Extended Abstract
Introduction
This study aims to compare spatial interpolation methods for continuous modeling of the Earth's crustal velocity field across the Iranian Plateau and to investigate the effect of station distribution on prediction accuracy. The performance of four methods, namely Kriging, Spline, Inverse Distance Weighting (IDW), and Radial Basis Function (RBF), was evaluated using data from 104 GNSS stations covering the period from 2007 to 2010. To account for tectonic heterogeneity, regional analyses were conducted across five spatial subsets under three training–testing scenarios: 90/10, 80/20, and 70/30. The Root Mean Square Error (RMSE), Mean Absolute Error (MAE), and Mean Bias Error (MBE) were employed to assess prediction errors and bias. In addition to the initial evaluations, 70 repeated spatial partitions based on geographic block withholding were performed to assess the sensitivity of the results to station configurations, while the uncertainty of method comparisons was estimated using block bootstrap resampling. Under the 80/20 scenario, the station-equally weighted aggregated RMSE values for Kriging and IDW were 7.967 and 7.835 mm/yr, respectively. The close agreement between these values, together with the inclusion of zero in the confidence interval for the difference in MAE between the two methods, indicates that no statistically conclusive superiority of either method can be inferred from the available data. The resulting maps exhibit differences in terms of smoothness and spatial patterns; however, without quantitative comparison against fault geometry and a meaningful physical model, visual agreement with fault locations does not establish the physical superiority of any interpolation method. Velocities are reported in the Earth-Centered, Earth-Fixed (ECEF) reference frame and transformed into the East, North, and Up (ENU) coordinate system for the interpretation of local components. Given the temporal coverage of the dataset, the present findings are limited to a methodological assessment for the study period and should not be interpreted as evidence of current fault monitoring.
Materials and Methods
The dataset used in this study comprises measurements from 104 GNSS network stations distributed across Iran, collected and processed over the period from 2007 to 2010. Raw observations were processed using Bernese GNSS Software, Version 5.2. During this process, daily coordinates were estimated for each station throughout the study period and stored as time series. These time series consist of the X, Y, and Z components in the Earth-Centered, Earth-Fixed (ECEF) reference frame. To estimate velocities, a linear model was fitted to the coordinate time series, allowing the velocity of each station to be determined along the three coordinate axes. These three components constitute the primary inputs to the interpolation methods. The spatial distribution of GNSS stations is not uniform across the different tectonic zones, and this heterogeneity represents a major challenge in modeling the crustal displacement field. In Zones 1 and 2 (northwestern Iran and the Central Alborz region), station density is relatively adequate, with a more spatially homogeneous distribution. Zone 3 (northeastern Iran), characterized by moderate station density and a relatively regular distribution pattern, presents different spatial conditions. In contrast, Zone 4 (southern Iran), with a more irregular station distribution and greater inter-station distances, represents an area of higher spatial complexity. Zone 5 encompasses all 104 stations across the country, providing a comprehensive perspective on velocity-field variations at the national scale.
Discussion
Selecting an appropriate interpolation method for modeling the Earth's crustal velocity field across the Iranian Plateau depends not only on numerical errors but also on station distribution, spatial heterogeneity, and the limitations of GNSS data. The results of the initial scenarios indicate differences in the performance of the methods across the various zones. However, repeated spatial validation demonstrated that Kriging and IDW achieve comparable accuracy at the national scale under the 80/20 testing scenario. Accordingly, the conclusions are based on comparisons of errors at withheld stations, block-bootstrap confidence intervals, and the limitations of physical interpretation, rather than relying solely on a single training–testing split or the visual agreement of the resulting maps.
In this study, the main findings are interpreted in consideration of the spatial distribution of stations and the repeated structure of the dataset. The point estimate of an error metric alone does not establish the superiority of any method, and the repeated occurrence of stations across different zones should not artificially inflate the effective independent sample size. To quantify uncertainty, pairwise comparisons between interpolation methods were conducted using geographic block bootstrap resampling.
The dataset covers the period from 2007 to 2010, and the conclusions drawn herein concern the performance of interpolation methods during this specific period rather than the present-day monitoring of fault activity. Three levels of results should be distinguished. The first concerns prediction accuracy at test stations, evaluated using RMSE and MAE. The second concerns the stability of method rankings under the removal of spatial blocks, assessed through repeated partitions and resampling. The third concerns the physical validity of the relationship between velocity-field gradients and fault structures, which requires independent tectonic validation.
In this study, the first two levels are quantitatively assessed. However, at the third level, fault geometries were neither explicitly incorporated into the model as spatial barriers nor were predictions compared against independent observations from both sides of the faults. Furthermore, the unknown realization of the reference frame limits the interpretation of absolute or plate-relative velocities. Therefore, the role of the present findings is to assess the capabilities of interpolation methods using historical data, rather than to predict new seismic events.
Conclusion
The selection of an interpolation model should not rely solely on achieving the minimum RMSE in a single data partition or on the visual agreement between an interpolated map and geological structures. In Iran's heterogeneous GNSS network, prediction error, sensitivity to the spatial configuration of stations, and velocity uncertainties all influence the results. Based on the supplementary validation, Kriging and IDW exhibit comparable errors under the national-scale 80/20 scenario, and the physical superiority of either model along fault structures cannot be established from the present findings. These conclusions are valid for the 2007–2010 dataset, and their extension to the current period requires new observations.