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Problem Statement
Problem Statement
Precipitation amounts vary widely across Arizona as elevation changes between desert valleys to mountain ranges. Rain gauges provide precipitation measurements at discrete locations, but station coverage may be uneven or records may include gaps. Spatial interpolation provides an estimate between gauges using nearby observations to generate continuous raster surfaces. This study compares three interpolation methods: Inverse Distance Weighing (IDW), Spline, and Kriging, to determine which produces the most reasonable surface of average August precipitation (2000-2025) across Arizona.
Data
Data
Study Area
Arizona is located in the southwestern United States and includes a wide range of elevations and landforms, from desert basins to mountain ranges. The North American Monsoon season in Arizona typically occurs between June through September, when warm, moist air rises and condenses, creating favorable conditions for storms (ASU). During this season, rainfall can be highly localized, with higher amounts often occurring near mountain ranges where moist air is forced upwards. As a result, precipitation is not uniform across the state, making Arizona a challenging area for predicting and mapping rainfall patterns.
Data Description
Monthly precipitation data were obtained from the National Centers for Environmental Information (NCEI) Global Summary of the Month (GSOM), Version 1 dataset for stations located in Arizona (Figure 1). For this study, only August precipitation totals were analyzed. After extracting the August values, and selecting gauges with relatively few missing observations, a long-term mean for each station was calculated by averaging values from 2000-2025. The final dataset includes the each station's ID, its latitude/longitude point location, and average August precipitation value. A metadata table summarizing the dataset is provided below (Table 1).
Figure 1: Map of precipitation gauges across Arizona.
Table 1: Metadata table listing information from the precipitation dataset.
Methods
Methods
To ensure the results from the interpolation methods were limited to Arizona, an Arizona state boundary feature layer was added to the map. When importing the precipitation CSV as an XY point layer, the defined projection was selected as NAD 1983 StatePlane to ensure the gauge points would plot accurately over the boundary layer. Using a consistent projected coordinate system across layers prevents misalignment and ensures interpolation tools operate on correctly located inputs.
A total of 60 rain-gauge station points were included in this analysis. Each point represents the geographic location of a precipitation gauge in Arizona and contains an attribute value equal to the station’s average August precipitation (in inches) calculated from 2000–2025. Stations were selected to include those with sufficient data coverage over the study period and represent spatial variability across the state rather than clustering in a single region.
To evaluate interpolation performance, the 60 stations were split into training and testing sets using an approximately 80/20 ratio. Fifty (50) stations were used to generate the IDW, Spline, and Kriging raster surfaces, and the remaining ten (10) stations were withheld for testing. The predicted values were extracted at the testing locations and compared to the observed station averages to assess how well each method estimated precipitation at these points.
IDW Method
Inverse Distance Weighing (IDW) estimates values at unknown locations using nearby measured points. This method assumes that things closer together are more similar than things farther apart; therefore, nearby rain gauges influence the estimate more than distance gauges. The predicted values are essentially a weighted-average of surrounding points, where the weights get smaller as distance increases. A "power" parameter controls how quickly the influence of a station drops off as distance increases. The input parameters and assumptions used in this study are described below.
IDW Method Assumptions:
Environment parameters were set first to ensure that the output result stayed within the Arizona state boundary layer. This automatically updates the Output Cell Size to match that of the boundary layer. The Power parameter influences how much the distance from points influence the results, with a higher value resulting in less influence. A value of 2 was chosen as reasonable value to put less influence on distance between points. Variable was selected as the Search radius with the Number of points being 12, where variable search radius finds a specific number of input sample points for the interpolation.
Spline Method
Spline interpolation fits a smooth surface through the input points, similar to bending a flexible sheet so it passes through each provided value. This method emphasizes gradual changes, creating a surface that gently rises and falls between points rather than changing abruptly, making it an ideal choice areas with smoother spatial trends. However, when values change sharply over short distances, the method can overshoot and create unrealistically high or low predictions. The input parameters and assumptions used in this study are described below.
Spline Method Assumptions:
Environment parameters were set first to ensure that the output result stayed within the Arizona state boundary layer. This automatically updates the Output Cell Size to match that of the boundary layer. The Spline type was chosen as Regularized since it aims to produce smoother surface rasters. For the Weight parameter, higher values typically produce smoother surfaces, with 0.1 being a common choice. Number of points was kept the same as IDW method.
Kriging Method
Kriging estimates values at unsampled locations using both distance and the spatial pattern of similarity in the data. It first models how values change with distance using a variogram, then uses that relationship to assign weights to nearby points when making predictions. In other words, Kriging produces a “smart weighted average” guided by the dataset’s spatial structure, not just proximity. This makes it well-suited for situations where spatial autocorrelation is present. The input parameters and assumptions used in this study are described below.
Kriging Method Assumptions:
Environment parameters were set first to ensure that the output result stayed within the Arizona state boundary layer. This selection updates the Output Cell Size to match that of the boundary layer. The Semivariogram properties were chosen as Ordinary Kriging with a Spherical model. These parameters assume the nearby gauges are strongly related, and the relationship with distance. The Search radius and Number of points were kept consistent with the other methods.
Results
Results
Summary statistics for the observed test-gauge values and the interpolated predictions are presented in Tables 2 and 3. Observed gauge precipitation results ranged from 15.74 to 99.14 inches, with a mean of 54.89 and standard deviation of 23.27. IDW produced a similar range, 15.77 to 99.14 inches, with a mean of 52.07 and standard deviation of 18.28. Kriging results also closely matched the gauge distribution, ranging from 15.75 to 99.14 inches, with a mean of 50.63 and standard deviation of 20.46. In contrast, Spline produced the widest and most extreme range, ranging from -42.14 to 168.69 inches, with a mean of 53.98 and standard deviation of 36.06.
Table 3 compares predicted values to the observed precipitation at the ten withheld test stations. To visualize these results clearer, a scatter plot was created to show how each interpolation method value plotted against the observation values (Figure 2). Locations (#) 1, 4, 5, and 10 had the closest accuracies with estimated values plotting near the true reported value. For the remaining locations, the three methods often produced similar estimates, generally within about ±10 inches of each other, but still differed noticeably from the observed gauge values. RMSE values were calculated to summarize overall performance: IDW = 16.97, Spline = 21.68, and Kriging = 18.58.
Interpolated raster surfaces produced from IDW, Spline, and Kriging methods are shown in Figure 3. The color ramp represents the average total amount of rainfall for the month of August, where lighter blue areas indicate lower precipitation and darker blue indicates higher amounts of precipitation.
Table 2: Comparing statistics between interpolated models to data points.
Table 3: Validating the predicted precipitation amount from each interpolated raster to the observed (Test) gauges.
Figure 2: Scatter plot showing a visual relation between test gauge values and reported estimates from IDW, Spline, and Kriging interpolation methods.
Figure 3: Comparison of interpolated surfaces of IDW, Spline, and Kriging methods.
Conclusion
Conclusion
IDW produced the lowest RMSE and the most stable value range compared to the observed gauges, suggesting it is the best method for creating a continuous precipitation surface for this dataset. Kriging produced similar spatial patterns, with a slightly higher RMSE, making it a strong alternative. Spline performed worst in this case, likely because Arizona's August precipitation is highly localized and can change abruptly, which increases the chance of unrealistic highs or lows when the model tries to enforce a smooth surface.
One limitation of this analysis is the relatively small number of stations and uneven spatial coverage across Arizona, with gaps that likely reduced accuracy in areas far from gauges. Incomplete station records may also bias the 2000–2025 averages if missing years are not consistent across locations. In addition, the test gauges were clustered more heavily in southern Arizona, which may have influenced the evaluation and reduced confidence in how well the methods perform in other parts of the state.
Overall, IDW produced the most reasonable surface for average August precipitation in Arizona based on the lowest RMSE and a distribution consistent with the gauge data. Kriging was a close second, while Spline was least suitable for these conditions. Expanding the station network and using a more evenly distributed set of test points would strengthen the comparison and help confirm the best method statewide.
References
References
Arizona State University, School of Geographical Sciences and Urban Planning. (n.d.). Basics of the Arizona monsoon & desert meteorology. Retrieved February 14, 2026, from https://sgsup.asu.edu/basics-arizona-monsoon-desert-meteorology
Esri. (n.d.). IDW (Spatial Analyst). ArcGIS Pro documentation. Retrieved February 15, 2026, from https://pro.arcgis.com/en/pro-app/latest/tool-reference/spatial-analyst/idw.htm
Esri. (n.d.). Kriging (Spatial Analyst). ArcGIS Pro documentation. Retrieved February 15, 2026, from https://pro.arcgis.com/en/pro-app/latest/tool-reference/spatial-analyst/kriging.htm
Esri. (n.d.). Spline (Spatial Analyst). ArcGIS Pro documentation. Retrieved February 15, 2026, from https://pro.arcgis.com/en/pro-app/latest/tool-reference/spatial-analyst/spline.htm
Lawrimore, J. H., Ray, R., Applequist, S., Korzeniewski, B., & Menne, M. J. (2016). Global Summary of the Month (GSOM), Version 1 [Data set]. NOAA National Centers for Environmental Information. https://doi.org/10.7289/V5QV3JJ5 (Retrieved February 14, 2026, from https://www.ncei.noaa.gov/access/search/data-search/global-summary-of-the-month?dataTypes=PRCP&pageNum=1
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