This study presents a new method, called the Radial Interpolation Method, to interpolate data characterized by an approximately radial pattern around a relatively constrained central zone, such as the ground deformation patterns shown in many active volcanic areas. The method enables the fast production of short-term deformation maps on the base of spatially sparse ground deformation measurements and can provide uncertainty quantification on the interpolated values, fundamental for hazard assessment purposes and deformation source reconstruction. The presented approach is not dependent on a priori assumptions about the geometry, location and physical properties of the source, except for the requirement of a locally radial pattern, i.e., allowing multiple centers of symmetry. We test the new method on a synthetic point source example, and then, we apply the method to selected time intervals of real geodetic data collected at the Campi Flegrei caldera during the last 39 years, including examples of leveling, Geodetic Precise Traversing measurements and Global Positioning System. The maps of horizontal displacement, calculated inland, show maximum values lying along a semicircular annular region with a radius of about 2–3 km in size. This semi-annular area is marked by mesoscale structures such as faults, sand dikes and fractures. The maps of vertical displacement describe a linear relation between the maximum vertical uplift measured and the volume variation. The multiplicative factor in the linear relation is about 0.3 × 10 6 m 3 /cm if we estimate the proportion of the Δ V that is captured by the GPS network onland and we use this to estimate the full Δ V . In this case, the 95% confidence interval on K because of linear regression is ± 5%. Finally, we briefly discuss how the new method could be used for the production of short-term vent opening maps on the base of real-time geodetic measurements of the horizontal and vertical displacements.

中文翻译：

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GPS 的径向插值和复活的火山口中地面变形的水准数据：应用到 Campi Flegrei（意大利）

本研究提出了一种称为径向插值法的新方法，用于对以相对受约束的中心区域周围的近似径向模式为特征的数据进行内插，例如许多活动火山区中显示的地面变形模式。该方法能够在空间稀疏的地面变形测量的基础上快速生成短期变形图，并且可以提供内插值的不确定性量化，这是危险评估目的和变形源重建的基础。所提出的方法不依赖于关于源的几何形状、位置和物理特性的先验假设，除了局部径向图案的要求，即，允许多个对称中心。我们在合成点源示例上测试新方法，然后，我们将该方法应用于过去 39 年在 Campi Flegrei 火山口收集的真实大地测量数据的选定时间间隔，包括水准测量、大地精确穿越测量和全球定位系统的示例。内陆计算的水平位移图显示的最大值位于半径约为 2-3 公里的半圆形环形区域。这个半环状区域以断层、沙堤和裂缝等中尺度结构为标志。垂直位移图描述了测量的最大垂直隆起与体积变化之间的线性关系。如果我们估计陆地 GPS 网络捕获的 Δ V 的比例并使用它来估计完整的 Δ V ，则线性关系中的乘数约为 0.3 × 10 6 m 3 /cm。在这种情况下，由于线性回归，K 的 95% 置信区间为 ± 5%。最后，我们简要讨论了如何在水平和垂直位移的实时大地测量的基础上，将新方法用于制作短期通风口开口图。