Modeling in Geophysics
It is obvious that models for various structures are build-up so that to gain better understanding of the studies on hand. In simple terms a model can be defined as a representation of something. So modeling can be regarded as the scientific technique to aid scientific study.
It is well known that in geophysics, when there is a certain problem that needs to be solved, a geophysical survey is designed and field data are acquired and plotted. In some situations through these plots we may obtain directly information about the problem on hand, however in most cases other more informations are needed to solve the problem about the subsurface.
Geophysical models can be grouped based on the type of geophysical parameter being measured such as gravity models, seismic models, magnetic models, resistivity models just to mention a few.
However in broad terms geophysics subsurface Modeling, utilizes two (2) modeling methods namely forward modeling and inversion modeling
Geophysical models can be used to aid interpretation of geophysical results.
Today this piece of post will lead you to gain basics on types of geophysical models with their examples which are frequently utilized.
The other practical side on how to deal with these models will be covered on other geophysical lab related posts so never lose hope, this is just the basics!
Geophysical Modeling
Geophysical modeling can be defined as the process whereby various geophysical parameters are applied to construct a representation of the subsurface so as to enhance the understanding of the subsurface geology.
In contrast to geological modeling at which the insight of the subsurface geology is derived from geological rock units and structures.
Let us look here down the two (2) types of geophysical modeling and how they work out,
1. Forward modeling
Forward modeling simply termed as modeling, begins with a model of earth properties, then mathematically simulates a physical experiment or process.
For example, electromagnetic, acoustic, nuclear, on the earth model, and finally outputs a modeled response. If the model and the assumptions are accurate, the modeled response looks like real data.
It starts off with a geological model describing it in terms of physical parameters and calculating the expected response. It means you have to construct the model, let's say synthetic model by using various physical parameters. See figure 1.
In forward modeling, the synthetic data that arise from a particular set of sources and receivers is computed when the distribution of physical properties is known.
The representation of forward modeling using data from a geophysical experiment can be generically written as,
F[m] = D
Where
F is a Forward modeling operator. This operator incorporates the details of the survey design and the relevant physical equations that must be solved numerically. The operator is often an integral or differential operator.
m is a generic symbol for a physical property distribution.
D is the observed data. This consists of the true data plus additive noise.
2. Inversion modeling
Inverse modeling simply termed as Inversion. It is considered as the inverse of forward modeling. Simply meaning that you reverse the working flow of the forward modeling.
It starts with actual measured data, applies an operation that steps backward through the physical experiment, and delivers an earth model. If the inversion is done properly, the earth model looks like the real earth. See figure 1.
Inversion modeling is the process of modifying the model to get a better fit for calculation data and better observational data which is done automatically (Grandis, 2009).
It start with field data and want to derive the geological model from it.
It can be treated either mathematically or automatically without the interactive input of a geophysicist or geologist.
It means you construct a model by using observed data then compare it with some standard geological models that best fit it.
N.B: The model must be run in iterative manner , by varying physical parameters again and again until it best fits the predefined statistics.
The principle of inversion is to find model parameters that produce a response that matches (fits) the data. This is why inversion is referred to as data fitting.
However, the inverse problem is non-unique and unstable means that it is an ill-posed problem (Tikhonov and Arsenin, 1977). A reformulation is required and all reformulations require that additional information, or prior knowledge about the model, be supplied
The goal of the inverse modeling is to recover the physical property distribution that gave rise to the observed data.
The representation of an inversion modeling using measured (observed) data from a geophysical measurements (observation) can be generically written as,
F-1[D] = m
Where,
F-1 is an Inversion modeling operator.
D is the observed data. This consist of the true data plus additive noise
m is a generic symbol for a recovered physical property distribution which can either be 1D, 2D or 3D.
So here we are finding a physical body or structure that would approximate the observed data.
So inverse modeling operates in reverse to that of forward modeling, see figure 1 below.
And this is why most statisticians say models are just statistics when asked about models.
Examples of Geophysical Modeling
Magnetics
One of the examples of forward modeling in magnetics (magnetic modeling) is that the magnetic field anomaly map can be used directly to infer and approximate horizontal locations of an existence buried object.
While in case of inverse modeling, (magnetic inversion) the data in the magnetic field anomaly map is inverted to generate a 3D subsurface distribution of the magnetic material that gives an approximation regarding either the depth of the buried object or details of its shape.
This means that, we would like to recover the earth’s magnetic susceptibility from the noise contaminated total field anomaly data measured at the surface.
In this example the Earth’s main magnetic field act as the source, while the model is the 3D distribution of magnetic susceptibility and the observed data are the total-field anomaly that are plotted as a magnetic field anomaly map.
Seismic Reflection
Another example of forward modeling is in seismic reflection (seismic modeling) which takes a model of formation properties such as acoustic impedance developed from well logs combines it with a seismic wavelet (pulse) and produces an output as a synthetic seismic trace.
Conversely, in case of inversion modeling (seismic inversion) begins with a recorded seismic data trace and removes the effect of an estimated wavelet, while creating the values of acoustic impedance at every time sample.
Seismic Refraction
Another example of inversion modeling is the 1D seismic refraction problem where the velocity increases continuously with depth.
Knowledge of travel time for all offsets is sufficient to recover down to a critical depth dependent upon the largest offset using Herglotz-Wiechert formula (see Aki and Richards, 2002).
Gravity
Another example of inverse modeling (3D - gravity inversion), at which the initial modeling can be done to get the synthetic gravity data by adding a Gaussian distribution error, then the synthetic gravity data is inverted using programs such as GRAV3D to obtain the recovered model.
All in All geophysical modeling can be regarded as one of interpretation techniques of geophysical data. So depending on the nature of the geophysical problem, geology and availability of other related data one can decide which of the two techniques can be utilized to produce the best results.
References
➔ Aki K. and Richards, P. G., (2002), Quantitative seismology, University Science Book
➔ Douglas W. O and Yaoguo L. (2014), Inversion for Applied Geophysics, A Tutorial, department of Earth and Ocean Sciences, University of British Columbia and department of Geophysics, Colorado School of Mines, Golden, Colorado.
➔ Grandis, H. (2009). Pemodelan Inversi Geofisika. (Himpunan Ali Geofisika Indonesia, Jakarta).
➔ Rina D. I., Mochamad A. A., Danastri L.T., and Sorja K (2022), 3D - Inversion of Gravity Data Modelling using the Chi Fact Algorithm for revealing the subsurface structure in Semarang City. Indonesian Journal of Applied Physics (IJAP), Volume 12, no.1, p.67.
➔ Tikhonov, A. V. and Arsenin, V. Y., (1977). Solution of ill-posed problems, Ed. Fritz, J., John-Wiley & Sons, New York.
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