Developing Rainfall Spatial Distribution for Using Geostatistical Gap-Filled Terrestrial Gauge Records in the Mountainous Region of Oman

Abstract

Arid mountainous regions are vulnerable to extreme hydrological events such as floods and droughts. Providing accurate and continuous rainfall records with no gaps is crucial for effective flood mitigation and water resource management in these and downstream areas. Satellite data and geospatial interpolation can be employed for this purpose and to provide continuous data series. However, it is essential to thoroughly assess these methods to avoid an increase in errors and uncertainties in the design of flood protection and water resource management systems. The current study focuses on the mountainous region in northern Oman, which covers approximately 50,000 square kilometers, accounting for 16% of Oman’s total area. The study utilizes data from 279 rain gauges spanning from 1975 to 2009, with varying annual data gaps. Due to the limited accuracy of satellite data in arid and mountainous regions, 51 geospatial interpolations were used to fill data gaps to yield maximum annual and total yearly precipitation data records. The root mean square error (RMSE) and correlation coefficient (R) were used to assess the most suitable geospatial interpolation technique. The selected geospatial interpolation technique was utilized to generate the spatial distribution of annual maxima and total yearly precipitation over the study area for the period from 1975 to 2009. Furthermore, gamma, normal, and extreme value families of probability density functions (PDFs) were evaluated to fit the rain gauge gap-filled datasets. Finally, maximum annual precipitation values for return periods of 2, 5, 10, 25, 50, and 100 years were generated for each rain gauge. The results show that the geostatistical interpolation techniques outperformed the deterministic interpolation techniques in generating the spatial distribution of maximum and total yearly records over the study area.

1. Introduction

Spatial rainfall distribution plays a significant role in hydrological studies. It has important impacts on, e.g., floods occurrence, crop production, urban drainage systems, and hydraulic structures design. It also influences decision-making and risk assessment in environmental management. Arid regions are especially vulnerable to extreme hydrological events such as floods and droughts [1]. Providing accurate and gap-free rainfall records is vital for effective flood mitigation and water resource management in these and the downstream areas. However, these areas often suffer from insufficient precipitation records. To overcome this challenge, more advanced techniques such as satellite data and geostatistical interpolation need to be utilized. These methods, however, should be assessed and validated to optimize the design processes in water resource management.

Water scarcity is a global challenge for human development and the achievement of economic goals. The world is suffering from increasing demand of water resources due to the increase in urbanization, industrial activities, the world’s human population, and the overexploitation of aquifers [2,3]. In total, 700 million people suffer from the shortage of drinking water, and more than 40% of the world’s population suffers from water scarcity [4]. The increasing stress of water resources in arid regions due to the increasing population and climate change causes increasing water scarcity [2]. Moreover, the occurrence of floods poses a threat to lives and economics. On 10 September 2023, storm Daniel caused landfall in Libya. The massive flooding killed more than 4300 people, while more than 8500 people are still missing. On 7 March 2024, deadly floods struck West Sumatra in Indonesia, leading to the damage of homes and forcing people to migrate. The maximum yearly precipitation forms the backbone for the design of flood mitigation measures, while the total yearly precipitation is fundamental for sustainable water resource management. Filling the gaps in rainfall data can contribute to managing the water resources and meeting the increasing needs of fresh water resources.

Rainfall distribution is affected by mountainous terrain. The complex topography of mountainous regions makes the spatial rainfall distribution different from plain regions [5]. There are different spatial interpolation methods used to model rainfall that depend on the data of sparse stations to predict rainfall distribution but differ in their mathematical concepts. Spatial interpolation predicts values at locations with no observations [6]. Interpolation methods can be divided into two main groups: deterministic and geostatistical approaches. Deterministic interpolation techniques, like inverse distance weighting and Thiessen polygons, specify values to locations according to the neighboring observed values and to mathematical formulas that set the degree of smoothness of the generated surface. Geostatistical methods like kriging depend on statistical models that contain autocorrelation (statistical relationship among the measured points). Hence, geostatistical approaches not only generate a prediction surface but also provide some measure of the accuracy of the estimations. Deterministic methods do not use probability theory or an indication of the extent of possible errors, while geostatistical methods provide probabilistic estimates or use the concept of randomness [7,8]. Kriging is the most used geostatistical method for spatial interpolation, in which the neighboring observed values are weighted to produce an estimated value for an unmeasured point. Weights depend on the distance between the observed locations, the estimation points, and the spatial arrangement among the observed points. The weights are calculated such that points nearby to target locations are given more weight than those farther away. Kriging methods do not only consider the distance between the measured values but also capture the spatial structure in the data; they depend on a spatial model between observations defined by a variogram. Kriging methods are divided into two categories: univariate and multivariate methods. The methods that can use secondary information are called “multivariate”, while those that ignore secondary information are known as “univariate” methods [9].

Table 1. Geostatistical and deterministic interpolation techniques.

Geostatistical	Deterministic
Kriging	Inverse distance weighting (IDW)
Co-kriging	Radial basis function (RBF)
Empirical Bayesian kriging	Thiessen polygon
	Local polynomial interpolation (LPI)
	Global polynomial interpolation (GPI)

shows a summary of different geostatistical and deterministic interpolation techniques, while

Table 2. Univariate and multivariate kriging methods.

Univariate	Multivariate
Simple kriging (SK)	Universal kriging
Ordinary kriging	SK with varying local means
Block kriging	Kriging with an external drift
Factorial kriging	Simple co-kriging
Dual kriging	Ordinary co-kriging

presents univariate and multivariate kriging methods.

A variogram is a visual representation of the covariance between each two points in the sampled data, also called a semivariogram. The semivariance (i.e., gamma value) is the value of the half mean-squared difference between observations. The semivariance is plotted against the distance (i.e., lag) between each two points in the sampled data [10]. There are different variogram models, while the most commonly used are exponential, spherical, linear, Gaussian, and circular [11].

Spatial interpolation of annual rainfall was performed in South Africa using different interpolation techniques (universal kriging, ordinary kriging, co-kriging, and IDW); however, the cross-validation was applied to determine the best interpolation approach [12]. The best performance was produced by ordinary kriging. The results showed that the kriging methods outperformed IDW. For the variogram models, the circular model was the best regardless of the technique used. Kyriakidis et al. [13] conducted a study to map rainfall distribution from rain gauge data using different interpolation techniques in the coastal region of northern California that has the characteristics of extreme seasonal variability in rainfall and complex terrain topography. The results showed that using secondary information such as terrain and atmospheric characteristics in predicting the rainfall distribution could improve estimates and result in more accurate representations of rainfall distribution. Moreover, the magnitude of estimation improvement depends on the spatial variability of the rainfall field, the density of rain gauge stations, and the degree of correlation between rainfall and predictors.

Sadeghi et al. [14] performed a study to predict rainfall distribution in Iran. The study used the average data of rainfall from 35 stations and rainfall observations for the period from 1982 to 2012 using different interpolation methods (i.e., kriging, co-kriging, local polynomial interpolation, global polynomial interpolation, RBF, and IDW). These techniques were evaluated using cross-validation. The results showed that the simple co-kriging (exponential) method was the most suitable method to predict rainfall distribution over the study area, while IDW with power 5 had the poorest performance. The study found that correlation existed between elevation variations and the precision of co-kriging method. Haberlandt [15] applied multivariate methods (i.e., indicator kriging with an external drift (IKED) and kriging with an external drift (KED)) to predict the spatial distribution of hourly rainfall from rain gauges with the use of secondary information from elevation, rainfall from a daily network, and radar, in a region of 25,000 km² in southeast Germany. The cross-validation method was utilized to compare the performances of IKED and KED incorporating secondary information with univariate techniques (Thiessen polygon, IDW, ordinary indicator kriging (IK), and ordinary kriging (OK)). The results showed that the multivariate techniques IKED and KED obviously outperformed the univariate techniques due to the use of secondary information from elevation, daily rainfall from the network, and radar. In addition, the best result was produced when all secondary information were utilized together with KED.

Rata et al. [16] compared three interpolation approaches to mapping the annual rainfall distribution of Cheliff watershed, Algeria. The mean annual rainfall data with elevation from 58 stations for the period from 1972 to 2012 were used during the study. Moreover, OK, KED, and regression kriging (RK) were considered for the study. The interpolation approaches used were compared utilizing the cross-validation method. It was found that KED was the best interpolation technique to represent mean annual rainfall distribution for the study area. Goovaerts [17] used three multivariate interpolation methods (co-located co-kriging, kriging with an external drift, and simple kriging with varying local means) to incorporate elevation into the spatial interpolation of rainfall in a region of 5000 km² in Portugal. The data of monthly and annual rainfall from 36 climatic stations were used, and cross-validation was also utilized to compare the multivariate methods with three univariate ones (ordinary kriging, inverse square distance, and Thiessen polygon). The results showed that the three multivariate interpolation methods outperformed the univariate ones, as higher prediction errors were produced by the univariate techniques that ignore rainfall measurements at neighboring stations and elevation.

Frequency analysis is utilized to estimate the periodic occurrence of quantities of rainfall that are predicted in the future. Data from rainfall frequency analysis are vital for many sectors of water resource engineering such as dam and sewage system design and flood mitigation [18,19]. Rainfall frequency analysis is often performed with the use of maximum yearly rainfall series [20]. The quantity of rainfall over a certain location in each period is characterized by a high variation in time. This variation is dependent on, e.g., the length of the given period, climate type, and topographic conditions. Arid areas usually display higher variation. Normally, management and design of flood mitigation systems, hydraulic structures, and irrigation water supply are dependent on rainfall amounts that are predicted for a certain return period. This rainfall is determined using frequency analysis of long time series [21]. The return period in the frequency analysis of precipitation is the average time interval between the occurrence of rainfall with a specific quantity or intensity [22]. Rainfall records in frequency analysis are considered random variables that are based on identical distribution and independent variables [21,22]. The frequency analysis is performed using different probability distributions such as two-parameter distributions (gamma and Gumbel) and three-parameter distributions (generalized Pareto (GPA), Pearson type III (PE-III), generalized normal (GNO), generalized extreme value (GEV), and log-Pearson type III (LP-III)) [23].

For the determination of the goodness of fit for probability distributions, the chi-squared test can be utilized [24,25]. The Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC) are also commonly utilized techniques for model selection [26,27]. Assessment of the performance of probability distributions for the frequency analysis of rainfall was carried out utilizing rainfall data for the period from 1973 to 2012 for Dois Vizinhos, Brazil [28]. The study showed that Weibull and gamma distribution were the most suitable. Waghaye et al. [29] compared probability distributions for monthly rainfall data from 1972 to 2001 at Adilabad district of Telangana, India. The chi-squared test was conducted, and the gamma, GEV, and Gumbel distributions were the best. Yuan et al. [30] identified five probability distributions (i.e., gamma, Gumbel, normal, log-Pearson type III, and log-normal) to predict the frequency analysis of annual maximum hourly rainfall for the period from 1981 to 2000 for 15 locations in Japan. The chi-squared test was considered for evaluating the goodness of fit. It was found that log-Pearson type III was the most suitable for the data.

Based on the above, it is obvious that geospatial interpolation is important to overcome insufficient rainfall gauge data, especially in arid mountainous regions. Therefore, the objectives of the current study were (1) using different geospatial interpolation models to fill the precipitation data gaps in the mountainous region in arid northern Oman based on data from 279 rain gauges spanning from 1975 to 2009, (2) comparing the performance of the investigated geospatial interpolation methods, (3) generating spatial distributions for annual maximum and total annual rainfall over the study area using different geospatial interpolation techniques, and (4) performing frequency analysis on the annual maximum rainfall considering different probability distributions to produce predictions for the annual maximum rainfall with different return periods over the study area.

2. Materials and Methods

2.1. Study Area

Oman is located in the southeastern quarter of the Arabian Peninsula with a 309,500 km² land area. Several catchments are characterized by a mountainous terrain. The selection of the current study catchments was based on the availability of flood volume field measurements and corresponding rain gauge records. The study area is over 80,000 km² and contains 279 rain gauges. The weather of Oman is hot and dry in summer, moderately cool in winter, hot and humid for coastal areas, and moderate and rainy for highlands. The temperature exceeds 40 °C for the period from April to August and decreases gradually from September to February, the temperature reaches 5 to 10 °C for mountainous regions in winter and decreases below these at extreme heights. Figure 1 shows a topographic map of the study area. Figure 2 shows the distribution of rain gauge stations over the study area.

2.2. Rain Gauge Records

There are potentially 279 rain gauges in the study area that have been in operation from 1975 to 2009, but the actual available data in each year can be sparse.

Table 3. Available Omani rainfall data from gauge stations.

Year	Number of Working Stations	Year	Number of Working Stations
1975	31	1993	162
1976	37	1994	183
1977	44	1995	220
1978	48	1996	230
1979	56	1997	235
1980	53	1998	232
1981	56	1999	203
1982	62	2000	185
1983	80	2001	173
1984	83	2002	178
1985	67	2003	174
1986	102	2004	191
1987	113	2005	191
1988	108	2006	199
1989	109	2007	160
1990	110	2008	121
1991	113	2009	80
1992	122

shows the available data from rain gauge stations for each year.

2.3. Geospatial Interpolation Methods

Geospatial interpolation was utilized to generate the spatial distribution of precipitation. These methods include two categories: geostatistical and deterministic techniques. The selected geospatial interpolation methods (Figure 3 and Figure 4) were utilized to fill gaps in data and produce the spatial distribution for total and annual daily maximum rainfall over the study area. Cross-validation was utilized to evaluate the interpolation methods, observations were excluded from the study area, and then the values at these points were estimated using the remaining data. Root mean square error (RMSE) and correlation coefficient (R) were calculated for the assessment of the performance of the interpolation techniques. The lower the value of RMSE, the higher the accuracy of the interpolation method. The higher the value of R, the higher the accuracy of the interpolation method [7,31]. We used two deterministic methods (inverse distance weighting (IDW) and radial basis function (RBF)) and seven geostatistical methods (empirical Bayesian kriging (EBK), ordinary kriging (OK), ordinary co-kriging (OCK), simple kriging (SK), simple co-kriging (SCK), universal kriging (UK), and universal co-kriging (UCK)). For IDW, the power values 2, 3, 4, and 5 were used. For the RBF, the basic functions spline with tension (ST) and thin-plate spline (TPS) were used. For EBK, the variograms linear (L), power (P), and thin-plate spline (TPS) were used. For kriging, the variogram models circular, exponential, Gaussian, J Bessel, K Bessel, spherical, and stable were used. For co-kriging, elevation was incorporated as a secondary variable to improve estimates. ArcGIS 10.8 software was utilized to generate the spatial rainfall distribution maps for the study area. After interpolation, missing rainfall value gaps were filled in. Spatial rainfall distribution maps of annual and annual maximum daily were generated for each year.

2.3.1. Inverse Distance Weighting (IDW)

IDW is one of the most used deterministic interpolation methods. The main assumption in this method is that the weight values are inversely proportional to the powers of the distance between the interpolation points and the measurement point, and the measured values at a closer distance have a greater weight than those further away [32,33]. IDW calculates cell values based on a linearly weighted combination of a group of sampled points as [32]:

$$\mathrm{Z}({\mathrm{x}}_0)={\displaystyle \frac{\sum _{i=1}^n\frac{x_i}{d_{ij}^r}}{{\sum }_{i=1}^n\frac{1}{d_{ij}^r}}}$$

(1)

where Z(x₀) is the predicted value, x is the set of spatial coordinates (x₁, x₂), and r is the weight related to the distance d_ij between the prediction of the n data points.

2.3.2. Radial Basis Function (RBF)

The RBF is a deterministic method in which the interpolated surface requires passing through every measured point. Examples of basic functions are spline with tension (ST), thin-plate spline (TPS), inverse multiquadric function (IMQ), multiquadric function (MQ), and completely regularized spline (CRS) [32,34]. The main difference between IDW and the RBF is that IDW is based on the extent of similarity, while the RBF is based on the degree of smoothness [35]. IDW cannot estimate values above maximum measured values or below minimum measured values, while the RBF estimates values higher than the highest measured values and lower than the minimum measured values [36,37].

2.3.3. Ordinary Kriging (OK)

OK is the most used kriging method. It estimates the value of a point where a variogram is known with the use of the data in the neighborhood of the predicted point. A block value can be estimated using OK in lieu of a point value [38]. It uses an average of a group of neighboring points in order to produce a particular interpolation point. OK is based on the assumption that there is spatial autocorrelation between the points in the domain. Based on this relationship, the measured values at the points separated by short distances are more similar than those measured at the points separated by larger distances [38,39]. Using OK supposes estimating a value at x₀ using the values of the data points. The neighboring sampled points xα are combined linearly with weights w_α as:

$$z_{OK}^{\mathrm{*}}({\mathrm{x}}_0)={\sum }_{\alpha =1}^n{w}_{\alpha }z\left({\mathrm{x}}_{\alpha }\right)$$

(2)

where $z_{OK}^{\mathrm{*}}$ is the predicted value at the target point x₀, z is the measured value at the data point x_α, w_α is the weight attached to the data point, and n is the number of the data points utilized for the estimation. The weights assigned to the data points should sum to unity (Equation (3)); hence, if all values are equal to a constant, this will result in the predicted value being constant [38,40]:

$${\sum }_{\alpha =1}^n{w}_{\alpha } = 1$$

(3)

2.3.4. Simple Kriging (SK)

SK utilizes the average of all the sampled data. This produces less accuracy compared to OK; however, it results in a smoother result. It considers that the data have a known constant mean value throughout the study area, and the weights assigned to values do not sum to unity [32,41,42]. Let Z(x_α) be a random variable at each of the data locations X_α and let Z(x₀) be a random variable at the target point X₀. These random variables are assumed as a subset of an infinite collection of random variables called a random function Z(x) and the random function is assumed as second-order stationary. The expectation and the covariance are both translation-invariant over the domain. Assuming a vector h linking any two points x and x + h in the domain:

E[z(x + h)] = E[z(x)]

(4)

cov[z(x + h), z(x)] = c(h)

(5)

The expected value E[z(x)] = m is the same at any point x of the domain. The covariance between any two points is based on the vector h that links them. The aim is to form a weighted average to produce an estimation of the value at the target location X₀ based on information at data locations X_α; α = 1:n. This estimation procedure requires determining the covariances between the random variables at the points of the domain. This process is like multiple regression transposed into a spatial context where Z(x_α) performs like a regressor in regression with Z(x₀). The spatial regression is defined as SK, which has a mean m that is constant through the domain. This mean m is calculated as the average of the data [38].

2.3.5. Empirical Bayesian Kriging (EBK)

EBK is different from other kriging methods as it accounts for the error in estimating the variogram through repeated simulations, it uses many variogram models instead of a single variogram. This process includes the following steps [43]:

• EBK estimates a variogram model using information on the data.
• EBK simulates a new value at input data locations based on the estimated variogram.
• The simulated data are used to estimate a new variogram model. Bayes’ rule is used to calculate the weight of this variogram.
• With every repetition of steps (2 and 3), a new group of values at the input data locations is simulated using the estimated variogram in step 1.
• A new variogram model and its weights are estimated using the simulated data. These weights are utilized to generate predictions and prediction standard errors at the unsampled points.

2.3.6. Universal Kriging (UK)

UK is mostly utilized with data that have a considerable spatial trend, like a sloping surface. It relaxes the assumption of stationarity as it allows the mean of values to differ in a deterministic way in different locations (through some kind of spatial trend), while only the variance remains constant through the whole field [10,38,44]. UK is an extension of OK. The mean in OK is assumed to be constant through the search window, but in some cases the mean may not be constant due to the existence of a local trend, which results in the mean being a function of the local trend (coordinates). UK incorporates the local trend through the neighborhood search window as a function of the coordinates. The estimated value at the target point X₀ is represented by UK as a deterministic function of spatial coordinates as [9,45,46]:

z^*(x₀) = μ(x₀) + ε(x₀)

(6)

where z^*(x₀) is the predicted value at the target point x₀, μ(x₀) is a deterministic function that varies smoothly, and ε(x₀) is stochastic residuals.

2.3.7. Co-Kriging

Co-kriging is a geostatistical method that is used when a secondary variable is cross-correlated with the target variable, then both variables are used for estimation using co-kriging. The additional observed variables (known as covariates, which are often correlated with the target variable) are utilized to improve the precision of the interpolation of the target variable. Information on secondary variables is utilized to decrease the estimation variance of the target variable. Co-kriging outperforms kriging considering the accuracy of the estimations and the costs in the case that the random fields under study are cross-correlated [47,48]. The expression of co-kriging can be represented as [9,49]:

$$z_1^{*}(x_0)-{\mathsf{\mu }}_1={\sum }_{i_1=1}^{n_1}{w}_{i_1}[{\mathrm{z}}_1(x_{i_1})-{\mathsf{\mu }}_1(x_{i_1})]+{\sum }_{j=2}^{n_v}{\sum }_{i_j=1}^{n_j}{w_i}_j[{\mathrm{z}}_{\mathrm{j}}({x_i}_j)-{\mathsf{\mu }}_{\mathrm{j}}({x_i}_j)]$$

(7)

where $z_1^{\mathrm{*}}$($x_0$) is the predicted value at the target point x₀, μ₁ is a known stationary mean of the target variable, z₁($x_{i_1}$) is the data of the target variable at point i₁, μ₁($x_{i_1}$) is the mean of sampled points through the search window, n₁ is the number of samples through the search window for point x₀ used to make the prediction, ${w_i}_1$ is the weights used to reduce the estimation variance of the target variable, n_v is the number of secondary variables, n_j is the number of jth secondary variable through the search window, ${w_i}_j$ is the weights selected to be attached to the $i_j^{th}$ point of jth secondary variable, z_j(${x_i}_j$) is the data at the $i_j^{th}$ point of the jth secondary variable, and μ_j(${x_i}_j$) is the mean of the sampled points of the jth secondary variable through the search window.

Simple Co-Kriging (SCK)

SCK basically depends on determining the means of the variables. Therefore, the value of a variable at a certain point is predicted with no need for data on the variable in the neighborhood [38]. By substituting μ₁($x_{i_1}$) with μ₁, and substituting μ_j(${x_i}_j$) with the stationary mean μ_j of secondary variables in Equation (7), the estimator of simple co-kriging can be obtained [9,49], which can be represented as:

$$\begin{array}{r}{z}_1^{*}(x_0)={\sum }_{i_1=1}^{n_1}{w}_{i_1}[{\mathrm{z}}_1(x_{i_1})-{\mathsf{\mu }}_1]+{\mathsf{\mu }}_1+{\sum }_{j=2}^{n_v}{\sum }_{i_j=1}^{n_j}{w_i}_j[{\mathrm{z}}_{\mathrm{j}}({x_i}_j) - {\mathrm{\mu }}_{\mathrm{j}}]\\ ={\sum }_{i_1=1}^{n_1}{w}_{i_1}[\mathrm{z}1(x_{i_1})]+{\sum }_{j=2}^{n_v}{\sum }_{i_j=1}^{n_j}{w_i}_j\mathrm{zj}({x_i}_j)+(1-{\sum }_{i_1=1}^{n_1}{w}_{i_1})\mathsf{\mu }1-{\sum }_{j=2}^{n_v}{\sum }_{i_j=1}^{n_j}{w_i}_j\mu \mathrm{j}\end{array}$$

(8)

The advantage of SCK is that there are no constraints on weights, which reduces the variance of estimation, and the estimates are unbiased [50].

Ordinary Co-Kriging (OCK)

OCK is based on the following equations:

$$z_1^{*}\left(x_0\right) = {\mathsf{\mu }}_1 + {\mathsf{\epsilon }}_1\left({\mathrm{x}}_0\right)$$

(9)

$$z_2^{*}\left(x_0\right) = {\mathsf{\mu }}_2 + {\mathsf{\epsilon }}_2\left({\mathrm{x}}_0\right)$$

(10)

where μ₁ and μ₂ are unknown constants, and ε₁(x₀) and ε₂(x₀) are two types of random errors. So, there is autocorrelation for each one and cross-correlation between them. OCK predicts $z_1^{*}$($x_0$) with the use of information on the covariate $z_2^{*}$($x_0$) in order to produce better prediction. The information on the target variable z₁, cross-correlations between z₁ and other variable types, and autocorrelation for z₁ are utilized by OCK to produce more accurate estimations [38,51].

Universal Co-Kriging (UCK)

UCK is utilized to make predictions when dealing with multivariate random functions. It is routinely used in environmental studies when there is a spatially variable mean or a trend in the secondary or primary variables. It can make predictions in a configuration that constitutes a single location, multiple points, or a gradient [52,53].

2.4. Frequency Analysis

By using the generated gap-free data obtained from the selected geospatial interpolation method, frequency analysis was performed using commonly used probability distributions, which can be divided into two groups: Three-parameter distributions are generalized extreme value (GEV), log-normal three-parameter (LN3), and Pearson type III (P-III)). Two-parameter distributions are extreme value type 1 (EV1), log-normal (LN2), exponential (EXPN), gamma, and Weibull (WEI2) [54,55]. Each distribution is identified by a probability density function.

Table 4. Functions of probability distribution used [56].

Name	Probability Distribution	Parameters
GEV	$f\left(x\right)={\displaystyle \frac{1}{\alpha }}{\left[1-{\displaystyle \frac{k}{\alpha }}\left(x-u\right)\right]}^{{\displaystyle \frac{1}{k}}-1}exp\left[-{\left(1-{\displaystyle \frac{k}{\alpha }}\left(x-u\right)\right)}^{{\displaystyle \frac{1}{k}}}\right]$	$\alpha ,k,u$
LN3	$f(x)={\displaystyle \frac{1}{(x-m)\sigma \sqrt{2\pi }}}exp\left[-{\displaystyle \frac{{\left(ln\left(x-m\right)-\mu \right)}^2}{2{\sigma }^2}}\right]$	$\mu ,\sigma ,m$
P-III	$f(x)={\displaystyle \frac{1}{{\alpha }^{\beta }\mathsf{\Gamma }\left(\beta \right)}}{(x-\gamma )}^{\beta -1}{e}^{-\left({\displaystyle \frac{x-\gamma }{\alpha }}\right)}$	$\alpha ,\beta ,\gamma$
EV1	$f(x)={\displaystyle \frac{1}{\alpha }}exp\left[-{\displaystyle \frac{x-u}{\alpha }}-exp\left(-{\displaystyle \frac{x-u}{\alpha }}\right)\right]$	$\alpha ,u$
LN2	$f(x)={\displaystyle \frac{1}{x\sigma \sqrt{2\pi }}}exp\left[-{\displaystyle \frac{{\left(ln\left(x\right)-\mu \right)}^2}{2{\sigma }^2}}\right]$	$\mu ,\sigma$
EXPN	$f(x)={\displaystyle \frac{1}{\alpha }}exp\left[-{\displaystyle \frac{x-\gamma }{\alpha }}\right]$	$\alpha ,\gamma$
Gamma	$f(x)={\displaystyle \frac{1}{{\alpha }^{\beta }\mathsf{\Gamma }\left(\beta \right)}}{x}^{\beta -1}{e}^{-\left({\displaystyle \frac{x}{\alpha }}\right)}$	$\alpha ,\beta$
WEI2	$f(x)={\displaystyle \frac{c}{\alpha }}{\left({\displaystyle \frac{x}{\alpha }}\right)}^{c-1}exp\left[-{\left({\displaystyle \frac{x}{\alpha }}\right)}^c\right]$	$\alpha ,c$

shows the functions of the probability distributions used. The maximum likelihood method (MLK) is utilized to choose the suitable parameters to optimize the likelihood of observations. Then, the chi-squared (x²) test is performed with a significance level of 5% for the assessment of the adequacy of the probability distribution.

The proposed probability distributions were utilized to make predictions for the maximum yearly rainfall in the future with different return periods. The Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC) were utilized to choose the suitable probability distribution for the data; the probability distribution that has the lowest AIC and BIC is the best to fit the data [27,57].

Figure 5 shows a schematic diagram for the applied procedure in the current study. First, the geospatial interpolation techniques were employed to fill the gaps in the data and generate the spatial distribution for annual maximum and total yearly precipitation over the study area. Secondly, the frequency analysis was performed using different probability distributions to make predictions for the maximum yearly rainfall in the future with different return periods.

3. Results and Discussion

The interpolation methods produced different spatial rainfall distribution patterns as shown in Figure 6; the colored legend goes from blue for low rainfall values to red for the highest rainfall values. Figure 6 shows a slight difference between the generated rainfall distribution pattern by each method, and the results showed that the OCK exponential method outperformed the other three methods (i.e., EBK L, OK stable, and RBF ST) in representing the spatial distribution of the maximum rainfall for year 1991 based on the cross-validation results, as it had a lower RMSE and a higher R.

For maximum yearly rainfall interpolation, the results showed that the kriging methods outperformed the deterministic ones (IDW and RBF). IDW with power 2 outperformed the other power values. The spline with the tension basis function was more efficient than the thin-plate spline. In average, universal co-kriging using the J Bessel variogram model was superior to the other methods. The exponential model performed as well as the circular model, while the stable model had the lowest performance. IDW with power 5 and the RBF method with the thin-plate spline basis function had the poorest performance in predicting the rainfall distribution over the study area. The results of the cross-validation for the interpolation methods used for maximum rainfall for the year 1999 are shown in

Table 5. Root mean square error (RMSE) and correlation coefficient (R) results for annual daily maximum rainfall for 1999.

Interpolation Method	RMSE	R	Interpolation Method	RMSE	R
EBK L	17.44	0.60	SCK J Bessel	18.11	0.56
EBK P	17.11	0.62	SCK K Bessel	17.58	0.62
EBK TPS	18.38	0.56	SCK spherical	17.60	0.59
IDW 2	17.31	0.61	SCK stable	17.89	0.58
IDW 3	17.55	0.60	SK circular	17.59	0.59
IDW 4	17.99	0.58	SK exponential	17.33	0.61
IDW 5	18.44	0.57	SK Gaussian	18.03	0.57
OCK circular	17.18	0.62	SK J Bessel	18.08	0.56
OCK exponential	17.35	0.61	SK K Bessel	17.26	0.62
OCK Gaussian	17.29	0.61	SK spherical	17.56	0.60
OCK J Bessel	17.32	0.61	SK stable	17.24	0.62
OCK K Bessel	17.21	0.62	UK circular	17.69	0.59
OCK spherical	17.27	0.61	UK exponential	17.52	0.60
OCK stable	17.14	0.62	UK Gaussian	17.59	0.60
OK circular	17.02	0.63	UK J Bessel	17.75	0.58
OK exponential	17.20	0.62	UK K Bessel	17.53	0.60
OK Gaussian	17.13	0.62	UK spherical	17.69	0.59
OK J Bessel	17.24	0.62	UK stable	17.51	0.60
OK K Bessel	17.20	0.62	UCK circular	17.56	0.60
OK spherical	17.13	0.62	UCK exponential	17.51	0.60
OK stable	17.22	0.62	UCK Gaussian	17.51	0.60
RBF ST	17.23	0.62	UCK J Bessel	17.20	0.62
RBF TPS	20.53	0.52	UCK K Bessel	17.43	0.61
SCK circular	17.62	0.59	UCK spherical	17.52	0.60
SCK exponential	17.37	0.61	UCK stable	17.44	0.60
SCK Gaussian	18.00	0.57

. Figure 7 and Figure 8 show the RMSE and R variability with interpolation methods for the maximum rainfall for the year 1999.

The results of maximum rainfall for 1999 showed that OK using circular variogram model was the best method at predicting the rainfall distribution as it had the lowest RMSE (17.02) and the highest R (0.63). Also, EBK P had the second best performance, with RMSE 17.11 and R 0.62. IDW with power 2 performed better than the other power values. For the RBF method, the spline with the tension basis function outperformed the thin-plate spline basis function. The RBF method with the thin-plate spline basis function had the poorest performance as it had the highest RMSE (20.53) and lowest R (0.52). Figure 7 and Figure 8 indicate that the RBF TPS method clearly produced less accuracy compared to other interpolation techniques, with the highest RMSE and lowest R. Figure 9 shows the spatial distribution of the maximum rainfall for the year 1999 produced by the OK circular method. The colored legend goes from blue for low rainfall values to red for the highest rainfall values. Figure 9 shows that the rainfall distribution is uneven, the western and northern areas show low rainfall values, while the high rainfall values are concentrated in the southern area.

Table 6. Best interpolation method for annual daily maximum rainfall for the period from 1975 to 2009.

Year	Best Interpolation Method	Year	Best Interpolation Method
1975	IDW 2	1993	OCK spherical
1976	SK J Bessel	1994	UCK J Bessel
1977	UCK J Bessel	1995	OCK exponential
1978	UCK circular	1996	UCK stable
1979	SK spherical	1997	OCK exponential
1980	UCK J Bessel	1998	OCK K Bessel
1981	SK Gaussian	1999	OK circular
1982	OCK K Bessel	2000	UCK Gaussian
1983	SCK circular	2001	RBF ST
1984	SK circular	2002	EBK P
1985	OK circular	2003	SK circular
1986	SCK K Bessel	2004	SCK J Bessel
1987	UCK J Bessel	2005	SCK Gaussian
1988	OCK K Bessel	2006	UCK exponential
1989	SK exponential	2007	UCK J Bessel
1990	OCK J Bessel	2008	SK J Bessel
1991	UCK J Bessel	2009	OK J Bessel
1992	SCK stable

shows the best interpolation method for annual daily maximum rainfall for each year for the period 1975–2009. In Table 6, it is clearly noted that geostatistical methods outperformed deterministic ones in predicting maximum yearly rainfall distribution over the study area.

For total annual rainfall interpolation, the kriging methods outperformed the deterministic ones (IDW and RBF). IDW with power 2 outperformed the other power values. The RBF method performed better than IDW. The spline with the tension basis function was more efficient than the thin-plate spline. On average, ordinary co-kriging and universal co-kriging using the K Bessel variogram model were superior to the other methods. The J Bessel and K Bessel variogram models performed well, while the Gaussian and exponential models had the lowest performance. IDW with power 5 and the RBF method with the thin-plate spline basis function had the poorest performance in predicting the rainfall distribution over the study area. The results of the cross-validation of the interpolation methods used for total rainfall for the year 2000 are shown in

Table 7. Root mean square error (RMSE) and correlation coefficient (R) results for the total rainfall in 2000.

Interpolation Method	RMSE	R	Interpolation Method	RMSE	R
EBK L	50.54	0.78	SCK J Bessel	49.57	0.79
EBK P	50.19	0.78	SCK K Bessel	48.47	0.80
EBK TPS	52.71	0.76	SCK spherical	48.31	0.80
IDW 2	53.50	0.77	SCK stable	48.56	0.80
IDW 3	56.25	0.75	SK circular	50.75	0.78
IDW 4	59.08	0.74	SK exponential	49.92	0.79
IDW 5	61.06	0.72	SK Gaussian	52.50	0.76
OCK circular	49.78	0.79	SK J Bessel	53.12	0.75
OCK exponential	48.74	0.80	SK K Bessel	49.87	0.79
OCK Gaussian	51.03	0.77	SK spherical	50.77	0.78
OCK J Bessel	51.14	0.77	SK stable	49.77	0.79
OCK K Bessel	48.51	0.80	UK circular	47.97	0.81
OCK spherical	49.55	0.79	UK exponential	47.81	0.81
OCK stable	48.57	0.79	UK Gaussian	47.86	0.81
OK circular	47.97	0.81	UK J Bessel	47.77	0.81
OK exponential	47.81	0.81	UK K Bessel	47.87	0.81
OK Gaussian	47.86	0.81	UK spherical	47.92	0.81
OK J Bessel	47.77	0.81	UK stable	47.82	0.81
OK K Bessel	47.87	0.81	UCK circular	49.78	0.79
OK spherical	47.92	0.81	UCK exponential	48.74	0.80
OK stable	47.82	0.81	UCK Gaussian	51.03	0.77
RBF ST	49.88	0.79	UCK J Bessel	51.14	0.77
RBF TPS	64.82	0.69	UCK K Bessel	48.51	0.80
SCK circular	48.29	0.80	UCK spherical	49.55	0.79
SCK exponential	48.80	0.80	UCK stable	48.57	0.80
SCK Gaussian	49.65	0.79

. Figure 10 and Figure 11 show the RMSE and R variability with interpolation method for the total rainfall for 2000.

The results for the total annual rainfall in 2000 showed that OK using the J Bessel variogram model and UK with the J Bessel variogram had the best performance in predicting the rainfall distribution as they had the lowest RMSE (47.77 mm/year) and the highest R (0.81). Also, OK using the exponential variogram and UK with the exponential variogram performed similarly well. IDW with power 2 performed better than the other power values. For the RBF method, the spline with the tension basis function outperformed the thin-plate spline basis function. The RBF method with the thin-plate spline basis function had the poorest performance as it had the highest RMSE (64.82 mm/year) and the lowest R (0.69). It is noted from Figure 10 and Figure 11 that the RBF TPS method obviously had less accuracy with respect to other interpolation techniques, with the highest RMSE and the lowest R. Figure 12 shows the spatial distribution of the total rainfall for 2000 produced by the UK J Bessel method. The colored legend goes from blue for low rainfall values to red for the highest rainfall values. It appears from Figure 12 that the rainfall distribution is uneven; the northern and southern areas are of low rainfall values, while the high rainfall values are concentrated in the western area.

Table 8. Best interpolation method for total annual rainfall for the period from 1975 to 2009.

Year	Best interpolation Method	Year	Best interpolation Method
1975	OCK circular/UCK circular	1993	OCK J Bessel/UCK J Bessel
1976	OCK exponential/UCK exponential	1994	OCK stable/UCK stable
1977	OCK K Bessel/UCK K Bessel	1995	OCK K Bessel/UCK K Bessel
1978	OCK circular/UCK circular	1996	OCK stable/UCK stable
1979	OCK J Bessel/UCK J Bessel	1997	OCK circular/UCK circular
1980	SCK Gaussian/SCK stable	1998	OCK stable/UCK stable
1981	RBF ST	1999	OCK Gaussian/UCK Gaussian
1982	OCK K Bessel/UCK K Bessel	2000	OK J Bessel/UK J Bessel
1983	OCK J Bessel	2001	RBF ST
1984	SCK exponential	2002	OK J Bessel/UK J Bessel
1985	UK J Bessel	2003	OCK spherical /UCK spherical
1986	OCK stable/UCK stable	2004	SCK J Bessel
1987	SK circular	2005	SCK J Bessel
1988	OCK K Bessel/UCK K Bessel	2006	EBK P
1989	OCK spherical /UCK spherical	2007	OCK circular/UCK circular
1990	OCK K Bessel/UCK K Bessel	2008	RBF ST
1991	OCK stable/UCK stable	2009	OCK J Bessel
1992	OCK spherical/UCK spherical

shows the best interpolation method for total yearly rainfall for each year for the period from 1975 to 2009. In general, the results obviously showed that geostatistical methods outperformed deterministic ones in predicting maximum yearly and total yearly rainfall over the study area. Figure 13 shows that for kriging methods, the J Bessel variogram model was superior to the other variogram models in predicting maximum yearly and total yearly rainfall over the study area regardless of the kriging method used, while the Gaussian model had the poorest performance.

A frequency analysis of the maximum yearly rainfall was performed for the original data and for the data after filling the gaps. For the original data analysis, 50 stations were excluded from the analysis as they had less than 10 records and they would not produce appropriate results.

Table 9. Frequency analysis of original sampled rainfall data (mm/year).

Station ID	Probability Distribution	Return Period (Years)						AIC	BIC
		2	5	10	25	50	100
CM996898AF	WEI2	25.57	46.03	59.10	74.67	85.55	95.87	306.06	309.17
	GEV	25.59	44.32	57.69	75.80	90.15	105.23	310.97	315.64
	EV1	26.39	44.37	56.27	71.32	82.48	93.55	309.23	312.34
	LN3	25.16	44.77	58.86	77.78	92.60	108.03	310.30	314.96
	GAMMA	24.69	45.64	60.16	78.69	92.47	106.16	306.77	309.88
	p-III	25.34	46.34	61.01	79.82	93.85	107.82	308.19	312.86
	EXPN	21.05	47.87	68.16	94.97	115.26	135.55	310.35	313.46
	LN2	21.90	47.69	71.62	110.51	146.24	188.16	312.89	316.00

shows the results of the frequency analysis of the original sampled rainfall data for return periods of 2, 5, 10, 25, 50, and 100 years.

Table 10. Frequency analysis of gap-free sampled rainfall data (mm/year).

Station ID	Probability Distribution	Return Period (Years)						AIC	BIC
		2	5	10	25	50	100
CM996898AF	WEI2	25.38	44.39	56.30	70.32	80.04	89.19	302.90	306.01
	GEV	26.04	43.37	54.77	69.06	79.60	90.00	308.02	312.69
	EV1	25.97	43.37	54.89	69.45	80.25	90.97	306.03	309.14
	LN3	25.47	43.61	55.97	71.94	84.05	96.34	307.55	312.21
	GAMMA	24.30	44.28	58.05	75.57	88.56	101.45	304.21	307.32
	p-III	24.94	44.95	58.84	76.60	89.82	102.95	305.85	310.52
	EXPN	20.55	46.67	66.43	92.56	112.32	132.08	308.52	311.63
	LN2	21.69	46.70	69.73	106.91	140.90	180.63	311.20	314.31

shows the frequency analysis of the data after filling the gaps. It appears that the WEI2 distribution was the best to fit the data for both the original data analysis and gap-free data analysis as it had the lowest AIC and BIC.

Figure 14 shows a comparison between the probability distributions in fitting the rainfall data. In general, the frequency analysis results showed that the WEI2 distribution was the best to fit the data, followed by the gamma distribution, while the LN3 distribution had the poorest performance.

Table 11. Ratio between average of predicted rainfall from mean gap-free and original data.

Return period (years)	2	5	10	25	50	100
Ratio between gap-free and original data	1.08	0.98	0.95	0.93	0.93	0.94

shows the ratio between the average of the predicted rainfall values by the gap-free data analysis and the average of the predicted values by the original data analysis, it was noticed that for the return period of 2 years, the average of the predicted rainfall values using the original data analysis was lower than the average of the predicted rainfall values using the gap-free data analysis. Also, for the other return periods, the average of the predicted values using the original data analysis was higher than the average of the predicted values using the gap-free data analysis. Figure 15 represents the trend in the frequency analysis results for the original sampled data analysis and gap-free data analysis. It appears that for return periods greater than 2 years, the predicted rainfall value from the original data analysis was higher than that obtained by the gap-free data analysis.

4. Conclusions and Recommendations

In the current research, the performance of various interpolation methods in predicting the spatial distribution of rainfall over the mountainous region of Oman was evaluated. The results demonstrated that geostatistical interpolation techniques (i.e., SK, SCK, OK, OCK, UK, UCK, and EBK) outperformed deterministic interpolation techniques (i.e., IDW and the RBF) in generating the spatial distribution of maximum and total yearly records over the study area in Oman.

Universal co-kriging was the most effective method at predicting the maximum yearly rainfall distribution across the study area. On the other hand, both ordinary co-kriging and universal co-kriging showed comparable performance in interpolating the total yearly rainfall distribution in Oman. These findings highlight the reliability of co-kriging methods, particularly when incorporating additional variables such as elevation, for accurate spatial rainfall estimation in complex terrains. The J Bessel variogram outperformed the other six evaluated variograms (i.e., exponential, circular, Gaussian, stable, spherical, and K Bessel) in representing the spatial variance of the rainfall records.

The frequency analysis was conducted using data from the available rain gauges, and the two-parameter Weibull distribution outperformed the other seven tested statistical distributions (i.e., GEV, LN3, P-III, EV1, LN2, EXPN, and gamma) in predicting the design storm for different return periods (i.e., 2, 5, 10, 25, 50, and 100 years). The results indicated that for return periods greater than two years, the corresponding rainfall depth derived from the raw rain gauge data (before filling data gaps) was higher than that obtained from the gap-filled data. This suggests that flood protection design based on rainfall depths derived from raw data without gap filling is on the conservative side, ensuring a higher margin of safety.