Evaluation of Geospatial Interpolation Techniques for Enhancing Spatiotemporal Rainfall Distribution and Filling Data Gaps in Asir Region, Saudi Arabia
Abstract
:1. Introduction
Interpolation Method  Validation Method  Recommended Method  Ref. 

Thiessen Polygon (TB), Inverse Distance Weighting (IDW), Linear Regression (LG), Kriging with External Drift (KED), Ordinary Kriging (OK)  Root Mean Squared Error (RMSE)  Ordinary Kriging (OK)  [42] 
Inverse Distance Weighting (IDW), Local Polynomial Interpolation (LPI) Global Polynomial Interpolation (GPI) Simple Kriging (SK) Universal Kriging (UK), Ordinary Kriging (OK), Radial Basis Function (RBF)  Mean Error (ME) Root Mean Squared Error (RMSE)  Ordinary Kriging (OK)  [43] 
Natural Neighbor Interpolation (NNI), Ordinary Kriging (OK) Cokriging (CK)  Root Mean Squared Error (RMSE)  Cokriging (CK)  [44] 
Kriging with External Drift (KED), Optimal Interpolation Method (OIM), Thiessen Polygons (TB)  Root Mean Squared Error (RMSE)  Optimal Interpolation Method (OIM)  [45] 
Inverse Distance Weighting (IDW), Radial Basis Function (RBF), Diffusion Interpolation with Barrier (DIB), Kernel Interpolation with Barrier (KIB), Ordinary Kriging (OK), Empirical Bayesian Kriging (EBK)  LeaveOneOut CrossValidation (LOOCV), Mean Square Error (MSE), Mean Absolute Error (MAE), Mean Absolute Percentage Error (MAPE), Symmetric Mean Absolute Percentage Error (SMAPE) Nash–Sutcliffe Efficiency Coefficient (NSE)  Kernel Interpolation with Barrier (KIB)  [46] 
Inverse Distance Weighting (IDW), Radial Basis Function (RBF), Local Polynomial Interpolation (LPI), Global Polynomial Interpolation (GPI), Simple Kriging (SK), Universal Kriging (UK), Ordinary Kriging (OK), Empirical Bayesian Kriging (EBK), Empirical Bayesian Kriging Regression Prediction (EBKRP)  Mean Error (ME), Root Mean Square Error (RMSE), Pearson R2 (R2), Mean Standardized Error (MSE), Root Mean Square Standardized Error (RMSSE), Average Standard Error (ASE)  Empirical Bayesian Kriging Regression Prediction (EBKRP)  [47] 
Inverse Distance Weighting (IDW), Kriging, ANUDEM, Spline  Mean Absolute Error (MAE), Mean Relative Error (MRE), Root Mean Squared Error (RMSE), Spatial and Temporal Distributions.  Inverse Distance Weighting (IDW)  [48] 
Inverse Distance Weighting, Natural Neighbor (NN), Regularized Spline (RS), Tension Spline (TS), Ordinary Kriging (OK), Universal Kriging (UK)  Root Mean Square Error (RMSE), Mean Absolute Error (MAE), Mean Bias Error (MBE), Coefficient of Determination (R2)  Ordinary Kriging (OK)  [49] 
Inverse Distance Weighting (IDW), Ordinary Kriging (OK), Ordinary Cokriging (OCK), Linear Regression (LR), Simple Kriging with varying Local Means (SKLM), Kriging with an External Drift (KED)  Mean Error (ME), and Root MeanSquare Error (RMSE)  Ordinary Cokriging (OCK)  [50] 
Circular Ordinary Kriging (COK), Spherical Ordinary Kriging (SOK), Exponential Ordinary Kriging (EOK), Gaussian Ordinary Kriging (GOK), Empirical Bayesian Kriging (EBK)  Mean Error (ME), Mean Standardized Error (MSDE), Root Mean Square Standardized Error (RMSSDE) Mean Standard Error (MSE), Root Mean Square Error (RMSE)  Exponential Ordinary Kriging (EOK), Empirical Bayesian Kriging (EBK)  [51] 
 Assessing various spatial interpolation techniques to ascertain the optimal approach for accurate rainfall prediction across diverse arid regions.
 Analyzing the sufficiency of rainfall station distribution and pinpointing optimal sites for installing new rain gauges within the study area.
 Providing an illustrative example elucidating the practical utilization of study outcomes in filling data gaps at any location and time within the study area for endusers.
2. The Study Area
3. Materials and Methods
3.1. Rain Gauges
3.2. Interpolation Techniques
3.2.1. Deterministic Techniques
3.2.2. Geostatistical Techniques
4. Results and Discussion
5. Use Case
6. Summary and Conclusions
Supplementary Materials
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Appendix A. Spatial Interpolation Equations and Main Characteristics
 (A)
 Deterministic Approaches
 (A.1.)
 Inverse Distance Weighting (IDW)
 ${Z}_{x}$: the predicted unknown value at point $\left(x\right)$.
 ${\lambda}_{i}$: the weight value of the sampled point $\left(i\right)$.
 ${Z}_{i}$: the value of the sampled point $\left(i\right)$.
 ${d}_{xi}$: the distance between the sampled point $\left(i\right)$ and the predicted point $\left(x\right)$.
 $P$: the power of decreasing weight with distance.
 (A.2.)
 Global Polynomial Interpolation
 ${Z}_{\left({x}_{i},{y}_{i}\right)}$: location $\left({x}_{i},{y}_{i}\right)$ value.
 $\epsilon \left({x}_{i},{y}_{i}\right)$: random error.
 $\beta $: parameter.
 (A.3.)
 Local Polynomial Interpolation
 (A.4.)
 Radial Basis Function
 ${K}_{o}:$ modified Bessel function
 (B)
 Geostatistical Approaches
 ${z}^{*}\left({u}_{o}\right)$: an estimate of the variable of interest at the location $\left({u}_{o}\right)$.
 $z\left({u}_{i}\right):$ the measured value of the variable of interest at the location $\left({u}_{i}\right)$.
 ${\lambda}_{i}:$ the Kriging weight of $z\left({u}_{i}\right)$.
 $n$: the total number of data locations.
 (B.1.)
 Simple Kriging
 (B.2.)
 Ordinary Kriging
 $m:$ the mean value of the stationary variable.
 (B.3.)
 Universal Kriging
 $L$: the number of unbiasedness conditions.
 ${f}_{P}^{L}$: the P^{th} basis function.
 (B.4.)
 Cokriging
 (B.5.)
 Empirical Bayesian Kriging
Interpolation Technique  Advantages  Disadvantages 

Inverse Distance Weighting 


Global Polynomial Interpolation 


Local Polynomial Interpolation 


Radial Basis Function 


Kriging 


Cokriging 


Empirical Bayesian Kriging 


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Gauge  Long (WGS 84)  Lat (WGS84)  Altitude (m)  Mean Max. Rainfall (mm)  Mean Total Rainfall (mm)  Gauge  Long (WGS 84)  Lat (WGS 84)  Altitude (m)  Mean Max. Rainfall (mm)  Mean Total Rainfall (mm) 

A004  43.10  18.17  2280  31.9  97.9  J141  39.92  22.50  661  49.7  96.4 
A005  42.48  18.20  2186  57.3  282.8  J204  40.20  21.35  685  39.7  154.0 
A006  42.60  18.25  2121  40.3  238.5  J205  40.22  21.35  750  46.3  171.0 
A007  42.15  19.10  2376  61.8  344.4  J214  39.98  21.98  641  33.5  97.0 
A103  42.78  18.10  2224  39.5  237.9  J219  39.43  22.20  192  22.2  38.4 
A104  43.37  17.93  2281  32.1  127.2  J220  39.82  22.37  473  23.4  62.2 
A105  43.18  18.23  2206  23.9  70.1  J221  39.35  21.92  88  27.5  45.0 
A106  42.48  18.27  2356  38.5  253.2  J239  39.68  21.97  269  23.9  64.1 
A107  42.57  18.60  2016  31.9  109.8  N001  44.23  17.55  1278  19.8  44.5 
A108  42.38  18.52  2516  26.4  85.0  N103  43.63  17.68  2191  34.2  145.9 
A110  42.98  18.68  1802  33.9  98.3  N203  43.62  17.67  2036  26.9  94.3 
A112  42.57  18.37  2096  36.2  112.4  SA001  42.95  17.05  169  36.4  203.0 
A113  42.68  18.63  1840  17.5  69.3  SA002  42.62  17.17  32  28.2  80.2 
A117  42.27  18.62  2489  38.2  181.7  SA003  41.88  19.00  355  37.5  202.7 
A118  42.37  18.25  2197  56.7  333.7  SA004  41.40  18.73  41  27.4  61.2 
A120  42.17  18.88  2308  56.0  213.9  SA005  41.88  19.56  2263  27.2  172.5 
A121  42.75  18.03  2269  44.5  292.7  SA101  42.83  16.97  68  41.6  173.0 
A123  42.87  18.32  2039  34.3  128.9  SA102  42.23  17.70  51  30.1  65.2 
A124  42.33  18.42  2577  41.0  185.0  SA104  43.08  17.04  574  50.3  403.9 
A126  43.22  18.53  1850  12.9  26.5  SA105  41.97  18.93  458  48.6  364.1 
A127  42.25  18.78  2531  48.3  247.3  SA106  42.53  17.37  73  37.2  137.5 
A128  42.70  18.47  1927  16.5  60.2  SA107  42.78  17.12  78  44.6  170.0 
A130  42.32  18.33  2470  36.1  216.5  SA108  41.92  18.33  533  41.8  255.8 
A201  42.52  18.42  2083  25.9  103.0  SA110  43.13  17.27  1210  47.1  401.0 
A206  42.25  18.68  2603  39.1  206.6  SA111  43.12  17.05  574  54.5  444.1 
A213  42.83  18.17  2114  30.4  132.0  SA113  42.03  18.53  455  41.3  298.1 
B001  41.29  20.18  1932  58.9  294.0  SA115  41.67  18.00  0  28.1  54.6 
B004  42.60  20.02  1155  21.3  71.8  SA116  42.20  18.25  1421  49.0  321.7 
B005  42.53  19.87  1202  18.6  73.6  SA120  41.83  19.43  611  46.2  243.5 
B006  43.52  19.53  1090  21.8  48.9  SA122  41.83  19.12  377  31.1  208.6 
B007  41.57  19.87  2047  52.6  269.3  SA125  42.45  17.13  7  25.8  72.5 
B008  42.67  20.08  1139  24.3  80.1  SA126  42.88  17.45  559  45.9  502.3 
B009  41.90  19.53  2279  46.0  272.6  SA129  42.90  17.17  163  49.9  313.7 
B101  41.58  19.90  2040  50.8  224.9  SA132  42.78  17.02  61  29.5  127.0 
B103  41.65  20.25  1571  20.0  51.5  SA135  43.23  16.80  287  43.7  330.6 
B110  42.88  18.80  1742  22.4  73.2  SA136  43.13  16.68  159  45.6  251.7 
B111  42.85  21.25  922  18.4  48.8  SA137  42.95  16.60  61  29.5  116.3 
B114  42.23  20.02  1286  25.5  98.2  SA138  42.02  18.63  393  39.7  267.1 
B208  42.73  19.02  1717  20.3  69.6  SA139  42.03  19.05  900  26.2  293.3 
B216  41.98  19.47  2239  42.9  234.8  SA140  43.03  17.32  688  49.7  448.6 
B217  41.93  19.75  1756  43.0  170.4  SA142  41.58  18.77  104  32.3  101.9 
B219  42.80  19.33  1475  18.4  53.1  SA143  43.13  16.90  259  46.3  420.7 
B220  42.04  19.20  1571  36.0  142.6  SA144  42.25  18.17  1013  46.9  366.6 
J001  41.05  19.53  53  36.8  77.3  SA145  42.80  17.62  699  46.0  306.8 
J002  39.34  22.16  72  18.3  33.3  SA147  41.47  19.03  115  30.9  66.3 
J102  39.70  21.43  355  27.8  50.3  SA148  42.53  16.92  0  35.5  79.0 
J106  39.33  22.15  66  18.8  35.8  SA204  42.60  17.57  188  43.3  160.2 
J107  40.45  20.32  95  34.1  63.6  TA002  40.50  21.30  1590  33.0  148.2 
J108  40.28  20.15  7  32.8  59.8  TA004  40.45  21.40  1553  28.6  119.4 
J111  38.83  23.10  12  17.4  33.7  TA005  41.67  21.18  1148  14.0  49.4 
J113  40.12  21.37  455  47.8  146.0  TA006  41.28  20.62  1389  52.6  117.9 
J114  39.82  21.43  298  37.9  78.6  TA007  41.47  19.98  2256  48.6  194.0 
J116  39.63  22.58  394  28.2  63.3  TA104  40.80  21.32  1394  31.4  97.3 
J121  41.05  20.23  338  40.9  162.1  TA106  40.32  21.33  1888  44.8  185.5 
J124  41.28  19.95  586  30.0  114.4  TA109  40.37  21.07  2145  45.4  269.6 
J126  41.43  19.77  353  37.3  236.8  TA125  40.42  21.26  1713  29.7  51.7 
J127  41.53  19.67  657  52.7  184.0  TA206  40.40  21.28  1675  35.7  165.8 
J131  41.60  19.47  474  41.9  182.6  TA233  40.65  21.13  1691  48.5  205.6 
J134  39.20  21.50  15  28.7  51.4  TA250  40.45  21.67  1241  22.7  91.2 
J137  41.33  19.97  632  36.4  222.3  TA251  40.37  21.37  1822  30.1  120.1 
J139  41.03  19.73  93  38.6  63.9  TA255  40.36  21.24  1730  34.1  118.5 
J140  39.03  22.82  10  21.1  39.4 
Geostatistical Interpolation Techniques  

Ordinary Kriging—Circular Variogram (KOC)  Ordinary Cokriging—Circular Variogram (CKOC) 
Ordinary Kriging—Spherical Variogram (KOS)  Ordinary Cokriging—Spherical Variogram (CKOS) 
Ordinary Kriging—Exponential Variogram (KOE)  Ordinary Cokriging—Exponential Variogram (CKOE) 
Ordinary Kriging—Gaussian Variogram (KOG)  Ordinary Cokriging—Gaussian Variogram (CKOG) 
Ordinary Kriging—KBessel Variogram (KOK)  Ordinary Cokriging—KBessel Variogram (CKOK) 
Ordinary Kriging—JBessel Variogram (KOJ)  Ordinary Cokriging—JBessel Variogram (CKOJ) 
Ordinary Kriging—Stable Variogram (KOT)  Ordinary Cokriging—Stable Variogram (CKOT) 
Simple Kriging—Circular Variogram (KSC)  Simple Cokriging—Circular Variogram (CKSC) 
Simple Kriging—Spherical Variogram (KSS)  Simple Cokriging—Spherical Variogram (CKSS) 
Simple Kriging—Exponential Variogram (KSE)  Simple Cokriging—Exponential Variogram (CKSE) 
Simple Kriging—Gaussian Variogram (KSG)  Simple Cokriging—Gaussian Variogram (CKSG) 
Simple Kriging—KBessel Variogram (KSK)  Simple Cokriging—KBessel Variogram (CKSK) 
Simple Kriging—JBessel Variogram (KSJ)  Simple Cokriging—JBessel Variogram (CKSJ) 
Simple Kriging—Stable Variogram (KST)  Simple Cokriging—Stable Variogram (CKST) 
Universal Kriging—Circular Variogram (KUC)  Universal Cokriging—Circular Variogram (CKUC) 
Universal Kriging—Spherical Variogram (KUS)  Universal Cokriging—Spherical Variogram (CKUS) 
Universal Kriging—Exponential Variogram (KUE)  Universal Cokriging—Exponential Variogram (CKUE) 
Universal Kriging—Gaussian Variogram (KUG)  Universal Cokriging—Gaussian Variogram (CKUG) 
Universal Kriging—KBessel Variogram (KUK)  Universal Cokriging—KBessel Variogram (CKUK) 
Universal Kriging—JBessel Variogram (KUJ)  Universal Cokriging—JBessel Variogram (CKUJ) 
Universal Kriging—Stable Variogram (KUT)  Universal Cokriging—Stable Variogram (CKUT) 
Empirical Bayesian Kriging (EBK)  
Deterministic interpolation techniques  
Global Polynomial Interpolation (GPI)  Inverse Distance Weighting—P = 2 (IDWP2) 
Local Polynomial Interpolation (LPI)  Inverse Distance Weighting—P = 3 (IDWP) 
Radial Basis Function (RBF)  Inverse Distance Weighting—P = 4 (IDWP4) 
Inverse Distance Weighting (P = 1) (IDWP1)  Inverse Distance Weighting—P = 5 (IDWP) 
Year  Total Yearly  Maximum Daily  Year  Total Yearly  Maximum Daily 

1966  KOJ  CKOK  1990  CKUK  CKOJ 
1967  KUJ  KUJ  1991  CKSJ  KOJ 
1968  KSJ  CKUC  1992  CKOJ  KSG 
1969  LPI  CKOJ  1993  CKUJ  CKSK 
1970  KSJ  KUJ  1994  CKOS  KOJ 
1971  CKOC  CKOC  1995  CKOC  CKUC 
1972  CKOC  KOJ  1996  CKSE  CKOG 
1973  CKOJ  KOJ  1997  CKUT  CKOT 
1974  KOJ  CKOC  1998  CKUK  CKOK 
1975  CKOJ  CKUJ  1999  CKSG  CKOC 
1976  CKOJ  LPI  2000  KSG  KOE 
1977  CKOG  CKUC  2001  KOT  KSJ 
1978  CKOJ  KUJ  2002  CKSC  CKOG 
1979  CKOT  CKOS  2003  KUJ  CKSJ 
1980  KOJ  CKOJ  2004  KOJ  CKOJ 
1981  CKSG  CKSJ  2005  CKSC  GPI 
1982  CKSG  CKUJ  2006  KOG  CKOK 
1983  CKUC  KOJ  2007  KOG  IDWP1 
1984  CKOC  CKSK  2008  CKSJ  CKSG 
1985  CKOE  CKOK  2009  CKSE  KUJ 
1986  CKOC  KUJ  2010  KSG  CKOC 
1987  KOJ  CKSK  2011  KSJ  KST 
1988  KUK  CKOC  2012  CKOS  CKUJ 
1989  CKUE  LPI  2013  CKOJ  CKOC 
Maximum Rainfall Error (%)  Total Rainfall Error (%)  Error Summation (%)  

Existing stations  14.60%  27.08%  41.68% 
Existing stations + Proposed stations (I)  12.71%  22.34%  35.05% 
Existing stations + Proposed stations (I) and (II)  11.65%  20.95%  32.61% 
Existing stations + Proposed stations (I), (II), and (III)  11.51%  20.79%  32.31% 
Aridity Level  Aridity Index (AI) 

Desert  AI < 0.03 
Hyper arid  0.03 < AI < 0.05 
Arid  0.05 < AI < 0.20 
Semiarid  0.20 < AI < 0.50 
Dry  0.50 < AI < 0.65 
Subhumid  0.65 < AI < 0.75 
Humid  AI > 0.75 
Cold  PET ≤ 400 mm 
J124  B001  J137  J126  TA007  J127  B007  B101  

2008  23.0  32.0  missing  26.78991  26.0  missing  46.0  19.0 
2009  34.0  63.0  missing  33.0  54.5  missing  42.5  25.0 
2010  60.0  56.0  missing  41.0  168.0  missing  54.0  45.0 
2011  28.0  82.5  missing  22.0  60.0  missing  44.0  27.0 
2012  40.0  missing  missing  17.0  34.5  missing  62.0  56.0 
2013  26.5  missing  missing  28.0  24.5  missing  54.0  40.0 
2014  7.3  30.8  70.4  35.5  37.8  40.5  36.1  41.5 
2015  6.5  89.0  51.3  51.5  missing  50.8  59.0  missing 
2016  32.0  69.0  50.5  50.8  missing  50.5  26.0  52.0 
2017  1.3  78.8  24.5  21.6  missing  45.3  26.2  15.0 
2018  5.0  34.4  42.5  21.5  missing  40.5  35.8  34.5 
J124  B001  J137  J126  TA007  J127  B007  B101  

2008  23.0  32.0  27.2  26.8  26.0  28.2  46.0  19.0 
2009  34.0  63.0  46.4  33.0  54.5  51.4  42.5  25.0 
2010  60.0  56.0  46.4  41.0  168.0  42.6  54.0  45.0 
2011  28.0  82.5  30.4  22.0  60.0  22.9  44.0  27.0 
2012  40.0  19.4  26.8  17.0  34.5  32.4  62.0  56.0 
2013  26.5  30.0  30.9  28.0  24.5  32.5  54.0  40.0 
2014  7.3  30.8  70.4  35.5  37.8  40.5  36.1  41.5 
2015  6.5  89.0  51.3  51.5  50.4  50.8  59.0  52.9 
2016  32.0  69.0  50.5  50.8  47.2  50.5  26.0  52.0 
2017  16.0  78.8  24.5  21.6  24.4  45.3  26.2  15.0 
2018  5.0  34.4  42.5  21.5  30.6  40.5  35.8  34.5 
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Helmi, A.M.; Elgamal, M.; Farouk, M.I.; Abdelhamed, M.S.; Essawy, B.T. Evaluation of Geospatial Interpolation Techniques for Enhancing Spatiotemporal Rainfall Distribution and Filling Data Gaps in Asir Region, Saudi Arabia. Sustainability 2023, 15, 14028. https://doi.org/10.3390/su151814028
Helmi AM, Elgamal M, Farouk MI, Abdelhamed MS, Essawy BT. Evaluation of Geospatial Interpolation Techniques for Enhancing Spatiotemporal Rainfall Distribution and Filling Data Gaps in Asir Region, Saudi Arabia. Sustainability. 2023; 15(18):14028. https://doi.org/10.3390/su151814028
Chicago/Turabian StyleHelmi, Ahmed M., Mohamed Elgamal, Mohamed I. Farouk, Mohamed S. Abdelhamed, and Bakinam T. Essawy. 2023. "Evaluation of Geospatial Interpolation Techniques for Enhancing Spatiotemporal Rainfall Distribution and Filling Data Gaps in Asir Region, Saudi Arabia" Sustainability 15, no. 18: 14028. https://doi.org/10.3390/su151814028