= Solution
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>:
$$
\mathbf M=\sum_{j=1}^6m_j\mathbf B_j,
\qquad
\mathbf d=\mathbf G(\mathbf q)\mathbf m+\boldsymbol\epsilon,
\qquad \mathbf q=(x_s,y_s,z_s,t_s).
$$
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 is
$$
\chi^2(\mathbf q,\mathbf m)
=[\mathbf d-\mathbf G(\mathbf q)\mathbf m]^T
\mathbf C_d^{-1}[\mathbf d-\mathbf G(\mathbf q)\mathbf m],
$$
with 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. \b[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.
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