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"source": [
"What is inversion?"
]
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"For those with little or no background or experience, the concept of inversion can be intimidating. Many available resources assume a certain amount of background already, and for the uninitiated, this can make the topic difficult to grasp. Likewise, if material is too general and simplistic, it is of little use to someone who has some understanding but is looking for deeper insight. This suite of tutorials will begin in earnest from the first module, but if you have no background, then this is the place to start. I will assume that you have had some basic linear algebra and matrix theory, and little else. I will also assume, at least at first, that you may have forgotten a few of these things along the way. So let's begin from zero, with the absolute simplest terms:
\n",
"\n",
"In algebra, when we want to solve an equation for an unknown variable $x$, say in the equation:
\n",
"\n",
"\\begin{equation}\n",
"ax=b \n",
"\\end{equation}
\n",
"where $a$ and $b$ are real numbers, we can do so, quite simply, by multiplying both sides of the equation by the *inverse* of $a$, that is $a^{-1}$. Our result $x = b/a$, is, in a very simple way, the solution to an inverse problem. And even though this example is trivial, inversion is simply an expansion of this process to larger systems of equations with more unknowns. Expanding ever so slightly, we do an analogous process by obtaining the inverse of a matrix. For example, say we have a system of equations that we represent in a square matrix $A$, a vector of unknowns $x$, and a vector of known values $b$. What we constuct is the matrix equation
\n",
"\n",
"\\begin{equation}\n",
"Ax=b \n",
"\\end{equation}
\n",
"\n",
"and provided an inverse for $A$ exists (that is, if $det(A) \\neq 0$, or if you prefer $A^{-1}A = I$), the solution can be found by multiplying both sides by the inverse of $A$\n",
"\n",
"\\begin{equation}\n",
"A^{-1}Ax= Ix = x = A^{-1}b \\\\\n",
"x = A^{-1}b\n",
"\\end{equation}
\n",
"\n",
"If you are the type of person that likes to see things through simple examples, consider the following. Say we have a set of two equations and two unknowns:\n",
"\n",
"\\begin{equation}\n",
"2x +0y = 4 \\\\\n",
"0x +3y = 6 \\\\\n",
"\\end{equation}
\n",
"\n",
"Now, because this is a trivial example, we can see from inspection that $x=2$ and $y=2$. But for the purpose of seeing how things are put together, let's solve this using matrices anyway. We can assemble the coefficients for each equation into the matrix equation:\n",
"\n",
"\\begin{equation}\n",
"\\begin{bmatrix}\n",
" 2 &0\\\\\n",
" 0 &3\\\\ \n",
"\\end{bmatrix}\n",
"\\left[\n",
"\\begin{array}{c}\n",
"x\\\\\n",
"y\n",
"\\end{array}\n",
"\\right]\n",
"=\n",
"\\left[\n",
"\\begin{array}{c}\n",
"4\\\\\n",
"6\n",
"\\end{array}\n",
"\\right]\n",
"\\end{equation}
\n",
"\n",
"This is an equation of the form $Ax=b$, and we can solve this by multiplying both sides of the equation by $A^{-1}$:\n",
"\n",
"\\begin{equation}\n",
"\\begin{bmatrix}\n",
" \\frac{1}{2} &0\\\\\n",
" 0 &\\frac{1}{3}\\\\ \n",
"\\end{bmatrix}\n",
"\\begin{bmatrix}\n",
" 2 &0\\\\\n",
" 0 &3\\\\ \n",
"\\end{bmatrix}\n",
"\\left[\n",
"\\begin{array}{c}\n",
"x\\\\\n",
"y\n",
"\\end{array}\n",
"\\right]\n",
"=\n",
"\\begin{bmatrix}\n",
" \\frac{1}{2} &0\\\\\n",
" 0 &\\frac{1}{3}\\\\ \n",
"\\end{bmatrix}\n",
"\\left[\n",
"\\begin{array}{c}\n",
"4\\\\\n",
"6\n",
"\\end{array}\n",
"\\right]\n",
"\\end{equation}
\n",
"\n",
"Multiplying this out on both sides gives us (unsurprisingly):
\n",
"\n",
"\\begin{equation}\n",
"\\left[\n",
"\\begin{array}{c}\n",
"x\\\\\n",
"y\n",
"\\end{array}\n",
"\\right]\n",
"=\n",
"\\left[\n",
"\\begin{array}{c}\n",
"2\\\\\n",
"2\n",
"\\end{array}\n",
"\\right] \n",
"\\end{equation}
\n",
"\n",
"Again, this is a trivial example. Now we will build up the notation a little. Thus far I have used the standard matrix-vector equation ($Ax=b$) that you have likely seen in a basic linear algebra textbook. In this set of tutorials I will use slightly different notation, and our matrix equation will take the form $Gm=d$. Of course, the choice of symbols for matrices and vectors is arbitrary, but in the context that we are going to proceed, $G$, $m$, and $d$ will take on new meaning. We represent the right hand side of the equation by $d$, which is the *data* that we measure in the field, $m$ which represents our *model parameters* (the unknowns that we seek), and $G$, which is our *forward operator* that operates on the model to produce the data. Generally, the governing equations that go into the physical system are represented by our matrix $G$. And even more generally, our matrix $G$ is a special case of a more general idea of a forward operator, $F[m]$. For a geophysical survey the a broader representation of our problem, for the ith data point would be\n",
"\n",
"\\begin{equation}\n",
"F_i[m] = d_i + n_i\n",
"\\end{equation}\n",
"\n",
"where $F_i[m]$ is the forward operator, $d_i $ is the measured data and $n_i$ is additive noise. This distinction is important as it adds a degree of richness to our scenario. Here are a few points to consider:\n",
"\n",
"