Compact Inversion of Gravity and Magnetic Data Using Huber Loss Function

Introduction

Gravity and magnetic methods are widely used in geophysics to investigate subsurface structures. These methods analyze variations in density and magnetic susceptibility to map geological formations. However, inverting this data to obtain meaningful subsurface models presents challenges due to non-uniqueness, noise, and outliers in data.

In this blog, we introduce a compact inversion technique based on the Huber loss function, which improves model stability and reduces the influence of outliers. This method achieves better convergence and higher accuracy compared to traditional inversion techniques.

1. Forward Modeling

Forward modeling calculates the geophysical response caused by subsurface features. It is based on the mathematical relationship between observed data and model parameters.

For gravity data, the relationship is given by:

[ f(r) = \int_V A(r, r_0) \kappa(r_0) d^3r_0 ]

where:

• ( f(r) ): Observed gravity field.
• ( A(r, r_0) ): Kernel function (Green’s function).
• ( \kappa(r_0) ): Density or magnetization.

For magnetic data, the kernel is:

[ A(r, r_0) = \frac{\mu_0}{4\pi} \frac{\partial^2}{\partial \alpha \partial \beta} \left( \frac{1}{|r - r_0|} \right) ]

2. Inverse Modeling

Inversion estimates the source parameters (density or magnetization) by minimizing the difference between observed and predicted data. The general form of the linear inverse problem is:

[ d = A m + e ]

where:

• ( d ): Observed data.
• ( A ): Sensitivity matrix.
• ( m ): Model parameters.
• ( e ): Error vector.

To stabilize the inversion, we minimize an objective function:

[ \Phi = \Phi_d + \beta \Phi_m ]

where:

• ( \Phi_d ): Data misfit.
• ( \Phi_m ): Model smoothness constraint.
• ( \beta ): Trade-off parameter.

3. Huber Loss Function

A major limitation of traditional inversion is its sensitivity to outliers. The Huber loss function overcomes this by combining L1 (absolute error) and L2 (squared error) norms.

[ L_\delta(r) = \begin{cases} \frac{1}{2} r^2 & \text{for } |r| \leq \delta
\delta(|r| - \frac{1}{2}\delta) & \text{otherwise} \end{cases} ]

This function behaves quadratically for small errors (smooth convergence) and linearly for large errors (robust to outliers).

4. Depth and Compactness Weighting

Depth weighting is applied to compensate for the natural decay of sensitivity with depth. It is defined as:

[ w(z) = (z + z_0)^{-\beta/2} ]

where:

• ( z ): Depth of the model cell.
• ( z_0 ): Small constant to prevent singularities.
• ( \beta ): Decay rate, typically 2 for gravity and 3 for magnetic data.

Compactness weighting promotes smaller, localized anomalies:

[ w(m_j) = (\epsilon + |m_j|^2)^{(\alpha - 2)/4} ]

where ( \alpha ) controls the compactness.

5. Real-World Application

The method was applied to gravity data from the Pyhäsalmi mine in Finland, a site with complex ore bodies. The Huber loss function effectively reduced noise and produced sharper boundaries compared to traditional methods, demonstrating its robustness in field applications.

6. Advantages of Huber Loss Function

• Robustness Against Outliers: Reduces the impact of noise and large residuals.
• Faster Convergence: Fewer iterations required compared to L1 and L2 norms.
• Improved Model Accuracy: Produces geologically meaningful models with sharper features.
• Flexibility: Applicable to both synthetic and real datasets with varying noise levels.

Conclusion

This blog demonstrated the principles behind forward and inverse modeling for gravity and magnetic data, highlighting the advantages of using the Huber loss function. The method offers significant improvements in robustness, speed, and accuracy, making it an essential tool for geophysical modeling.

For more details, download the full paper here.