Some basic concepts of inverse problem

1. 1. 1 the necessity of inverse problem

When the groundwater movement is simulated by numerical method, the mathematical model established should objectively reflect the hydrogeological conditions of the actual aquifer system and the basic characteristics of groundwater movement. That is to say, when natural or man-made influences are exerted, the reflection of the mathematical model should be consistent with or very close to that of the actual aquifer. Only in this way can the model be used to predict the state of aquifer system and carry out reasonable management and control. Therefore, it is necessary to test the established mathematical model, that is, to simulate the known actual process (natural groundwater level dynamics or pumping test process) by using the established differential equation and definite solution conditions, and compare the calculated value with the measured value. If the difference is too big, it is necessary to modify the differential equation or boundary conditions or parameters, and re-simulate until they are basically the same.

In fact, the core of the inversion problem is the identification of hydrogeological parameters. Because of the selection of differential equation types and the determination of definite solution conditions, it is relatively easy to solve it through hydrogeological exploration and experiment. In the inversion process, even if these two items are modified, the problem can be easily solved because of the limited types available. However, for the forward problem of numerical calculation, we can't directly use the hydrogeological parameters obtained from the analytical formula based on field test data. This is because the mathematical model of each analytical parameter formula in groundwater dynamics is quite different from the actual situation, such as the assumptions of homogeneous, isotropic, equal thickness and infinite extension of aquifer. Strictly speaking, the hydrogeological parameters obtained from the analytical formula are only applicable to the forward problem of the mathematical model corresponding to the parameter formula. In fact, the parameters correspond to the model, which is the so-called concept of model parameters. On the other hand, the hydrogeological parameters obtained from the field pumping test only represent a small area near the test point. Therefore, we must use the corresponding model to invert the hydrogeological parameters.

The concept of model parameters of 1. 1.2

The groundwater flow system in nature is very complicated in time and space. In order to solve practical problems, it is necessary to generalize the aquifer system, that is, ignore some factors that have nothing to do with the current problems or have little to do with them, and establish corresponding mathematical models. Generally speaking, it includes the following aspects:

1) regional geometric generalization of groundwater flow system;

2) Generalization of boundary properties and simulation of initial flow field;

3) Generalization of parameter properties;

4) Generalization of groundwater dynamics.

Due to the limitation of mathematical tools, hydrogeologic models that are often generalized to be very reasonable cannot be described by existing mathematical tools, or can be described but their solutions cannot be obtained. Therefore, the expedient measure is to "make do", even if the generalization is not lost, the mathematical model is simple and solvable. From the analysis here, we can see that the mathematical model established by generalizing the complex groundwater flow system is only an approximate expression of the actual groundwater flow system. As a part of the mathematical model, parameters are also the approximation and synthesis of the inherent parameters of the aquifer itself, but not the inherent parameters of the aquifer itself. In order to distinguish, we define the parameters obtained from the mathematical model as "model parameters". Obviously, when establishing a hydrogeological model for the same groundwater flow system through exploration, due to the limitations of exploration and the differences of workers' experience, different workers can establish different hydrogeological models and have different mathematical models and "model parameters" accordingly. Therefore, in practical work, it will be difficult to judge whether the parameters are correct or not, and it is impossible to generalize. For example, for the phreatic aquifer system, we can choose Bolton model or Newman model for simulation. Obviously, the number and types of parameters of the two models are different; Even parameters of the same type will have different values due to different models.

1. 1.3 well-posedness of inverse problem

For the mathematical model describing groundwater flow, if the following three items are satisfied:

1) must find its solution (existence of solution);

2) The obtained solution is unique (uniqueness of solution);

3) The solution is continuously dependent on the original data, that is, when the parameters or conditions of the solution change slightly, the change of the solution is also small (the stability of the solution), so the mathematical model is said to be well-posed.

Generally speaking, positive questions are well-posed, while inverse questions are often ill-posed. From the hydrogeology itself, there is no doubt about the existence of the solution of the inverse problem, so the following only discusses the uniqueness and stability of the solution.

1. 1.3. 1 uniqueness of the solution to the rhetorical question

Because when the hydrogeological parameters are different, it is still possible to produce the same head distribution, so it is often impossible to determine the hydrogeological parameters only by observing the head value, such as the one-dimensional steady flow model with pressure: