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Improve the theoretical background section of the documentation
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claudiodsf committed Oct 16, 2024
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16 changes: 8 additions & 8 deletions docs/theoretical_background.rst
Original file line number Diff line number Diff line change
Expand Up @@ -188,13 +188,8 @@ coefficient and convert it to seismic moment units:
M^{p|s}_x(f) \equiv
\mathcal{G}(r) \times
\frac{4 \pi \rho_h^{1/2} \rho_r^{1/2} c_h^{5/2} c_r^{1/2}}
{2 R_{\Theta\Phi}}
\times S^{p|s}_x(f) =
M_O \times
\frac{1}{1+\left(\frac{f}{f^{p|s}_c}\right)^2}
\times
e^{- \pi f t^*}
{F R_{\Theta\Phi}}
\times S^{p|s}_x(f)
Then the spectrum is converted in units of magnitude:

Expand All @@ -216,11 +211,16 @@ three components (e.g., Z, N, E) through the root sum of squares:
\left( Y^{p|s}_e(f) \right)^2
}
The data vector is compared to the theoretical model:
The data vector is compared to the theoretical model :math:`M^{p|s}_{theo}(f)`
which incorporates the Brune's spectrum of the seismic moment and the inelastic
attenuation. This model is also converted in magnitude units:

.. math::
Y^{p|s}(f) =
\frac{2}{3}
\left[ \log_{10} \left( M^{p|s}_{theo}(f) \right) - 9.1 \right]
=
\frac{2}{3}
\left[ \log_{10} \left(
M_O \times
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