Inversion & uncertainty
Turning surface measurements back into a picture of the subsurface is called inversion. It never has just one answer, so the skill lies in finding the models that fit, choosing among them honestly, and saying how uncertain the result is.
- Tell a forward problem from an inverse problem
- Explain why many different subsurface models can fit the same data (try it yourself with gravity)
- Explain how noise sets the limit on what the data can tell you
- Describe how regularisation, prior geology and drilling turn an impossible question into a useful answer
This module builds on Geophysical methods, which explains what gravity, magnetics, electrical and seismic surveys measure. Here the question is what those measurements can and can't tell you.
Forward and inverse problems
Every geophysical method involves two problems that run in opposite directions.
The forward problem asks: given a model of the subsurface, what would we measure? This is just physics. Give me a sphere of a known size, depth and density, and Newton's law of gravity tells me exactly what a gravity meter will read at every point along the surface. A forward problem has one answer, and computers solve it easily.
The inverse problem asks the question we actually care about: given what we measured, what is down there? Working backwards like this is called inversion. In practice it is solved by guess-and-check on repeat.
The loop sounds simple, but the inverse problem is much harder than the forward one, for three reasons:
Non-uniqueness
Different subsurface models can produce exactly the same measurements. For gravity this follows from the physics itself, as the figure below shows.
Noise
Every reading includes error from the instrument, the terrain, traffic, tides and the operator. A model that fits every wiggle is fitting noise.
Too few data
A survey might have a few thousand readings. A 3D model might have a million cells. With more unknowns than measurements the problem is underdetermined: infinitely many models fit.
A worked inversion: gravity
Gravity is the most intuitive method and the clearest example of inversion. A gravimeter, essentially an extraordinarily sensitive spring balance, measures the tiny extra pull of buried dense rock, in milligals (mGal). A large ore body changes gravity by about one part in a million, around 1 mGal, and a good instrument detects changes ten to a hundred times smaller.
Below is a real-style inverse problem. A gravity survey has recorded readings every 50 m along a 2 km line (black dots). Somewhere underneath is a buried, dense, roughly spherical body, but its details are hidden. Adjust your model until its predicted curve (blue line) runs through the data. Then press Let the computer fit it to watch an automatic inversion run the guess, predict, compare, adjust loop for you.
Under the hood the forward problem is a single formula. For a sphere of radius R and density contrast Δρ (how much denser it is than the surrounding rock), with its centre at depth z, the extra gravity at horizontal distance x from the point above its centre is:
Here G is the gravitational constant (6.674 × 10⁻¹¹ in SI units). The answer comes out in m/s², and 1 mGal = 10⁻⁵ m/s². The misfit readout is the root-mean-square (RMS) difference between your curve and the dots. The readings start with about 0.02 mGal of random noise, so that is the best misfit you can honestly hope for. You can change the noise level too.
The data come from a sphere plus random noise (the same random draw for everyone, scaled by the noise slider). "Every model that fits" searches all positions and depths and keeps those whose misfit is within what noise alone would produce. Real bodies aren't spheres, which widens the range further.
Depth comes from the shape. A deep body gives a broad, gentle anomaly and a shallow one gives a sharp, narrow spike. For a sphere, the curve falls to half its peak value at a horizontal distance of about 0.77 × z either side of the peak. This distance is the half-width. If the observed curve drops to half its peak about 230 m from the centre, the sphere's centre must be about 230 ÷ 0.77 ≈ 300 m deep, whatever its size.
Mass comes from the height. Once depth is fixed, the peak value fixes the total excess mass, Δρ × (4/3)πR³. But a small, very dense sphere and a big, slightly dense one with the same mass give exactly the same curve. Press "Show another model that fits" to see this. The data constrain depth and mass. They cannot separate size from density at all. Only extra information, such as a rock density measured on drill core, can do that.
Noise sets the error bars. Turn the noise up and press "Show the range that fits": the box of possible centres grows, because a wider range of curves now fits within the scatter. Better data narrow the answer; they never remove the size–density trade-off.
Every method has its own blind spot
Gravity's size-versus-density trade-off is not a quirk. Every method has a combination of properties that its data cannot pull apart, and knowing it is the first step to reading any result. The simulators in Geophysical methods show each of these in action.
| Method | The data pin down well | The data can't separate |
|---|---|---|
| Gravity | Position and total excess mass; depth only once a shape is assumed (otherwise just a maximum depth) | Size from density contrast |
| Magnetics | Position and depth to the top of shallow bodies | Size from susceptibility; today's induced magnetism from "fossil" magnetism |
| MT and EM | A conductive layer's total conductance (thickness ÷ resistivity) | A thin, very conductive layer from a thicker, less conductive one |
| Seismic | Travel times, very precisely | Depth from sound speed: echo timing only loosely constrains speed, so depths need drill-hole ties |
For a thin conductive layer, MT measures the layer's conductance, in siemens. A 1 km layer at 10 Ωm (1,000 ÷ 10 = 100 siemens) and a 2 km layer at 20 Ωm (also 100 siemens) give almost the same curve. It is the same problem as size versus density in gravity, and the same cures apply: drill holes, other methods, and honest uncertainty.
Taming non-uniqueness
If infinitely many models fit, how does anyone get a useful answer? By adding information that isn't in the data, and being honest that you've done it.
First, decide what "fits" means. The standard measure is the data misfit, often written χ² ("chi-squared"). For each reading, take the gap between observed and predicted, divide it by that reading's expected noise, square it, and add them all up:
If the model is right and the noise estimate is honest, each term averages about 1, so χ² should come out roughly equal to the number of readings. Much larger means the model misses real signal. Much smaller means the model is chasing noise. That is overfitting, and it produces structures that aren't there.
Second, among all the models that fit to that level, choose the simplest. This is regularisation. The most common form, named after the mathematician Andrey Tikhonov, minimises a combined score:
Roughness measures how much the model changes from cell to cell, so smooth models score low. The regularisation parameter λ (lambda) sets the balance. Small λ means "fit the data at all costs". Large λ means "keep it smooth even if the fit suffers". Plotting one against the other for many values of λ traces a characteristic L-curve, and the best compromise usually sits near its corner. A common alternative is to pick the λ that gives χ² about equal to the number of readings.
Each reading below is a blurred average of the true signal around it, plus noise, much as a gravity reading blurs together the rocks beneath it. The model is the computer's attempt to un-blur the readings. It starts deliberately overfitted: turn the smoothing up and watch the model and the L-curve.
The smoothness preference explains model C in the earlier figure. Smooth inversions return blurry blobs, not because the ore body is blurry but because blur is what was asked for. Depth makes it worse: the signal from deeper structure is weaker and broader, so inversions resolve shallow features far better than deep ones. Geophysicists add further knowledge to narrow the range of acceptable models:
Prior geology
Mapped contacts, known rock densities and susceptibilities, and the expected shape of the deposit type are built into the starting model or the allowed ranges.
Joint inversion
Gravity, magnetics, EM and seismic respond to different properties of the same rocks. Joint inversion inverts them together and requires their boundaries to line up, which rules out many models that fit only one dataset.
Drill holes
Where a hole has measured the real rock, the model is pinned there. Each new hole is both a test of the last inversion and an input to the next one.
Inversion thinking is the mindset carbon-removal measurement depends on (see Measuring removal (MRV)). Estimating how much CO₂ spread basalt has removed means fitting a model to sparse, noisy soil and water samples, and many different stories about the field can fit the same samples. A credible answer fits to the noise level and no further, states the assumptions used to fill the gaps, and reports an honest uncertainty instead of a single confident number.
Inversion is also at the heart of geophysical exploration. A magnetic or gravity survey only measures the field at the surface. Turning it into a picture of what lies below is an inversion with many possible answers, and drilling decisions depend on knowing which parts of that picture the data actually support.