Questions: Stress Inversion and Focal Mechanism Analysis

Question 1

A seismologist obtains a precise focal mechanism for a single well-recorded earthquake. Can she determine the regional stress tensor from this alone?

• A. Yes, because the focal mechanism uniquely specifies the stress field that caused slip on the fault
• B. No, for two reasons: the focal mechanism has a fault-plane ambiguity (two planes fit equally), and even knowing the fault plane, infinitely many stress states could produce that observed slip direction
• C. Yes, if the earthquake has a pure double-couple mechanism, the principal stress axes are directly readable from the compressional and dilatational quadrants
• D. No, because focal mechanisms only constrain the P and T axes, which do not correspond to any physical stress axis

Answer: (B)

Stress inversion from a single focal mechanism is impossible for two compounding reasons. First, the focal mechanism has an inherent ambiguity: a double-couple source has two nodal planes, and seismology alone cannot identify which one actually slipped. Second, even knowing the fault plane, the Wallace-Bott hypothesis still allows infinitely many stress tensors to predict the same slip direction on that plane — there are six independent stress components but only two angles of slip. Stress inversion only becomes tractable when many focal mechanisms are combined, making the problem overdetermined.

Question 2

What does the stress ratio R = (σ₂ − σ₃)/(σ₁ − σ₃) recovered by stress inversion describe?

• A. The magnitude of the maximum compressive stress relative to lithostatic pressure
• B. The shape of the stress ellipsoid — whether the intermediate stress σ₂ is closer to the maximum or minimum principal stress
• C. The ratio of horizontal to vertical stress, which determines whether faulting is normal, reverse, or strike-slip
• D. The fraction of differential stress released as seismic energy versus heat

Answer: (B)

The stress ratio R (0 ≤ R ≤ 1) describes the shape of the stress ellipsoid: where the intermediate stress σ₂ falls between σ₁ and σ₃. When R ≈ 0, σ₂ ≈ σ₃; when R ≈ 1, σ₂ ≈ σ₁. R is one of the four parameters stress inversion can recover (along with the three principal stress orientations). Crucially, stress inversion from focal mechanisms cannot recover absolute stress magnitudes — only orientations and relative magnitudes encoded in R. Focal mechanisms constrain slip directions, not force magnitudes, making absolute magnitudes inaccessible from this data type alone.

Question 3

Stress inversion can recover the absolute magnitudes of the principal stresses (in units of MPa) from focal mechanism data.

• T. True
• F. False

Answer: False

Focal mechanisms record the geometry of fault slip — the direction of slip — not the magnitude of forces involved. Under the Wallace-Bott hypothesis, slip direction is invariant under uniform scaling of all stress components: scaling all three principal stresses by the same factor does not change any predicted slip direction. Therefore, focal mechanisms constrain only stress *orientations* and the *relative* magnitudes encoded in the stress ratio R, not absolute values in Pascals. Absolute stress magnitudes require independent measurements such as borehole breakouts, hydraulic fracturing, or in-situ strain gauges.

Question 4

The Wallace-Bott hypothesis is the foundational assumption of stress inversion: each earthquake's fault slipped in the direction of maximum resolved shear stress on that fault plane.

• T. True
• F. False

Answer: True

This is the central assumption that makes stress inversion tractable. Under the Wallace-Bott hypothesis, the observed slip vector on each fault plane equals the predicted slip vector — the direction the tectonic stress would push a frictionless fault of that orientation. The inversion then minimizes the angular misfit between predicted and observed slip directions across many earthquakes. The assumption breaks down when faults have heterogeneous properties, when earthquake interactions significantly perturb local stress, or when the catalog mixes earthquakes from multiple stress regimes.

Question 5

Why is bootstrap resampling used to assess stress inversion results, and what does a scattered bootstrap distribution indicate?

Model answer: Bootstrap resampling tests solution robustness by repeatedly solving the inversion on different random subsets of the focal mechanism catalog and examining how much the recovered stress parameters vary. If the stress tensor is well-constrained — consistent information across all earthquakes — bootstrap solutions cluster tightly, confirming the result is not driven by a few unusual focal mechanisms. A scattered distribution means the solution is poorly constrained: data may be insufficient, contain large measurement errors, or mix earthquakes from regions with different stress states. In the latter case, systematic spatial patterns in the misfits guide subdivision into distinct stress zones.

Statistical testing is essential because focal mechanisms contain measurement uncertainties and the regional stress assumption is an idealization. Bootstrap resampling provides an honest uncertainty assessment for the stress tensor solution — determining whether results can reliably be used to interpret tectonic loading directions or Coulomb stress transfer calculations.