Past exam of the physics of the Earth as a Planet course of the University of Cambridge / 2015 / Paper 1 / 1 / c

For seismic source inversion, collect calibrated, consistently filtered samples from all available stations and components into a data vector $\mathbf{d}$. For each trial source position and origin time, compute the corresponding elastodynamic Green tensor in an Earth model, including the instrument response used for the data. Write the symmetric seismic moment tensor in six independent symmetric basis tensors:Each column of $\mathbf{G}$ is the synthetic seismogram for one basis tensor. An off-diagonal basis tensor contains both symmetric entries, so the double contraction and its factors of two remain consistent.

A suitable objective iswith noise covariance ${\mathbf{C}}_d$; a justified penalty can be added for regularization. At fixed $\mathbf{q}$, solve for the six entries of $\mathbf{m}$ by weighted least squares, preferably using a QR decomposition or singular value decomposition rather than explicitly inverting the matrix of the normal equations. Then search over the four nonlinear parameters using a coarse location/time grid followed by local refinement. This variable projection reduces a ten-parameter search to a four-parameter nonlinear search with an inner six-parameter linear fit. Merely having more than ten data samples does not ensure identifiability: the design matrix must have adequate rank and the network must resolve the source's radiation patterns.

Potential errors include inaccurate velocity, density, anisotropy and attenuation in the assumed Earth model; neglected finite rupture and source-time effects; timing, instrument calibration or component-orientation errors; correlated noise; uneven station coverage; and tradeoffs between depth, origin time and the moment tensor. Imposing a double-couple seismic source mechanism without evidence can hide genuine non-double-couple components. Regularization also changes the recovered source and should be reported.

Inspect residuals, singular values, recovery of synthetic sources and uncertainty under resampling or noise perturbations. Linearized covariance or a posterior distribution can quantify uncertainty within the assumed forward model, but does not automatically account for errors in that model. A good waveform fit alone is insufficient evidence that all ten parameters are uniquely recovered.