Geophysical Problem is the one among key issues that either geophysicist or geophysical team should address by providing the real solution so as to achieve the overall objective of a given geophysical project. It must define what should be solved! Since it is very challenging to solve something you have not identified (defined) and this is why most geophysical research projects should get started up with defining the geophysical problem.
It is the key issue as it carries the overall objective of the geophysical project. Some problems are more challenging than others depending on objective and the complexity of the geological terrain. And the sad news is that some geophysical problems are ambiguous meaning that they produce more than one solution, this may cause difficulty to produce one decisive pros statement without consulting other project data sources.
Geophysical problems are typically of two types. Forward problems and Inverse problems.
Forward Problems
These are straightforward problems as you should start with a certain generalized equation for what you are interested in, then solve for some parameters. Example you can use generalized Hooke's law to solve the travel time and amplitude of energy of P and S waves.
This means that you will have a Forward model at which model parameters are applied to sake for the best fit, then at this condition other parameters can be determined as the solution.
The assumed input geology model is fed into a computer equation to simulate the geophysical data.
Model (model parameters m, initial source s) -------> Geophysical data
Then geophysical data = AF * (Model)........................(1)
Where,AF= Forward Problem operator.
In layman terms, let us say here we start with an assumed geology model, then the geophysical observation (data) that can explain and best fit our model is simulated, then the parameters that satisfy this model are regarded as solutions.
Inverse Problems
These problems are solved in reverse of those of Forward problems. These are more challenging as many lead to more than one solution. So the term ambiguous in interpretation rises in these problems as always data is not sufficient to explain the reality.
Generally it is easy to speak about inverse problems, but it is very challenging to solve these problems as they are directly related to causality (causes).
The solution is the model that can be used to explain the recorded geophysical observation (data). Because here we are finding a physical body, structure or any interesting response that would approximate and best fit our geophysical observation.
Let say we have the unknown initial source and the recorded geophysical observation {data), and we want to find the initial source, provided (it assumed) that the model parameters related to the physical properties of the medium are known, then this is the Inverse Source Problem.
A good example of this problem is the gravity inverse problem and seismological inverse problem at which in gravity inverse problem the density distribution of the rock is the source of the gravity field. While in a seismological (earthquake) inverse problem, the goal is to estimate the location and type of seismic source from the observed seismic field.
Then, Geophysical data ----> model {initial source}
Then, Initial source, s = As-1(data)............................2(a)
Where, As-1 = Inverse problem operator.
But If we want to find model parameters related to physical properties of the medium, provided that the initial source is known, then the problem will be the Inverse Model Problem. Such that,
Geophysical data ----> model {parameters, m}
Model parameters, m = Am-1(data)...........................2(b)
Where, Am-1 = Inverse problem operator
A typical example of this problem is related to electromagnetic field or acoustic field propagation.
You can see the figure 1 below to have a concept of the flow direction, how geophysical problems are solved.
Figure 1: A simple flowchart for geophysical problems.
The sad news is that most properties of geological materials vary greatly with various factors. It is very difficult to generalize without limiting yourself to some conditions.
And the fact that different geological structures (responses) may produce the same geophysical observations. For example in gravity the bodies with the same shape and mass, but differ in size may produce the same geophysical gravity anomaly.
This means that it is difficult to make one decisive solution of inverse geophysical problems without consulting other external data sources such as geology, boreholes. At the end of the day geophysics needs intensive applied mathematics and statistics as the principal tools for analysis of geophysical datasets and solving challenging geophysical problems.
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