Classical regularization method for solving ill-posed inverse problems in satellite gravimetry, balancing data fit against solution smoothness to obtain stable gravity field estimates.
\[J(\mathbf{x}) = \|\mathbf{Ax} - \mathbf{y}\|^2 + \alpha\|\mathbf{Lx}\|^2\]
Minimizing yields: \(\mathbf{x}_\alpha = (\mathbf{A}^T\mathbf{A} + \alpha\mathbf{L}^T\mathbf{L})^{-1}\mathbf{A}^T\mathbf{y}\)
Standard Tikhonov regularization penalizing solution norm. Best for problems where small solution values are expected. Equivalent to ridge regression.
Penalizes first derivative, promoting smooth solutions. Good for gravity field recovery where gradients should be continuous.
Penalizes curvature of solution. Produces very smooth fields, appropriate when signal is expected to be slowly varying.
Degree-dependent regularization based on expected gravity field power spectrum: σₙ² ∝ 1/n⁴. Physically motivated for Earth's gravity.
Plot \(\log\|\mathbf{Ax}_\alpha - \mathbf{y}\|\) vs \(\log\|\mathbf{Lx}_\alpha\|\) for varying α
Optimal α found at corner of L-shaped curve (maximum curvature)
• Direct solution, no iterations
• Requires full matrix storage
• Best for moderate-size problems
• Easy error propagation
• Discrete regularization
• Sharp spectral cutoff
• Good for rank-deficient problems
• No smoothing between components
• Memory efficient
• Early stopping as regularization
• Suitable for large-scale problems
• Harder to quantify uncertainty