7. Probability

MTfit has PDF s for two data types, with two different approaches to measuring the polarity.

However, it is possible to add PDF s for other data-types (see Extending MTfit)

7.1. Polarity PDF

Pugh et al, 2016a has a derivation of the polarity PDF used in MTfit. It is given by:

\[\mathrm{p}\left(Y=y\,|\, A,\sigma_{y},\varpi\right)=\frac{1}{2}\left(1+\mathrm{erf}\left(\frac{yA}{\sqrt{2}\sigma_{y}}\right)\right)\left(1-\varpi\right)+\frac{1}{2}\left(1+\mathrm{erf}\left(\frac{-yA}{\sqrt{2}\sigma_{y}}\right)\right)\varpi\]

The different symbols in this equation are:

Symbol	Meaning
\(Y\)	Polarity
\(A\)	Amplitude
\(\sigma_{y}\)	Uncertainty
\(\varpi\)	Mispick Probability

and \(\mathrm{erf}\left(x\right)\) is the error function, given by \(\mathrm{erf}\left(x\right)=\frac{2}{\sqrt{\pi}}\int_{0}^{x}\mathrm{e}^{-t^{2}}\mathrm{d}t\).

This approach requires an estimate of the uncertainty \(\sigma_{y}\). This is not the noise at the arrival, since it does not scale correctly in comparison to the modelled amplitude due to the propagation effects. It could be estimated from the fractional amplitude uncertainty, but this will be greater than or equal to the true value, because the amplitude at a receiver is only ever less than or equal to the maximum theoretical amplitude (accounting for propagation effects). Consequently, this would most likely overestimate the uncertainty. It is clear that the uncertainty value should be station-specific as noise environments at different stations often vary, so the maximum estimate of the event signal-to-noise ratio (SNR) fails to account for the variation across the receivers.

The difficulty in estimating \(\sigma_{y}\) is increased further when polarity picking is done manually, so the uncertainty on the trace is perhaps not even known. Due to the difficulty in quantifying the uncertainty, it is best left as a user-defined parameter that reflects the confidence in the arrival polarity pick, which can be mapped to the pick quality. However, Pugh et al, 2016b proposes an alternate method for calculating polarity uncertainties that can be included in this framework (see Polarity Probability PDF)

7.3. Polarity Probability PDF

Pugh et al, 2016b introduces an alternate method for estimating the polarity, using an automated Bayesian probability estimate. This approach results in estimates of the postive and negative polarity probabilities. autopol provides a Python module for calculating these values (Pugh, 2016a), and may be available on request. These observations can be included in MTfit, although the data independence must be preserved. The PDF is:

\[\mathrm{p}\left(\psi|A,\sigma,\tau,\sigma_{\tau},\varpi\right)=1-\varpi+\left(2\varpi-1\right)\left[\mathrm{H}\left(A\right)+\psi-2\mathrm{H}\left(A\right)\varpi\right]\]

The different symbols in this equation are:

Symbol	Meaning
\(\psi\)	Polarity Probability
\(A\)	Amplitude
\(\sigma\)	Trace Noise
\(\tau\)	Pick Time
\(\sigma_{\tau}\)	Pick Time Noise
\(\varpi\)	Mispick Probability

and \(\mathrm{H}\left(x\right)\) is the Heaviside step function, given by \(\mathrm{H}\left(x\right)=\int_{-\infty}^{x}\delta\left(s\right)\mathrm{d}s\).

7.5. Amplitude Ratio PDF

The amplitude ratio PDF used in MTfit is based on the ratio PDF for two gaussian distributed variables (Hinkley, 1969):

\[\begin{split}P\left(r\right)=\frac{b\left(r\right)d\left(r\right)}{\sigma_{x}\sigma_{y}a^{3}\left(r\right)\sqrt{2\pi}}\left[\Phi\left(\frac{b\left(r\right)}{a\left(r\right)\sqrt{1-\rho^{2}}}\right)-\Phi\left(\frac{-b\left(r\right)}{a\left(r\right)\sqrt{1-\rho^{2}}}\right)\right]\\ +\frac{\sqrt{1-\rho^{2}}}{\pi\sigma_{x}\sigma_{y}a^{2}\left(r\right)}e^{\left(-\frac{c}{2\left(1-\rho^{2}\right)}\right)}\end{split}\]

With coefficients \(a\left(r\right)\), \(b\left(r\right)\), \(c\) and \(d\left(r\right)\) given by :

\[\begin{split}a\left(r\right) = \sqrt{\frac{r^{2}}{\sigma_{x}^{2}}-2\rho\frac{r}{\sigma_{x}\sigma_{y}}+\frac{1}{\sigma_{y}^{2}}}\\ b\left(r\right) = \frac{\mu_{x}r}{\sigma_{x}^{2}}-\rho\frac{\mu_{x}+\mu_{y}r}{\sigma_{x}\sigma_{y}}+\frac{\mu_{y}}{\sigma_{y}^{2}}\\ c = \frac{\mu_{x}^{2}}{\sigma_{x}^{2}}-2\rho\frac{\mu_{x}\mu_{y}}{\sigma_{x}\sigma_{y}}+\frac{\mu_{y}^{2}}{\sigma_{y}^{2}}\\ d\left(r\right) = e^{\left(\frac{b^{2}\left(r\right)-ca^{2}\left(r\right)}{2\left(1-\rho^{2}\right)a^{2}\left(r\right)}\right)}\\\end{split}\]

The resultant PDF is (unsigned amplitude ratios):

\[P\left(R=r\,|\, A_{x},A_{y},\sigma_{x},\sigma_{y}\right)=\mathcal{R_{N}}\left(r,A_{x},A_{y},\sigma_{x},\sigma_{y}\right)+\mathcal{R_{N}}\left(-r,A_{x},A_{y},\sigma_{x},\sigma_{y}\right)\]

With \(\mathcal{R_{N}}\left(r,\mu_{x},\mu_{y},\sigma_{x},\sigma_{y}\right)\) referring to the ratio PDF above, since \(\rho\), the correlation between the variables, is zero.

The different symbols in this equation are:

Symbol	Meaning
\(R\)	Amplitude Ratio
\(A\)	Amplitude
\(\sigma\)	Amplitude Noise