Update MT_N_layered_Earth.rst

content/geophysical_surveys/mt/physics.rst

@@ -19,13 +19,13 @@ The natural source fields travel as plane waves and the physics behind these are
             :file: images/Reflection_Transmission.html
    *  - Powered by: `SimPEG <http://simpeg.xyz/>`_
 
-.. todo:: step-by-step from plane source to apparent resistivity (radio button widghet): (1) plane waves, (2) skin depth), (3) phase difference between E and H (tie back to 1D stuff Thibaut already did)
+.. todo:: step-by-step from plane source to apparent resistivity (radio button widget): (1) plane waves, (2) skin depth), (3) phase difference between E and H (tie back to 1D stuff Thibaut already did)
 
 The fields propogate at different frequencies and each frequency provides different information about the subsurface. Low frequencies penetrate deeper while high frequencies provide information near the surface. The plane wave attenuates with depth and a skin depth is the distance at which the amplitude has decreased by :math:`1/e`. This is explained in detail :ref:`on the fundamentals page <MT_skindepthdoi>`. :ref:`The movie below <mt_sd_envelopes>` shows the magnetic field at different frequencies and how it attenuates in the subsurface.
 
 .. _mt_sd_envelopes:
 
-.. list-table:: : Given the same conductivity model, the skindepth decreases as the frequency increases.
+.. list-table:: : Given the same conductivity model, the skin depth decreases as the frequency increases.
    :header-rows: 0
    :widths: 10
    :stub-columns: 0
@@ -65,7 +65,7 @@ and :math:`\mathbf{E} = \left( \begin{matrix} \mathbf{E}_{x}\\ \mathbf{E}_{y} \e
 
 For a halfspace earth, the impedance simplifies to a single component of the matrix: :math:`Z_{xy} = \frac{\mathbf{E}_x}{\mathbf{H}_y}`. From the impedance, we can calculate the apparent resistivity :math:`\rho_a` and the phase :math:`\Phi`:
 
-.. math:: \rho_a = \frac{1}{\omega \mu} \left| Z_{xy} \right| ^2
+.. math:: \rho_a = \frac{1}{\omega \mu_0} \left| Z_{xy} \right| ^2
 
 .. math:: \Phi = \tan^{-1} \left( \frac{Im(Z_{xy})}{Re(Z_{xy})} \right) = -\frac{\pi}{4}
 

content/maxwell3_fdem/natural_sources/MT_N_layered_Earth.rst

@@ -225,10 +225,12 @@ 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^{i 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}}
+    \delta = \sqrt{ \frac{2}{\omega \mu_0 \sigma}} \simeq \frac{500}{\sqrt{\sigma f}}
     :label: Skin Depth
 
 We see the skin depth is highly dependent on both the frequency of our signal and the conductivity of the Earth material. In air , the conductivity is almost 0, so we do not notice important decreased of the electromagnetic wave. In the ground, this is different.
@@ -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:
 
@@ -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}}
@@ -382,7 +384,7 @@ 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::
@@ -392,7 +394,7 @@ acquired usually at the surface. We define an apparent impedance:
 
 Notice this is a complex number, with a norm and an angle.
 
-Impendance matrix
+Impedance matrix
 *****************
 
 We saw that in 1D, the horizontal orthogonal components of the electric and