Merge pull request #474 from geoscixyz/PhysPropTitlePage

Organized title page and stated things more explicitly

content/physical_properties/index.rst

@@ -5,53 +5,67 @@ Physical Properties
 
 .. purpose::
 
-    Generate a resource that has a commpilation of values for physical
+    To generate a resource that has a compilation of values for physical
     properties as a function of rock type, mineral composition and saturating
     fluids. Background about how the laboratory measurements are made and the
     connection with constitutive relationships are presented. Each section
     will begin with examples for where the particular physical property has
     been diagnostic.
 
-
-
-In geophysics, we characterize materials by their physical properties. The
-relevant  properties for electromagnetics are:
+A physical property quantifies how a rock responds to a particular physical input. In electromagnetic geophysics, the relevant physical properties are:
 
 - :ref:`electrical_conductivity_index`: :math:`\sigma` (or its reciprocal, resistivity, :math:`\rho`)
 - :ref:`magnetic_permeability_index`, :math:`\mu`
 - :ref:`dielectric_permittivity_index`, :math:`\varepsilon`
 
-A physical property quantifies how a rock responds to a particular input and
-it therefore connects a forcing field to a resulting flux. Typically the
-laboratory experiments are carried out in the frequency domain and the
-resultant distributive relationships are given by:
+Electromagnetic physical properties are defined using a set of constitutive relationships. Each constitutive relationship characterizes the resulting flux due to a causative forcing field. In most cases, laboratory measurements are performed in the frequency domain, but they can be performed in the time domain. If the physical properties are non-dispersive, the relationship between the field and the corresponding flux is independent of frequency. Thus the constitutive relationships are given by:
 
 - :math:`\mathbf{J}(\omega)= \sigma \mathbf{E}(\omega)`  (Ohm's Law)
 - :math:`\mathbf{B}(\omega)= \mu \mathbf{H}(\omega)`
 - :math:`\mathbf{D}(\omega)= \varepsilon \mathbf{E}(\omega)`
 
-Physical properties can be tensors, for example :math:`J=\Sigma E` where
-:math:`\Sigma` is a 3x3 tensor. Or the properties can be dispersive, that is,
-frequency dependent. If expressions of the constitutive relations in time are
-needed then application of the Fourier transform yields
+By taking the inverse Fourier transform, the corresponding time domain relationships are given by:
+
+- :math:`\mathbf{j}(t)= \sigma \mathbf{e}(t)`  (Ohm's Law)
+- :math:`\mathbf{b}(t)= \mu \mathbf{g}(t)`
+- :math:`\mathbf{d}(t)= \varepsilon \mathbf{e}(t)`
+
+**Dispersion**
+
+Dispersion implies that a given physical property is frequency-dependent. Electrical conductivity, magnetic permeability and dielectric permittivity all exhibit dispersive properties under various conditions. For dispersive physical properties, the constitutive relationships are given by:
+
+- :math:`\mathbf{J}(\omega)= \sigma (\omega) \mathbf{E}(\omega)`  (Ohm's Law)
+- :math:`\mathbf{B}(\omega)= \mu (\omega) \mathbf{H}(\omega)`
+- :math:`\mathbf{D}(\omega)= \varepsilon (\omega) \mathbf{E}(\omega)`
+
+Once again, the time domain equivalent for the constitutive relationships can be obtained by taking the inverse Fourier transform. If the physical properties are dispersive, the relationship between each field and its corresponding flux becomes a convolution:
 
 - :math:`\mathbf{j}(t)=\sigma(t) \ast \mathbf{e}(t)`
 - :math:`\mathbf{b}(t)=\mu(t) \ast \mathbf{h}(t)`
 - :math:`\mathbf{d}(t)=\varepsilon(t) \ast \mathbf{e}(t)`
 
-The constitutive relations, along with :ref:`Maxwell's equations <quick_guide_maxwell>`,  form a complete set of equations for electromagnetics.
+**Anisotropy**
 
-In this section, we present basic material regarding the various physical properties. For each property, we provide:
+Anisotropy means that the physical response of a rock to an applied field is not the same in all directions. In this case, each field and is corresponding flux is related by a 3X3 tensor:
 
-- Examples to applications
-- Constitutive relationship and laboratory experiment
-- Useful tables
-- Additional information
-- References
+- :math:`\mathbf{J}= \Sigma \mathbf{E}`
+- :math:`\mathbf{B}= \Gamma \mathbf{H}`
+- :math:`\mathbf{D}= \Upsilon \mathbf{E}`
 
+where
 
+.. math::
+    \Sigma = \begin{bmatrix} \sigma_{xx} & \sigma_{xy} & \sigma_{xz} \\ \sigma_{yx} & \sigma_{yy} & \sigma_{yz} \\ \sigma_{zx} & \sigma_{zy} & \sigma_{zz} \end{bmatrix} , \; \;
+    \Gamma = \begin{bmatrix} \mu_{xx} & \mu_{xy} & \mu_{xz} \\ \mu_{yx} & \mu_{yy} & \mu_{yz} \\ \mu_{zx} & \mu_{zy} & \mu_{zz} \end{bmatrix} \; \; \textrm{and} \; \;
+    \Upsilon = \begin{bmatrix} \varepsilon_{xx} & \varepsilon_{xy} & \varepsilon_{xz} \\ \varepsilon_{yx} & \varepsilon_{yy} & \varepsilon_{yz} \\ \varepsilon_{zx} & \varepsilon_{zy} & \varepsilon_{zz} \end{bmatrix} , \; \;
 
+The constitutive relations, along with :ref:`Maxwell's equations <quick_guide_maxwell>`,  form a complete set of equations for electromagnetics. In this section, we present basic material regarding the various physical properties. For each property, we provide:
 
+- the constitutive relationship, a physical descriptions and its relevance to electromagnetic geoscience
+- a description of common laboratory measurement techniques
+- tables containing a typical range of values
+- a list of factor which control the physical property value
+- references
 
 **Contents:**