Lecture 6a - Forward and Inverse Modelling

Forward and Inverse Modeling

Introduction

• Converting apparent resistivity measurements into estimates of actual subsurface resistivity variations.
• Two main modeling approaches:
  • Forward modeling
  • Inverse modeling
• These approaches are general and applicable beyond resistivity or geophysics.

Clarifying Forward and Inverse Models

• Defining and exploring the similarities and differences between forward and inverse models.
• Applying a forward model to resistivity sounding measurements to assess vertical resistivity changes.
• Highlighting key considerations and limitations of the resistivity method.
• Recap of forward modeling concept from the gravity section.

Forward Modeling

• A representation of the subsurface used to simulate expected measurement values.
• Example (Resistivity):
  • Model represents spatial variability of resistivity.
  • Output provides apparent resistivity values measured at the surface.
• Requires knowledge of the electrode geometry.
• Symbolized as estimating data from the model.
• Adjustments are typically done manually.
• In resistivity sounding, the model represents subsurface resistivity distribution, simulating the sounding curve with knowledge of array geometry.

Inverse Modeling

• Starts with measurement data to generate the subsurface model.
• Estimating the model from the data.
• Can be challenging and may require high-quality data.
• Model represents the subsurface and how it is observed, described by a number of parameters.
• Uses an automated optimization algorithm to find parameter values that allow the model output to best fit the data.

Inverse Modeling as a Process of Model Parameter Optimization

• Usually carried out automatically via numerical algorithms.
• Analogy: Fitting a trend line in Excel.
  • Data is provided.
  • Computer derives a model by automatically calculating the best parameter values in the trend line equation.

Forward Modeling Revisited (Gravity Example)

• Model of the subsurface representing density variations.
• Model contains parameters describing density differences and geometries.
• Use equations to estimate the gravity anomaly measurable at the surface.
• Model includes consideration of measuring vertical gravity changes.
• Assess how well simulated anomaly values match measured data (typically by eye).
• Adjust model parameters to better fit the measured data.

Applying Forward Modeling to Resistivity Sounding Data

• Resistivity model represents the vertical distribution of resistivity.
• Assumptions:
  • Subsurface layering is perfectly horizontal.
  • No lateral variations in resistivity.
• Model parameters to estimate:
  • Number of layers.
  • Resistivity of each layer.
  • Thickness of each layer.
• Limitations on detail due to assumptions.
• Measurement sensitivity decreases with depth.
• Models should be more detailed near the surface and less detailed with depth.

Measurement Sensitivity and Resistivity Structure

• Different resistivity layers have different detectabilities depending on their resistivity values.
• Sensitivity depends on resistivity distribution in the subsurface.
• Example: Three-layer model with varying resistivity for the lowest layer.
• $\text{Simulation: Three Layer Model}$
• $\rho_1 = 500 \Omega \cdot m$
• $\rho_2 = \text{Variable}$
• $\rho_3 = \text{Variable}$
• If the resistivity of the lower layer is high (e.g., $1000 \Omega \cdot m$), the sounding curve shows a dip, indicating three layers.
• If the resistivity of the lower layer is low (e.g., $250 \Omega \cdot m$ or less), the dip disappears, and the curve becomes sigmoidal, indicating only two layers.

Representing Subsurface Models

• Typically represented as a resistivity depth plot.
• Log scale used for the horizontal resistivity axis.
• Step line shows resistivity variations with depth.
• Vertical sections indicate resistivity for each layer.
• Horizontal sections indicate depths of layer boundaries.
• Four-layer model example:
  • Thin uppermost layer: $\approx 1 m$ thickness, $\approx 60 \Omega \cdot m$
  • Second layer: $\approx 10 m$ thickness, $\approx 200 \Omega \cdot m$
  • Third layer: $\approx 20 m$ thickness, $\approx 600 \Omega \cdot m$
  • Deepest layer: $\approx 40 \Omega \cdot m$
• Model represented by seven parameters (four resistivity values and three depth values).

Model Design Considerations

• Data may show clear evidence for only three layers, but prior experience may suggest a fourth layer.
• Include wider understanding from alternative sources (e.g., known topsoil layer).
• Avoid overparameterization (inclusion of too many parameters).
• Model complexity should reflect the volume quality and resolving power of the available measurements.
• Changes in one parameter can be accommodated by changes in another parameter without affecting the overall model fit to the data.
• Adopt Occam's razor: accept the simplest model that satisfies the data.

Summary

• Forward models use a subsurface model to estimate measured values.
• Models are improved by manually adjusting parameters.
• Inverse modeling is a computer-intensive approach that estimates the model from the data.
• Resistivity sounding models represent horizontal layers of different resistivity.
• Models are simplified representations subject to assumptions and sensitivity limitations.
• Models shouldn't be overcomplicated.
• Awareness of limitations is crucial, however well models fit the data.