Probability Model and Symbol Mapping

This page defines the SPIDER probabilistic model in symbols, then maps each symbol to configuration keys.

Equations intentionally use symbols only (no config key names inside math).

Quick equation summary

Use this block as a one-screen reference for the full model:

\[ r_n = d_n - \hat d_n, \qquad u_n = \frac{r_n}{\sigma_n} \]

\[ \mathcal{L}_{\text{ind}} = \frac{1}{B}\sum_{n=1}^{B}\Big(\rho(u_n) + \log \sigma_n\Big) \]

where \(\rho(\cdot)\) is selected from Gaussian, Laplace, Student-\(t\), or Huber.

\[ \mathcal{L}_{\text{corr}} = \frac{1}{B}\left[\frac12\sum_g \mathbf{r}_g^\top \boldsymbol{\Sigma}_g^{-1}\mathbf{r}_g\right] + \frac{1}{B}\sum_{n=1}^{B}\log \sigma_n \]

\[ \mathcal{J} = \mathcal{L}_{\text{data}} + \frac{1}{N}\big(-\log p(\Delta \mathbf{Z})\big), \qquad \mathcal{L}_{\text{data}}\in\{\mathcal{L}_{\text{ind}},\mathcal{L}_{\text{corr}}\} \]

Operational phase policy:

• Phase 1 (MAP warmup) uses robust independent likelihoods to damp outlier influence and support outlier identification.

• Phases 2-4 (sampling) use the correlated Gaussian likelihood with shared-event structure.

Quick-summary notation:

• \(d_n\): observed differential time for datum \(n\)

• \(\hat d_n\): model-predicted differential time for datum \(n\)

• \(r_n\): residual, \(r_n=d_n-\hat d_n\)

• \(u_n\): standardized residual, \(u_n=r_n/\sigma_n\)

• \(\sigma_n\): phase-dependent scale for datum \(n\)

• \(\rho(\cdot)\): per-datum robust penalty (Gaussian, Laplace, Student-\(t\), or Huber form)

• \(B\): number of observations in the current likelihood batch

• \(g\): station-phase group index in the correlated model

• \(\mathbf{r}_g\): residual vector for group \(g\)

• \(\boldsymbol{\Sigma}_g\): group covariance in the correlated model

• \(N\): total number of observations in the full dataset

• \(\Delta \mathbf{Z}\): stacked event perturbations across all events

Notation conventions

The symbols below are used throughout the page:

Symbol	Definition
\(n\)	Observation index (differential-time row)
\(i,j\)	Event indices
\(M\)	Number of events
\(B\)	Number of observations in a batch used by the likelihood term
\(N\)	Total number of observations in the full dataset
\(\mathbf{z}_i\)	Event state for event \(i\) (space + origin-time component)
\(\Delta \mathbf{z}_i\)	Perturbation for event \(i\)
\(\Delta \mathbf{Z}\)	Collection of all event perturbations \(\{\Delta \mathbf{z}_i\}_{i=1}^M\)
\(\mathbf{x}_i\)	Spatial part of event state for event \(i\)
\(t_i\)	Origin-time part of event state for event \(i\)
\(T(\mathbf{x}, s, \varphi)\)	Travel-time surrogate evaluated at event location \(\mathbf{x}\), receiver \(s\), phase \(\varphi\)
\(\sigma_P,\sigma_S\)	Phase-specific residual scales
\(\tau_P,\tau_S\)	Phase-specific shared-event random-effect scales
\(\mathbf{I}\)	Identity matrix of appropriate dimension

1) Forward model and residuals

For each differential-time datum \(n\), let:

• \((i_n, j_n)\) be the event pair

• \(s_n\) be the receiver

• \(\varphi_n \in \{P,S\}\) be the phase

• \(d_n\) be the observed differential time

Write each event state as \(\mathbf{z}_i=[\mathbf{x}_i^\top\; t_i]^\top\), where \(\mathbf{x}_i\) is spatial location and \(t_i\) is origin time.

Event state is represented as:

\[ \mathbf{z}_i = \mathbf{z}_i^{(0)} + \Delta \mathbf{z}_i, \qquad \Delta \mathbf{z}_i = \begin{bmatrix} \Delta x_i & \Delta y_i & \Delta z_i & \Delta t_i \end{bmatrix}^{\!\top} \]

Predicted differential time:

\[ \hat d_n = \Big(T(\mathbf{x}_{j_n}, s_n, \varphi_n) + t_{j_n}\Big) - \Big(T(\mathbf{x}_{i_n}, s_n, \varphi_n) + t_{i_n}\Big) \]

Residual:

\[ r_n = d_n - \hat d_n \]

Phase-dependent scale:

\[\begin{split} \sigma_n = \begin{cases} \sigma_P, & \varphi_n = P \\ \sigma_S, & \varphi_n = S \end{cases} \end{split}\]

2) Independent residual likelihood family (Phase 1 robust path)

Define standardized residual \(u_n = r_n / \sigma_n\).
The per-observation negative log-likelihood is:

Gaussian

\[ \ell_n^{\text{Gauss}} = \tfrac12 u_n^2 + \log \sigma_n \]

Laplace

\[ \ell_n^{\text{Lap}} = |u_n| + \log \sigma_n \]

Student-\(t\) (fixed degrees of freedom \(\nu\))

\[ \ell_n^{t} = \tfrac{\nu+1}{2}\log\!\left(1+\frac{u_n^2}{\nu}\right) + C(\nu) + \log \sigma_n \]

where \(C(\nu)\) is the Student-\(t\) normalization constant (depends only on \(\nu\)).

Huber (threshold \(\delta_H\))

\[ \ell_n^{\text{Huber}} = h_{\delta_H}(u_n) + \log \sigma_n \]

with

\[\begin{split} h_{\delta_H}(u)= \begin{cases} \tfrac12 u^2, & |u| \le \delta_H \\ \delta_H\left(|u|-\tfrac12\delta_H\right), & |u|>\delta_H \end{cases} \end{split}\]

Batch-average independent likelihood term:

\[ \mathcal{L}_{\text{ind}} = \frac{1}{B}\sum_{n=1}^{B}\ell_n \]

3) Correlated shared-event likelihood (Phases 2-4 sampling path)

For each station-phase group \(g\), let \(\mathbf{r}_g\) be grouped residuals and \(\mathbf{b}_g\) latent event effects:

\[ \mathbf{r}_g = \mathbf{B}_g \mathbf{b}_g + \boldsymbol{\varepsilon}_g \]

Here \(\mathbf{B}_g\) is the signed incidence operator mapping event-level latent terms to edge-level residual contributions within group \(g\).

\[ \mathbf{b}_g \sim \mathcal{N}(\mathbf{0}, \tau_g^2 \mathbf{I}), \qquad \boldsymbol{\varepsilon}_g \sim \mathcal{N}(\mathbf{0}, \sigma_g^2 \mathbf{I}) \]

With optional edge weighting, using weighted incidence \(\widetilde{\mathbf{B}}_g\):

\[ \boldsymbol{\Sigma}_g = \sigma_g^2 \mathbf{I} + \tau_g^2 \widetilde{\mathbf{B}}_g \widetilde{\mathbf{B}}_g^{\!\top} \]

Current collapsed quadratic term:

\[ \mathcal{Q}_{\text{corr}} = \frac12\sum_g \mathbf{r}_g^{\!\top}\boldsymbol{\Sigma}_g^{-1}\mathbf{r}_g \]

Implemented sampling loss contribution:

\[ \mathcal{L}_{\text{corr}} = \frac{1}{B}\mathcal{Q}_{\text{corr}} + \frac{1}{B}\sum_{n=1}^{B}\log \sigma_n \]

Note: this path currently uses the correlated quadratic form (whitening/PCG solve) and does not include a correlated log-determinant term.

4) Priors

Event perturbation prior (default diagonal Gaussian)

\[ \Delta \mathbf{z}_i \sim \mathcal{N}(\mathbf{0}, \mathbf{S}), \qquad \mathbf{S}=\operatorname{diag}(s_x^2,s_y^2,s_z^2,s_t^2) \]

Optional centroid prior

\[ \bar{\Delta \mathbf{z}} = \frac{1}{M}\sum_{i=1}^{M}\Delta \mathbf{z}_i, \qquad \bar{\Delta \mathbf{z}} \sim \mathcal{N}(\mathbf{0}, \mathbf{C}) \]

with

\[ \mathbf{C}=\operatorname{diag}(c_x^2,c_y^2,c_z^2,c_t^2) \]

Optional hierarchical event precision prior

\[ \Delta \mathbf{z}_i \mid \mathbf{\Lambda}_{k(i)} \sim \mathcal{N}\!\left(\mathbf{0}, \mathbf{\Lambda}_{k(i)}^{-1}\right) \]

\[ \mathbf{\Lambda}_{k} \sim \operatorname{Wishart}(\nu_0,\mathbf{V}_0) \]

where \(k(i)\) maps event \(i\) to its cluster index, \(\mathbf{V}_0\) is constructed from scale hyperparameters, and \(\nu_0\) is the Wishart degrees of freedom.

5) Training objective (negative log posterior)

SPIDER uses a per-observation normalized objective:

\[ \mathcal{J} = \mathcal{L}_{\text{data}} + \frac{1}{N}\big(-\log p(\Delta \mathbf{Z})\big) \]

where:

• \(\mathcal{L}_{\text{data}} = \mathcal{L}_{\text{ind}}\) for independent-likelihood runs

• \(\mathcal{L}_{\text{data}} = \mathcal{L}_{\text{corr}}\) when collapsed shared-event correlation is enabled

• \(N\) is the total number of observations in the full dataset

6) Symbol-to-config mapping

Likelihood symbols

Symbol	Meaning	Config key(s)
\(\sigma_P, \sigma_S\)	Phase-dependent residual scales	model.likelihoods.locate_map.phase_unc, model.likelihoods.sample.phase_unc
\(\nu\)	Student-\(t\) degrees of freedom	model.likelihoods.locate_map.student_t.nu, model.likelihoods.sample.student_t.nu
\(\delta_H\)	Huber threshold	model.likelihoods.locate_map.huber_delta, model.likelihoods.sample.huber_delta
Likelihood family selector	Choice among Gaussian/Laplace/Student-\(t\)/Huber	model.likelihoods.locate_map.type
Correlated sampling likelihood selector	Enables correlated Gaussian sampling path	model.likelihoods.sample.type

Shared-event correlated symbols

Symbol	Meaning	Config key(s)
\(\tau_P, \tau_S\)	Phase-specific latent RE scales	model.likelihoods.sample.shared_event_re.model.tau_s
Group definition	Grouping strategy for correlated solve	model.likelihoods.sample.shared_event_re.model.group_by
Edge-weight model (inside \(\widetilde{\mathbf{B}}_g\))	Distance-based weighting mode	model.likelihoods.sample.shared_event_re.edge_weights.mode
Weight length scale	RBF weight scale	model.likelihoods.sample.shared_event_re.edge_weights.ell_km
Weight power/scale	Power-law weight controls	model.likelihoods.sample.shared_event_re.edge_weights.power, model.likelihoods.sample.shared_event_re.edge_weights.scale_km
Global weight multiplier	Global edge-weight factor	model.likelihoods.sample.shared_event_re.edge_weights.global_scale
Numerical jitter	Stabilization added to the node system	model.likelihoods.sample.shared_event_re.numerics.jitter0, model.likelihoods.sample.shared_event_re.numerics.jitter_max
Solver tolerance / iterations	PCG stopping controls	model.likelihoods.sample.shared_event_re.solver.tol, model.likelihoods.sample.shared_event_re.solver.max_iters, model.likelihoods.sample.shared_event_re.solver.min_iters

Prior symbols

Symbol	Meaning	Config key(s)
\(s_x,s_y,s_z,s_t\)	Event prior standard deviations	model.priors.event.params.std
\(c_x,c_y,c_z,c_t\)	Centroid prior standard deviations	model.priors.centroid.params.std
\(\nu_0\)	Wishart hyperprior degrees of freedom	model.priors.event.hyper.params.df
\(\mathbf{V}_0\) scale controls	Wishart scale hyperparameters	model.priors.event.hyper.params.scale_std
Hyperprior update cadence	Epoch cadence for precision updates	model.priors.event.hyper.update.every_epochs

7) Which likelihood is active in each stage

• Phase 1 (MAP warmup/outlier-screening stage) uses the locate_map likelihood block.

• In this stage, robust independent residual families (Laplace, Huber, or Student-\(t\)) are used to reduce sensitivity to outliers and help identify problematic residuals.

• Phases 2-4 use the sample likelihood block and are intended to run with correlated Gaussian structure (correlated_gaussian) and shared-event whitening.

• If shared-event correlation is disabled in the sampling block, phases 2-4 fall back to the independent residual form.