to assess the predicted values of seismic hazard based on Figure 7 and Figure 8 against the actual seismic hazard values (determined based on the method indicated in Figure 5) at a specific set of sites. For this purpose, sites associated with a GZC of 0.02 were considered as reference sites, chosen due to the sufficient data available for this coefficient. The predicted value for seismic hazard at each site is determined based on the spatial distributions obtained from each of the two interpolation methods along with a different GZC, from 0.02 to 0.3. To evaluate the discrepancies between the actual and predicted values at the reference sites, it is essential to identify appropriate indices. A comprehensive statistical evaluation is essential to ensure a more rigorous and objective assessment of the interpolation methods. For this evaluation, five statistical metrics, including ME, MAE, RMSE, P.R², and MaxE, were employed to address the objectives of this section. A higher value of P.R² and lower values of MaxE, RMSE, MAE, and ME generally demonstrate a good match between the actual and predicted values. Finally, considering the outcomes of statistical metrics, optimal locations for sites can be identified for SHA, along with the suitable spatial interpolation method. This process was accomplished by computing A.I for each of the spatially interpolated maps. In the following section, the statistical results and A.I based on them were assessed according to simultaneously considering GZC and type of the interpolation method.

7.1. ME Indicator

The ME indicator for both methods is negligible as long as GZC is 0.14 or less than, as displayed in Figure 9. However, with an increase in the GZC values, the ME indicator is increased. In certain GZC categorizations, it is witnessed that the predicted values are lower than the actual values, yet this underestimation is acceptable given the relatively small magnitude of the difference. Generally, major errors are observed in the GZC values greater than 0.16, and a significant increase in errors is observed with a decrease in the number of sites and interpolation with them. The highest ME indicator occurs at a GZC value of 0.3, reaching 1.11% for the IDW method. In contrast, for the OKI method, this value is 0.8%. Moreover, both spatial interpolation methods show instability patterns in estimating according to the ME indicator. For example, at a GZC value of 0.26, both interpolation methods underestimate the values compared with the actual values, whereas at a GZC of 0.3, they overestimate the estimates.

7.2. MAE Indicator

According to Figure 10, a stable increasing trend in the error of the MAE indicator is seen with an increase in the GZC values. Across all GZC values, the error values for the IDW method are greater than those for the OKI method. The average value of the MAE indicator is 35% higher in the IDW method. A GZC value of up to 0.16 illustrates an error that is less than 50% of the maximum error observed among all GZC values. However, as the GZC values increase beyond this threshold, errors tend to escalate.

7.3. RMSE Indicator

As depicted in Figure 11, a pattern of stability is observed in the RMSE indicator for both spatial interpolation methods. However, with an increase in the GZC values, there is an evident escalation in the RMSE indicator. Changes in the RMSE indicator between the IDW and OKI methods decrease significantly with an increase in GZC, reducing the difference from 56% to 1%. Indeed, this observation (based on Figure 12) indicates the fact that with an increase in the distance between sites (with an increase in GZC), both proposed methods exhibit errors and performances that become closer, although they are not entirely equal. There is, however, one exception related to GZC of 0.02. In this case, the relative error in the OKI method surpasses that in the IDW method. This point suggests that in cases where the distance between sites is extremely close, the IDW method demonstrates superior performance.

7.4. P.R² Indicator

According to Figure 13, with an increase in the values of GZC and a decrease in the number of sites, the P.R² indicator decreases. This decreasing trend is noticeable in both spatial interpolation methods, but it is more pronounced in the IDW method. In cases where the number of sites is high and close to each other (smaller values for GZC), the P.R² indicator for both spatial interpolation methods is similar and close.

In Figure 13, the P.R² indicator is calculated based on the reference sites, but the details were not sufficiently clear. Therefore, in subsequent analyses, the P.R² indicator was computed for a specific set of sites (fewer than the reference site). This was performed to evaluate the predictions at a specific set of points. At each step of the analysis, the minimum acceleration was considered. Then, sites with acceleration exceeding the defined minimum threshold were used to assess the P.R² indicator. With changes in the minimum acceleration during subsequent analyses and reevaluations, the P.R² indicator was reassessed. In Figure 14 and Figure 15, the results of this analysis are presented for both spatial interpolation methods. As long as GZC is less than 0.14, the P.R² indicator in OKI provides a good approximation. However, for the GZC values exceeding 0.14, especially when PGA exceeds 0.4 g, the approximations appear to be insufficient. However, for the GZC values exceeding 0.14, especially when PGA exceeds 0.4 g, the OKI and IDW methods have similar patterns.

7.5. MaxE Indicator

The MaxE indicator in interpolation models based on each GZC is displayed in Figure 16. With an increase in the value of GZC, an increasing trend in error is evident in both interpolation methods, albeit the increase is more pronounced for the IDW method compared with the OKI method. However, with increasing distance from the sample point (for a GZC greater than 0.2), the different MaxE indicators of both methods are reduced.

7.6. Comparison of Spatial Interpolation Method with A.I

In this research work, to quantify the mentioned statistical indicators and assess their utility in identifying appropriate interpolation methods and suitable sample point distances, the A.I was employed. The results of A.I are shown in Figure 17. The results of A.I in Figure 17 indicate that as the values of GZC increase, the value of A.I decreases, but the decrease in values is more pronounced in the IDW method compared with the OKI method. In accordance with Figure 17, the closer the sample points are to each other, the better the predicted model quality. The maximum A.I is equal to 0.9 at a GZC of 0.02 for both interpolation methods, but the decreasing trend of this index continues to 0.31 for the OKI method and 0.18 for the IDW method. Based on the OKI method, A.I decreases by less than 20% up to a GZC of 0.14 compared with a GZC of 0.02, exhibiting suitable sample point distances within this range. While the error of a GZC of 0.14 is 7% higher than that of a GZC of 0.12, the computational volume decreases by more than 50%. According to this and engineering judgment, a GZC of 0.14 can be suitable. However, considering the analysis time and computational volumes based on the number of sites considered in each GZC value, a GZC of 0.14 can yield appropriate outputs with reduced computational time and complexity compared with other GZC values. While a benefit model may prioritize a higher A.I value, the decision-making process for selecting a suitable model often considers time and volume savings from the analysis. The A.I method combines statistical indicators but does not consider information time and computational volume.

Based on the statistical indicators in the current study (Figure 9 to Figure 17), it is evident that the OKI method reveals higher accuracy compared with the IDW method. This difference in the estimation accuracy is attributed to the algorithms used in the two interpolation methods. In IDW, values are calculated utilizing the weighted average of nearby points. This method may produce results with a lower accuracy and greater sensitivity to outliers. The OKI method models spatial correlation employing a semivariogram and describes this correlation by solving equations. Computationally, this method is more optimized and efficient. The two methods can be compared based on their complexity, optimality, uncertainty estimation, and flexibility. IDW offers simplicity and speed but does not provide uncertainty assessments. On the other hand, OKI delivers optimal predictions along with uncertainty estimations but is more intricate.

8. Conclusions

A seismic hazard map is one of the most crucial stages in assessing and managing seismic hazards and risk and is applied across various domains of earthquake engineering. A seismic hazard map is a continuous process in a studied region, and nowadays, classifications are generated based on the outcomes of a series of discrete data (or relevant sites) using various interpolation methods. Consequently, a meticulous consideration of data quality, site relevance, and interpolation techniques is fundamental for enhancing the efficacy and reliability of seismic hazard maps, ultimately contributing to more informed decision-making in earthquake engineering. Hence, the computational demands of SHA increase with a larger dataset for interpolation. Moreover, each spatial interpolation method carries inherent uncertainties, necessitating careful choice based on the dataset’s size. Achieving a balance between computational efficiency and uncertainty management is paramount for developing regional hazard classifications. In this research work, a method is proposed to determine the appropriate number of sites for seismic hazard mapping, along with recommending a suitable spatial interpolation method. This method divides the study area into smaller cells (using GZC) to increase the number of sites. By considering the IDW and OKI models, it determines the spatial distribution of seismic hazards in the specified region. A clear understanding of the spatial distribution of seismic hazards obtained from these methods is necessary. On this basis, the spatial interpolation methods are evaluated utilizing efficiency and error estimates. Efficiency is assessed based on a combination of statistical indicators, including ME, MAE, RMSE, P.R², and MaxE with A.I. The following results can be inferred from observations and comparisons between the interpolation methods, considering the sample data and assessment criteria:

• The proposed method demonstrates that the OKI method is superior to the IDW method, except in cases where the data points are very close to each other, where the IDW method reports better results.
• With a reduction in the number of sites (sample data) and an increase in the distance between them, the results from both spatial interpolation methods tend to converge. However, even in this scenario, the outcomes related to the OKI method reveal higher accuracy.
• The spatial distribution of a seismic hazard, employing the OKI method, displays smoother variations compared with the IDW method. This characteristic becomes more pronounced in situations where the distances between sites are greater.
• In a recent study, it has been possible to achieve a qualitatively sound spatial distribution of seismic hazard employing sample data related to a GZC of 0.14. This results in a reduction in computational time compared with the reference data (related with a GZC value of 0.02). With good approximation, it predicts results close to those of the reference data.

Earthquake hazard maps are a fundamental component of seismic risk assessments. This study provides infrastructure projects with seismic hazard maps using an optimal point selection-based spatial interpolation method. The results assist specialists in generating earthquake hazard analysis maps with minimal field costs through appropriate grid grading and interpolation. These maps enable experts to evaluate ground motion parameter intensities and select suitable routes for energy infrastructure projects based on infrastructure seismic vulnerability.