From c3c2fb3dc19b5eb7ebed2f3ec99d7119af21a3d3 Mon Sep 17 00:00:00 2001 From: PAULGOYES Date: Fri, 9 Oct 2020 21:18:29 -0500 Subject: [PATCH 1/3] Update skindepth.rst add minus to beta in A(z) --- .../harmonic_planewaves_homogeneous/skindepth.rst | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/content/maxwell1_fundamentals/harmonic_planewaves_homogeneous/skindepth.rst b/content/maxwell1_fundamentals/harmonic_planewaves_homogeneous/skindepth.rst index 496fb46e..734ee960 100644 --- a/content/maxwell1_fundamentals/harmonic_planewaves_homogeneous/skindepth.rst +++ b/content/maxwell1_fundamentals/harmonic_planewaves_homogeneous/skindepth.rst @@ -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} where absolute :math:`A` is the amplitude, :math:`A_0` is the absolute amplitude at :math:`z` = 0 m and: From f39879dbdc61aaad417871a4c34287174f54abfe Mon Sep 17 00:00:00 2001 From: PAULGOYES Date: Tue, 13 Oct 2020 18:46:47 -0500 Subject: [PATCH 2/3] Update MT_N_layered_Earth.rst update equation 422 --- content/maxwell3_fdem/natural_sources/MT_N_layered_Earth.rst | 4 +++- 1 file changed, 3 insertions(+), 1 deletion(-) diff --git a/content/maxwell3_fdem/natural_sources/MT_N_layered_Earth.rst b/content/maxwell3_fdem/natural_sources/MT_N_layered_Earth.rst index 54d6d7d8..874baaa8 100644 --- a/content/maxwell3_fdem/natural_sources/MT_N_layered_Earth.rst +++ b/content/maxwell3_fdem/natural_sources/MT_N_layered_Earth.rst @@ -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}} From bcb04f06d58d9e0765915dbfc190ffbc269f51f7 Mon Sep 17 00:00:00 2001 From: PAULGOYES Date: Tue, 13 Oct 2020 19:03:19 -0500 Subject: [PATCH 3/3] Update MT_N_layered_Earth.rst update de eq 144 and some typo changes --- .../natural_sources/MT_N_layered_Earth.rst | 12 ++++++------ 1 file changed, 6 insertions(+), 6 deletions(-) diff --git a/content/maxwell3_fdem/natural_sources/MT_N_layered_Earth.rst b/content/maxwell3_fdem/natural_sources/MT_N_layered_Earth.rst index 874baaa8..1d9d630d 100644 --- a/content/maxwell3_fdem/natural_sources/MT_N_layered_Earth.rst +++ b/content/maxwell3_fdem/natural_sources/MT_N_layered_Earth.rst @@ -245,7 +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: @@ -294,7 +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}} @@ -384,7 +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:: @@ -392,9 +392,9 @@ acquired usually at the surface. We define an apparent impedance: :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 @@ -419,7 +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}`.