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2.5-D Electrical Resistivity Forward Modelling with Undulating Topography using a Modified Half-Space Analytical Solution

This paper presents an improved 2.5-D electrical resistivity forward modelling method that utilizes a new analytical primary potential for V-shaped wedges to accurately handle undulating topography with sharp corners, thereby overcoming the geometric mismatches and significant errors inherent in traditional flat-half-space singularity removal techniques.

Original authors: Naveen K., Michael C. Koch, Kazunori Fujisawa, Arindam Dey, Sreedeep S

Published 2026-04-07

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CC BY 4.0

Original authors: Naveen K., Michael C. Koch, Kazunori Fujisawa, Arindam Dey, Sreedeep S

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to map what's happening deep underground—like finding buried treasure, groundwater, or old ruins—without digging a single hole. Geophysicists do this using Electrical Resistivity Tomography (ERT). They send an electric current into the ground through electrodes and measure how the voltage changes at the surface. The way the electricity flows tells them what the soil and rocks look like below.

However, the ground isn't always flat like a billiard table. Often, it's hilly, has deep valleys, or sharp cliffs. This creates a massive headache for the computer models used to interpret the data.

Here is a simple breakdown of the problem and the clever solution presented in this paper.

The Problem: The "Flat Earth" Mistake

To make sense of the electrical readings, computers use a technique called Forward Modelling. They simulate what the electricity should do in a specific model of the ground.

For decades, scientists have used a mathematical shortcut to make these simulations fast. They split the problem into two parts:

The Primary Potential: A simple, perfect mathematical formula for how electricity flows in a perfectly flat, empty world.
The Secondary Potential: The messy, complicated part that accounts for hills, valleys, and different types of rocks.

The Analogy: Imagine you are trying to predict how water flows down a hill.

The Old Way: You assume the ground is perfectly flat. You calculate the water flow for a flat floor, and then you try to "add on" the effect of the hill later.
The Flaw: If you stand right at the edge of a sharp cliff or a deep V-shaped trench, assuming the ground is flat is like trying to pour water on a flat table when you are actually standing on a steep roof. The math breaks down. The "flat floor" formula doesn't know how to handle the sharp angle of a cliff.

When the ground has sharp corners (like a V-shaped trench) or is very curvy, the old "flat floor" math creates huge errors. To fix this, previous methods tried to either:

Smooth out the cliff: Pretend the sharp edge is actually a gentle slope (which changes the real geography).
Use a super-fine mesh: Break the computer model into millions of tiny, tiny pieces to force the math to work (which takes forever to compute).

Both of these are inefficient or inaccurate.

The Solution: The "Custom-Fit" Formula

The authors of this paper realized that the error comes from a geometric mismatch. When electricity hits a sharp corner (like the bottom of a V-trench), it spreads out differently than it does on a flat surface.

They asked a simple question: "How much of the sky (or earth) can the electricity see from this specific spot?"

In geometry, this is called the Solid Angle.

On a flat surface, the ground covers exactly half the sky (a solid angle of $2\pi$ ).
In a deep V-trench, the ground covers more than half the sky because the walls are right there.
On a hilltop, the ground covers less than half the sky.

The Innovation:
Instead of forcing the computer to use the "flat surface" formula, the authors derived a new, custom formula that changes based on the angle of the ground right where the electrode is placed.

The Analogy:
Think of the old method as wearing one size fits all shoes. If you have flat feet, they are fine. If you have a clubfoot or a sharp angle, they hurt and don't fit.
The new method is like custom-molded shoes. The computer measures the exact shape of the ground at the electrode's location (the "solid angle") and instantly generates a mathematical formula that fits that specific shape perfectly.

Why This is a Big Deal

No More Smoothing: You don't have to pretend a sharp cliff is a gentle hill anymore. The math handles the sharp edge naturally.
Coarse Meshes Work: You don't need millions of tiny computer pieces to get an accurate answer. You can use a "chunkier" grid (like a low-resolution photo) and still get a crystal-clear result because the math formula is so smart.
Speed: Because the math is smarter, the computer doesn't have to work as hard. This means geophysicists can process data faster and more accurately.

The Results

The team tested their new method on:

Flat ground: It worked perfectly (as expected).
V-shaped trenches: The old method was off by 16% (a huge error), while the new method was off by less than 0.1% (basically perfect).
Wavy hills: Even with sharp curves, the new method stayed accurate, while the old method struggled unless the computer grid was made impossibly small.

The Bottom Line

This paper gives geophysicists a "magic formula" that understands the shape of the ground. It allows them to map the underground world accurately, even in rugged, mountainous, or cliff-filled terrain, without needing supercomputers or changing the actual landscape in their models. It's a smarter, faster, and more honest way to look beneath our feet.

1. Problem Statement

Direct Current (DC) resistivity imaging is widely used for subsurface profiling but is frequently conducted over undulating terrain. Accurate forward modeling is essential for reliable inversion, yet incorporating complex topography presents significant numerical challenges:

Singularity Removal: Standard numerical methods struggle with the singularity of the electric potential at point source electrodes. The "primary-secondary" potential split technique is commonly used, where the singular primary potential is solved analytically, and the smooth secondary potential is solved numerically.
Limitations of Existing Methods: Current approaches (e.g., Penz et al., 2013) rely on an analytical primary potential derived for a flat homogeneous half-space. This assumption is valid only when the terrain near the source is smooth and planar.
The Core Issue: In natural terrain with sharp corners, high curvature, or slope discontinuities, or even when smooth terrain is discretized using linear finite elements, the flat-surface assumption fails.
- Linear elements approximate curved surfaces as piecewise linear segments, creating artificial "V-shaped" corners at the source.
- The flat-half-space analytical solution does not satisfy the Neumann boundary condition (zero flux) on these V-shaped geometries.
- This geometric mismatch leads to substantial modeling errors (often >1%), which persist even with mesh refinement or geometric smoothing of the terrain.

2. Methodology

The authors propose a modified singularity removal strategy that derives a new analytical primary potential tailored to the local geometry of the source.

Governing Equations: The study utilizes the 2.5-D formulation of DC resistivity, applying a Fourier cosine transform to the 3-D Poisson equation to reduce it to a 2-D problem in the $x-z$ plane.
Primary-Secondary Split: The total potential $u$ is split into a primary potential ( $u_p$ ) and a secondary potential ( $u_s$ ).
Derivation of Modified Primary Potential:
- Instead of assuming a flat half-space (solid angle $S = 2\pi$ ), the authors derive an analytical solution for a point source located at the apex of a V-shaped wedge (a spherical lune).
- The solution accounts for the solid angle ( $S$ ) subtended by the local topography at the source.
- The new analytical expression for the primary potential is:
  $u_p(r) = \frac{I}{\sigma_0 S r}$
  In the 2.5-D Fourier domain:
  $\tilde{u}_p(x, k_y, z) = \frac{I}{\sigma_0 S} K_0(k_y r_{xz})$
  Where $S = 4\pi - 2\beta$ (with $\beta$ being the wedge angle), and for a flat surface, $S$ reverts to $2\pi$ .
Implementation:
- The modified primary potential is used to define the boundary conditions for the secondary potential.
- The method is implemented within a Finite Element Method (FEM) framework using linear (Q4) and higher-order (Q9, Q16) elements.
- The solid angle $S$ is calculated explicitly based on the geometry of the elements connected to the source node, accommodating both linear and curved discretizations.

3. Key Contributions

Identification of Geometric Mismatch: The paper rigorously demonstrates that modeling errors in undulating terrain stem from the mismatch between the flat-surface analytical primary field and the true solid angle subtended by the discretized topography (especially with linear elements).
Novel Analytical Formulation: Derivation of a closed-form analytical solution for the primary potential of a point source in a V-shaped wedge, generalizing the standard half-space solution.
Robustness with Linear Elements: The method allows for the use of coarse linear finite element meshes without requiring artificial terrain smoothing or excessive mesh refinement near the source, which are computationally expensive workarounds in existing methods.
Error Independence: Theoretical analysis shows that the error in the traditional flat-surface approach depends solely on the ratio of the true solid angle to $2\pi$ ( $|1 - S/2\pi|$ ), making the error constant regardless of the distance from the source.

4. Results

The proposed method was validated through numerical experiments on flat, V-shaped trench, and sinusoidal hill-valley models:

Flat Homogeneous & Layered Models: The method achieved relative errors < 0.1% against analytical solutions, confirming the implementation's correctness.
V-Shaped Trench Model:
- Traditional Method (Flat assumption): Produced errors of ~16.5% due to the sharp slope discontinuity at the source.
- Proposed Method: Achieved errors < 0.1% even with coarse meshes (1m element size) and linear elements.
- The error remained constant across the domain, validating the theoretical derivation.
Sinusoidal Hill-Valley Model:
- Traditional Method: Errors ranged from 1% to 2% with linear elements and up to 10% when the source was at a slope junction, even with higher-order elements.
- Proposed Method: Consistently achieved errors ≤ 0.1% across all mesh types (linear and curved) and source locations (valley, flat, and slope junction).
- Notably, the proposed method with linear elements outperformed the traditional method with curved higher-order elements in certain scenarios.

5. Significance and Conclusion

Computational Efficiency: By eliminating the need for dense mesh refinement or manual terrain smoothing near the source, the method significantly reduces the computational cost of forward modeling and inversion.
Accuracy in Complex Terrain: It provides a robust solution for DC resistivity modeling in areas with sharp topographic features (e.g., cliffs, trenches, ridges) where previous methods failed.
Practical Application: The ability to use linear elements (which are faster to compute and easier to generate) while maintaining high accuracy makes this approach highly suitable for large-scale 2.5-D inversion frameworks.
Future Work: The authors suggest extending this solid-angle-based approach to full 3-D forward modeling.

In summary, this paper presents a mathematically rigorous and computationally efficient advancement in geophysical forward modeling, resolving a long-standing issue of accuracy in undulating terrain by aligning the analytical primary field with the actual geometric solid angle of the discretized domain.