Editing Inverse problem (section)

==Conceptual understanding==
Since Newton, scientists have extensively attempted to model the world. In particular, when a [[mathematical model]] is available (for instance, Newton's gravitational law or Coulomb's equation for electrostatics), we can foresee, given some parameters that describe a physical system (such as a distribution of mass or a distribution of electric charges), the behavior of the system. This approach is known as mathematical modeling and the above-mentioned physical parameters are called the '''model parameters''' or simply the '''model'''. To be precise, we introduce the notion of '''state of the physical system''': it is the solution of the mathematical model's equation. In [[Optimal control|optimal control theory]], these equations are referred to as the [[State-space representation|state equations]]. In many situations we are not truly interested in knowing the physical state but just its effects on some objects (for instance, the effects the gravitational field has on a specific planet). Hence we have to introduce another operator, called the '''observation operator''', which converts the state of the physical system (here the predicted gravitational field) into what we want to observe (here the movements of the considered planet). We can now introduce the so-called '''forward problem''', which consists of two steps:
* determination of the state of the system from the physical parameters that describe it
* application of the observation operator to the estimated state of the system so as to predict the behavior of what we want to observe.
This leads to introduce another [[operator (mathematics)|operator]] <math>F</math> (''F'' stands for "forward") which maps model parameters <math>p</math> into <math>F(p)</math>, the data that model <math>p</math> predicts that is the result of this two-step procedure. Operator <math>F</math> is called '''forward operator''' or '''forward map'''.
In this approach we basically attempt at predicting the effects knowing the causes.

The table below shows, the Earth being considered as the physical system and for different physical phenomena, the model parameters that describe the system, the physical quantity that describes the state of the physical system and observations commonly made on the state of the system. 
{| class="wikitable"
|-
! Governing equations !! Model parameters (input of the model)!! State of the physical system !! Common observations on the system
|-
| [[Newton's law of universal gravitation|Newton's law of gravity]] || Distribution of mass || [[Gravitational field]] || Measurement made by [[gravimeter]]s at different surface locations
|-
| [[Maxwell's equations]] || Distribution of [[magnetic susceptibility]] || [[Magnetic field]] ||Magnetic field measured at different surface locations by [[magnetometer]]s (case of a steady state)
|-
| [[Wave equation]] || Distribution of wave-speeds and densities || Wave-field caused by artificial or natural [[seismic source]]s || [[Particle velocity]] measured by seismometers placed at different surface locations
|-
| [[Diffusion equation]] || Distribution of [[Mass diffusivity|Diffusion coefficient]] || Diffusing material concentration as a function of space and time || Monitoring of this concentration measured at different locations
|}
In the inverse problem approach we, roughly speaking, try to know the causes given the effects.