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Update skindepth.rst #563

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Original file line number Diff line number Diff line change
Expand Up @@ -18,7 +18,7 @@ Attenuation defines the rate of amplitude loss an EM wave experiences at it prop
.. _harmonic_planewaves_homogeneous_attenuation_formula:

.. math::
A(z) = A_0 e^{\beta z}
A(z) = A_0 e^{-\beta z}
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must be minus in order to attenuate the amplitude.

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If it is positive, the "z" values must be negative. I think we can clarify this in some sentences. For example "z subsurface is between zero and -infinity". That is because, in several books, the "z" is positive, thus, the beta value in the amplitude equation must be with a minus.


where absolute :math:`A` is the amplitude, :math:`A_0` is the absolute amplitude at :math:`z` = 0 m and:

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16 changes: 9 additions & 7 deletions content/maxwell3_fdem/natural_sources/MT_N_layered_Earth.rst
Original file line number Diff line number Diff line change
Expand Up @@ -225,7 +225,9 @@ The skin depth :math:`\delta` is defined as the depth where the signal has
decayed to a factor :math:`\frac{1}{e}(\simeq` 36%).

.. math::
e^{-i Im(k) \delta} = \frac{1}{e}
e^{-Im(k) \delta} = \frac{1}{e}

Assuming the Earth is non-magnetic (:math:`\mu \sim \mu_0 = 4\pi \times 10^{-7}` H/m):

.. math::
\delta = \sqrt{ \frac{2}{\omega \mu \sigma}} \simeq \frac{500}{\sqrt{\sigma f}}
Expand All @@ -243,7 +245,7 @@ We see the skin depth is highly dependent on both the frequency of our signal an
In :numref:`SkinDepth_MT` and in the movie, we can see that even at very high
frequency (20000 Hz), MT is still a deep exploration method in resistive
environment (:math:`10^{-5} S/m`) with a skin depth of about 1125m. Skin Depth
is often use as an estimator for the depth of investigation of a survey.
is often used as an estimator for the depth of investigation of a survey.

.. _MT_refl_transcoeff:

Expand Down Expand Up @@ -292,7 +294,7 @@ negligible, we also obtain from equation :eq:`Continuity of H` :
k_j E^i - k_j E^r = k_{j+1} E^t
:label: faraday continuity condition

Replacing the differents components of equation :eq:`faraday continuity condition` with equation :eq:`Continuity of E`, we obtain the reflection coefficient R and the transmission coefficient T:
Replacing the different components of equation :eq:`faraday continuity condition` with equation :eq:`Continuity of E`, we obtain the reflection coefficient R and the transmission coefficient T:

.. math::
R = \frac{E^r}{E^i} = \frac{k_j - k_{j+1}}{k_j + k_{j+1}}
Expand Down Expand Up @@ -382,17 +384,17 @@ Field Acquisition
-----------------

In MT, the source is unknown but we are avoiding the problem by measuring the
ratio of the fields, which cancel the amplitude of the source. The data are
ratio of the fields, which cancels the amplitude of the source. The data are
acquired usually at the surface. We define an apparent impedance:

.. math::
\hat{Z}_{xy} = \frac{E_x}{H_y}
:label: Apparent Impedance Definition


Notice this is a complex number, with a norm and an angle.
Notice this is a complex number, with a norm and an argument (angle).

Impendance matrix
Impedance matrix
*****************

We saw that in 1D, the horizontal orthogonal components of the electric and
Expand All @@ -417,7 +419,7 @@ The matrix linking the component of :math:`\mathbf{E}` and

On field, we do not know a priori the orientation of the source wave. This
orientation can also changes over times if the source wave is polarised. We
usally record both horizontals components of each field. If the Earth is
usually record both horizontals components of each field. If the Earth is
purely 1D, a simple rotation of the matrix would allow to find the
antisymetric matrix and thus obtain the apparent impedance
:math:`\hat{Z}_{xy}`.
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