Spherical-Spline-Interpolation

Overview

This project implements spherical spline interpolation to address approximation problems on the sphere, specifically for gravity potential data. The interpolation uses three distinct kernel functions—Abel-Poisson, Singularity, and Logarithmic—to interpolate data from a coarser 6-degree grid to a finer 3-degree grid. The work is particularly relevant to fields such as geophysics, physical geodesy, and environmental sciences, where high-resolution, spatially distributed data are critical.

The study also explores the effects of ill-conditioning in the design matrix and applies regularization methods (TSVD, Tikhonov, Cholesky decomposition) to achieve stable solutions. Finally, the implementation evaluates the interpolation performance under noise conditions.

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Kernel Functions

Three kernel functions are used:

1. Abel-Poisson Kernel:

2. Singularity Kernel:

3. Logarithmic Kernel:

Regularization Methods

• Cholesky Decomposition: For numerically stable solutions in well-conditioned cases.
• TSVD (Truncated Singular Value Decomposition): For handling ill-conditioned matrices by truncating small singular values.
• Tikhonov Regularization: Uses a regularization parameter derived from Variance Component Estimation (VCE).

Dataset

The gravity potential data used in this project was obtained from the International Center for Global Earth Models (ICGEM). The original dataset is sampled on a 6-degree global grid and interpolated to a finer 3-degree grid.

Results

Without Noise

The interpolation was performed using all three kernel functions. Below is a summary of the results:

Abel-Poisson Kernel

Method	Mean Error ((m^2/s^2))	Norm of Errors ((m^2/s^2))
Cholesky	0.1612	2014.1234
TSVD	0.1766	4175.8890
Tikhonov (VCE)	-2.4909	3931.1049

Singularity Kernel

Method	Mean Error ((m^2/s^2))	Norm of Errors ((m^2/s^2))
Cholesky	0.1814	2014.3146
TSVD	0.1696	2016.4105
Tikhonov (VCE)	2.0778	3931.1049

Logarithmic Kernel

Method	Mean Error ((m^2/s^2))	Norm of Errors ((m^2/s^2))
Cholesky	0.1503	2005.1889
TSVD	0.1746	2013.2417
Tikhonov (VCE)	0.4764	5436.0035

With Added Noise

White noise with a standard deviation of (200 , \text{m}^2/\text{s}^2) was added to the input data. Below are the results for the Abel-Poisson Kernel:

Method	Mean Error ((m^2/s^2))	Norm of Errors ((m^2/s^2))
Cholesky	-3.1558	5548.1711
TSVD	-3.1328	12137.6909
Tikhonov (VCE)	71.1711	8200.7749

Tikhonov regularization provided smoother and more realistic results in noisy conditions compared to Cholesky and TSVD.