C4 · Publication Volume 14

Inversion and Non-Uniqueness

forward models, objective functions, regularisation and equivalence

Learning goals

The learner should be able to distinguish forward and inverse problems; construct a data-misfit term with a defensible error model; explain bounds, regularisation and prior constraints; demonstrate equivalent models; inspect residuals and sensitivity; and communicate an inverse result as a family of data-compatible explanations rather than recovered truth.

Inversion is valuable because it makes assumptions explicit and tests whether a property model can reproduce observations. It becomes misleading when the model mesh is mistaken for resolution, a low objective value is mistaken for geological correctness or one regularised solution is presented without alternatives.

Forward models and inverse questions

The forward problem predicts \mathbf d=F(\mathbf m,\mathbf g) from property model \mathbf m and geometry \mathbf g. The inverse problem asks which models are compatible with observed data, error and prior information. It may estimate continuous properties, boundaries, layer parameters, source locations or a small set of hypotheses.

Before inversion, validate the forward code against analytic cases, reciprocal relations, mesh refinement and unit tests. Include actual topography and source–receiver geometry where material. A forward-model error can be absorbed as false property structure if it is omitted from the data-error model.

Parameterisation is an assumption. Cell size, layer boundaries, body shapes and coordinate axes determine which models can exist. Fine cells do not guarantee fine resolution; they can increase equivalence. Choose a parameterisation that can represent the hypotheses without pretending every parameter is independently constrained.

Data misfit and error models

A common weighted misfit is

$\Phi_d=(\mathbf d-F(\mathbf m))^TC_d^{-1}(\mathbf d-F(\mathbf m)),$

where C_d represents uncertainty and correlation. Independent equal errors are a special case, not a default truth. Line-level bias, shared calibration, navigation and filtered data create correlations. Ignoring them can over-count evidence.

An error floor prevents extremely precise small values from dominating when modelling or systematic errors remain. It must be justified from repeats, residual structure and forward accuracy, not tuned solely to make convergence easy. Use normalised residuals and inspect them by line, component, time, frequency, gate and location.

A model that underfits may omit relevant structure or use excessive regularisation. A model that fits below credible noise may be fitting error. Neither the target misfit nor number of iterations establishes geological validity. Residual pattern is often more diagnostic than a scalar objective.

Regularisation, bounds and prior information

Because many models fit the data, inversion adds preferences such as small property magnitude, smooth variation, compactness, sparse boundaries or closeness to a reference model. A general objective is

$\Phi(\mathbf m)=\Phi_d(\mathbf m)+\lambda\Phi_m(\mathbf m),$

where \Phi_m encodes the preference and \lambda balances fit and regularisation. The selected model is the result of data plus that preference.

Bounds can express physically plausible property ranges. A reference model can encode known cover or background. Structural coupling can encourage shared boundaries across properties, but different processes need not coincide. Hard constraints should be supported by independent evidence; otherwise they can force a persuasive but wrong result.

Run regularisation-path and starting-model tests. Compare smooth, compact and interface-based parameterisations when the decision depends on shape. Record optimisation algorithm, stopping criteria, bounds, transforms, mesh, starting and reference models, weights and random seed where applicable.

Equivalence, resolution and uncertainty

Non-uniqueness has several forms: too few observations, broad physical kernels, parameter trade-offs, limited bandwidth, acquisition gaps and noise. Potential fields are especially non-unique, but electrical, EM and seismic inversions also admit equivalent structures. A deterministic solver returning one array does not remove this fact.

Different property models can fit the same observations; regularisation selects one member of a compatible family
Different property models can fit the same observations; regularisation selects one member of a compatible family

Assess sensitivity, resolution matrices where tractable, perturbation recovery, ensembles, posterior distributions or profile likelihoods according to problem scale. Each method inherits assumptions. A synthetic checkerboard recovered under the same regularisation is not proof that an unexpected real body is resolved.

Communicate robust features separately from model-dependent details. For example, data may require excess conductance in a broad corridor while its depth–thickness split remains unresolved. Provide at least two materially different acceptable models and an observation that would separate them.

Worked synthetic example

Suppose a synthetic datum obeys the linear relation

$d=m_1+2m_2=10.$

Models (m_1,m_2)=(10,0), (6,2) and (2,4) all fit exactly. If both parameters must be non-negative, infinitely many pairs on the segment from (10,0) to (0,5) remain. The datum does not identify a unique model.

Minimising m_1^2+m_2^2 subject to the relation selects one minimum-norm solution. Substitution gives m_1=10-2m_2; minimising (10-2m_2)^2+m_2^2 yields m_2=4 and m_1=2. That is a consequence of the norm preference, not extra information in the datum. A second independent observation, for example 2m_1-m_2=0, would select (2,4) from the family.

Inversion audit workflow

  1. state the decision, hypotheses and model parameters with units.
  2. validate forward physics, mesh, geometry and boundary conditions.
  3. build an uncertainty model including systematic and correlated terms.
  4. declare parameterisation, bounds, starting and reference models.
  5. define regularisation and justify its relation to prior evidence.
  6. inspect normalised residuals by acquisition dimension.
  7. vary error floors, regularisation, mesh and starting model.
  8. produce materially different data-compatible models.
  9. report robust features, unresolved trade-offs and a discriminating test.

Practice and review

  1. Find three solutions to 2m_1+m_2=12 under non-negative bounds.
  2. Explain why a smaller mesh cell does not guarantee better resolution.
  3. Give examples of correlated errors in line, EM and seismic data.
  4. Compare smoothness and compactness preferences for a thin body.
  5. Design a second observation to break a depth–contrast trade-off.

Review questions: Is the forward model validated? What uncertainty weights the residuals? Which structure came from regularisation? Are residuals patterned? Which alternatives fit? What data—not preference—require the reported feature?

Sources