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"source": [
"Module 1: Setting up the problem"
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"source": [
"Introduction"
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"Geophysical surveys consist of a similar basic framework. An energy source is delivered into the earth, which can be natural (for example, the Earth's magnetic field) or human-made (current in the ground, acoustic wave energy, etc.), and this stimulates a response according to the variation in physical properties in the subsurface. At the surface, receivers pick up the signal and record this as data.
\n",
"\n",
"\n",
"
\n",
"\n",
"The goal of inversion is to find a model of the physical property distribution in the earth that produced the data. This is a difficult process because (1) information about a physical property for each datum is encoded in a complex way, and (2) we have a finite amount of data and cannot represent the physical property distribution everywhere.
\n",
"\n",
"Inversion is a multistep process, often represented as a workflow. \n",
"\n",
"
\n",
"\n",
"The goal of this module is to cover the first section of the workflow, which discretizes the data and places the values of the functions onto a mesh. This will be done using the following steps:
\n",
"\n",
"(1) Start with an expression that relates a kernel function with the continuous distribution of a physical property.
\n",
"(2) Discretize this expression, and introduce a simple example problem to illustrate the mathematics in detail.
\n",
"(3) Define a mesh that organizes our information.
\n",
"(4) Build up the matrix equation $d = Gm$.
\n",
"(5) Generalize the form of the problem from the example.
\n",
"(6) Implement the example problem in Python as a forward problem.\n",
"\n",
"But first, here are some fundamental definitions:
\n",
"\n",
"The general mathematical description of the inverse problem can be written as follows:
\n",
"\n",
"\\begin{equation} F_i[m]= d_i +n_i \\quad \\text{for} \\quad i=1,..,N \\; \\text{where}\\end{equation}
\n",
" \n",
"\n",
"