Inverse modeling is a mathematical technique where the objective is to determine the physical properties of the subsurface of an earth region that has produced a given
seismogram
A seismogram is a graph output by a seismograph. It is a record of the ground motion at a measuring station as a function of time. Seismograms typically record motions in three cartesian axes (x, y, and z), with the z axis perpendicular to the ...
. Cooke and Schneider (1983)
defined it as calculation of the earth's structure and physical
parameters
A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when ...
from some set of observed
seismic
Seismology (; from Ancient Greek σεισμός (''seismós'') meaning "earthquake" and -λογία (''-logía'') meaning "study of") is the scientific study of earthquakes and the propagation of elastic waves through the Earth or through other ...
data. The underlying assumption in this method is that the collected seismic data are from an earth structure that matches the cross-section computed from the inversion
algorithm
In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific Computational problem, problems or to perform a computation. Algorithms are used as specificat ...
.
Some common earth properties that are inverted for include acoustic velocity,
formation
Formation may refer to:
Linguistics
* Back-formation, the process of creating a new lexeme by removing or affixes
* Word formation, the creation of a new word by adding affixes
Mathematics and science
* Cave formation or speleothem, a secondar ...
and fluid
densities
Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek language, Greek letter Rho (letter), rho), although the Latin letter ''D'' ca ...
,
acoustic impedance
Acoustic impedance and specific acoustic impedance are measures of the opposition that a system presents to the acoustic flow resulting from an acoustic pressure applied to the system. The SI unit of acoustic impedance is the pascal-second per cub ...
,
Poisson's ratio
In materials science and solid mechanics, Poisson's ratio \nu ( nu) is a measure of the Poisson effect, the deformation (expansion or contraction) of a material in directions perpendicular to the specific direction of loading. The value of Pois ...
, formation compressibility, shear rigidity,
porosity
Porosity or void fraction is a measure of the void (i.e. "empty") spaces in a material, and is a fraction of the volume of voids over the total volume, between 0 and 1, or as a percentage between 0% and 100%. Strictly speaking, some tests measure ...
, and fluid saturation.
The method has long been useful for geophysicists and can be categorized into two broad types:
Deterministic
Determinism is a philosophical view, where all events are determined completely by previously existing causes. Deterministic theories throughout the history of philosophy have developed from diverse and sometimes overlapping motives and consi ...
and
stochastic
Stochastic (, ) refers to the property of being well described by a random probability distribution. Although stochasticity and randomness are distinct in that the former refers to a modeling approach and the latter refers to phenomena themselv ...
inversion. Deterministic inversion methods are based on comparison of the output from an earth model with the observed field data and continuously updating the earth model parameters to minimize a function, which is usually some form of difference between model output and field observation. As such, this method of inversion to which linear inversion falls under is posed as an minimization problem and the accepted earth model is the set of model parameters that minimizes the
objective function
In mathematical optimization and decision theory, a loss function or cost function (sometimes also called an error function) is a function that maps an event or values of one or more variables onto a real number intuitively representing some "cos ...
in producing a numerical seismogram which best compares with collected field seismic data.
On the other hand, stochastic inversion methods are used to generate constrained models as used in
reservoir
A reservoir (; from French ''réservoir'' ) is an enlarged lake behind a dam. Such a dam may be either artificial, built to store fresh water or it may be a natural formation.
Reservoirs can be created in a number of ways, including contro ...
flow simulation, using geostatistical tools like
kriging
In statistics, originally in geostatistics, kriging or Kriging, also known as Gaussian process regression, is a method of interpolation based on Gaussian process governed by prior covariances. Under suitable assumptions of the prior, kriging giv ...
. As opposed to deterministic inversion methods which produce a single set of model parameters, stochastic methods generate a suite of alternate earth model parameters which all obey the model constraint. However, the two methods are related as the results of deterministic models is the average of all the possible non-unique solutions of stochastic methods.
Since seismic linear inversion is a deterministic inversion method, the stochastic method will not be discussed beyond this point.
Linear inversion
The
deterministic
Determinism is a philosophical view, where all events are determined completely by previously existing causes. Deterministic theories throughout the history of philosophy have developed from diverse and sometimes overlapping motives and consi ...
nature of
linear inversion requires a
functional
Functional may refer to:
* Movements in architecture:
** Functionalism (architecture)
** Form follows function
* Functional group, combination of atoms within molecules
* Medical conditions without currently visible organic basis:
** Functional sy ...
relationship which models, in terms of the earth model
parameters
A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when ...
, the seismic variable to be inverted. This functional relationship is some mathematical model derived from the fundamental laws of physics and is more often called a forward model. The aim of the technique is to minimize a function which is dependent on the difference between the convolution of the forward model with a source
wavelet
A wavelet is a wave-like oscillation with an amplitude that begins at zero, increases or decreases, and then returns to zero one or more times. Wavelets are termed a "brief oscillation". A taxonomy of wavelets has been established, based on the num ...
and the field collected
seismic trace. As in the field of optimization, this function to be minimized is called the
objective function
In mathematical optimization and decision theory, a loss function or cost function (sometimes also called an error function) is a function that maps an event or values of one or more variables onto a real number intuitively representing some "cos ...
and in convectional inverse modeling, is simply the difference between the convolved forward model and the seismic trace. As earlier mentioned, different types of variables can be inverted for but for clarity, these variables will be referred to as the
impedance series of the earth model. In the following subsections we will describe in more detail, in the context of linear inversion as a minimization problem, the different components that are necessary to invert seismic data.
Forward model
The centerpiece of seismic linear inversion is the forward model which models the generation of the experimental data collected.
According to Wiggins (1972),
it provides a functional (computational) relationship between the model parameters and calculated values for the observed traces. Depending on the seismic data collected, this model may vary from the classical
wave equation
The (two-way) wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields — as they occur in classical physics — such as mechanical waves (e.g. water waves, sound waves and s ...
s for predicting
particle displacement
Particle displacement or displacement amplitude is a measurement of distance of the movement of a sound particle from its equilibrium position in a medium as it transmits a sound wave.
The SI unit of particle displacement is the metre (m). In mo ...
or fluid pressure for sound wave propagation through rock or fluids, to some variants of these classical equations. For example, the forward model in Tarantola (1984) is the wave equation for pressure variation in a liquid media during seismic wave propagation while by assuming constant velocity layers with plane interfaces, Kanasewich and Chiu (1985)
used the brachistotrone model of John Bernoulli for travel time of a
ray along a path. In Cooke and Schneider (1983),
the model is a synthetic trace generation algorithm expressed as in Eqn. 3, where R(t) is generated in the Z-domain by recursive formula. In whatever form the forward model appears, it is important that it not only predicts the collected field data, but also models how the data is generated. Thus, the forward model by Cooke and Schneider (1983)
can only be used to invert CMP data since the model invariably assumes no spreading loss by mimicking the response of a laterally
homogeneous
Homogeneity and heterogeneity are concepts often used in the sciences and statistics relating to the uniformity of a substance or organism. A material or image that is homogeneous is uniform in composition or character (i.e. color, shape, siz ...
earth to a plane-wave source
-
where t is ray travel time, x, y, z are depth coordinates and vi is the constant velocity between interfaces i − 1 and i.
where represent bulk modulus, density, the source of acoustic waves, and the pressure variation.
where ''s''(''t'') = synthetic trace, ''w''(''t'') = source wavelet, and ''R''(''t'') = reflectivity function.
Objective function
An important numerical process in inverse modeling is to minimize the objective function, which is a function defined in terms of the difference between the collected field seismic data and the numerically computed seismic data. Classical objective functions include the sum of squared deviations between experimental and numerical data, as in the
least squares
The method of least squares is a standard approach in regression analysis to approximate the solution of overdetermined systems (sets of equations in which there are more equations than unknowns) by minimizing the sum of the squares of the res ...
methods, the sum of the
magnitude
Magnitude may refer to:
Mathematics
*Euclidean vector, a quantity defined by both its magnitude and its direction
*Magnitude (mathematics), the relative size of an object
*Norm (mathematics), a term for the size or length of a vector
*Order of ...
of the difference between field and numerical data, or some variant of these definitions. Irrespective of the definition used, numerical solution of the inverse problem is obtained as earth model that minimize the objective function.
In addition to the objective function, other constraints like known model parameters and known layer interfaces in some regions of the earth are also incorporated in the inverse modeling procedure. These constraints, according to Francis 2006,
help to reduce non-uniqueness of the inversion solution by providing a priori information that is not contained in the inverted data while Cooke and Schneider (1983)
reports their useful in controlling noise and when working in a geophysically well-known area.
Mathematical analysis of generalized linear inversion procedure
The objective of mathematical analysis of inverse modeling is to cast the generalized linear inverse problem into a simple
matrix
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** ''The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
algebra by considering all the components described in previous sections. viz; forward model, objective function etc. Generally, the numerically generated seismic data are non-linear functions of the earth model parameters. To remove the non-linearity and create a platform for application of
linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and through matrices.
...
concepts, the forward model is
linearized by expansion using a
Taylor series
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor serie ...
as carried out below. For more details see Wiggins (1972),
Cooke and Schneider (1983).
Consider a set of
seismic field observations
, for
and a set of
earth model parameters
to be inverted for, for
. The field observations can be represented in either
or
, where
and
are vectorial representations of model parameters and the field observations as a function of earth parameters. Similarly, for
representing guesses of model parameters,
is the vector of numerical computed seismic data using the forward model of Sec. 1.3. Taylor's series expansion of
about
is given below.
On linearization by dropping the non-linear terms (terms with (p⃗ − ⃗q) of order 2 and above), the equation becomes
Considering that has components and and have components, the discrete form of Eqn. 5 results in a system of linear equation
In mathematics, a linear equation is an equation that may be put in the form
a_1x_1+\ldots+a_nx_n+b=0, where x_1,\ldots,x_n are the variables (or unknowns), and b,a_1,\ldots,a_n are the coefficients, which are often real numbers. The coefficien ...
s in variables whose matrix form is shown below.
is called the difference
vector
Vector most often refers to:
*Euclidean vector, a quantity with a magnitude and a direction
*Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism
Vector may also refer to:
Mathematic ...
in Cooke and Schneider (1983).
It has a size of
and its components are the difference between the observed trace and the numerically computed seismic data.
is the corrector vector of size
, while
is called the sensitivity matrix. It has a size of
and its comments are such that each column is the
partial derivative
In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Part ...
of a component of the forward function with respect to one of the unknown earth model parameters. Similarly, each row is the partial derivative of a component of the numerically computed seismic trace with respect to all unknown model parameters.
Solution algorithm
is computed from the forward model, while
is the experimental data. Thus,
is a known quality. On the other hand,
is unknown and is obtained by solution of Eqn. 10. This equation is theoretically solvable only when
is invertible, that is, if it is a square matrix so that the number of observations
is equal to the number
of unknown earth parameters. If this is the case, the unknown corrector vector
, is solved for as shown below, using any of the classical direct or iterative solvers for solution of a set of linear equations.
In most
seismic inversion In geophysics (primarily in oil-and-gas exploration/development), seismic inversion is the process of transforming seismic reflection data into a quantitative rock-property description of a reservoir. Seismic inversion may be pre- or post- stack, d ...
applications, there are more observations than the number of earth parameters to be inverted for, i.e.
, leading to a system of equations that is mathematically over-determined. As a result, Eqn. 10 is not theoretically solvable and an exact solution is not obtainable.
An estimate of the corrector vector is obtained using the least squares procedure to find the corrector vector
that minimizes
, which is the sum of the squares of the error,
.
The error
is given by
In the least squares procedure, the corrector vector that minimizes
is obtained as below.
Thus,
From the above discussions, the objective function is defined as either the
or
norm of
given by
or
or of
given by
or
.
The generalized procedure for inverting any experimental seismic data for
or
, using the mathematical theory for inverse modeling, as described above, is shown in Fig. 1 and described as follows.
An initial guess of the model impedance is provided to initiate the inversion process. The forward model uses this initial guess to compute a synthetic seismic data which is subtracted from the observed seismic data to calculate the difference vector.
# An initial guess of the model impedance
is provided to initiate the inversion process.
# A synthetic seismic data
is computed by the forward model, using the model impedance above.
# The difference vector
is computed as the difference between experimental and synthetic seismic data.
# The sensitivity matrix
is computed at this value of the impedance profile.
# Using
and the difference vector from 3 above, the corrector vector
is calculated. A new impedance profile is obtained as
# The
or
norm of the computed corrector vector is compared with a provided tolerance value. If the computed norm is less than the tolerance, the numerical procedure is concluded and the inverted impedance profile for the earth region is given by
from Eqn. 14. On the other hand, if the norm is greater than the tolerance, iterations through steps 2-6 are repeated but with an updated impedance profile as computed from Eqn. 14. Fig. 2
shows a typical example of impedance profile updating during successive iteration process. According to Cooke and Schneider (1983),
use of the corrected guess from Eqn. 14 as the new initial guess during iteration reduces the error.
Parameterization of the earth model space
Irrespective of the variable to be inverted for, the earth's impedance is a continuous function of depth (or time in seismic data) and for numerical linear inversion technique to be applicable for this continuous physical model, the continuous properties have to be discretized and/or sampled at discrete intervals along the depth of the earth model. Thus, the total depth over which model properties are to be determined is a necessary starting point for the discretization. Commonly, as shown in Fig. 3, this properties are sampled at close discrete intervals over this depth to ensure high resolution of impedance variation along the earth's depth. The impedance values inverted from the
algorithm
In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific Computational problem, problems or to perform a computation. Algorithms are used as specificat ...
represents the average value in the discrete interval.
Considering that inverse modeling problem is only theoretically solvable when the number of discrete intervals for sampling the properties is equal to the number of observation in the trace to be inverted, a high-resolution sampling will lead to a large matrix which will be very expensive to invert. Furthermore, the matrix may be singular for dependent equations, the inversion can be unstable in the presence of noise and the system may be under-constrained if parameters other than the primary variables inverted for, are desired. In relation to parameters desired, other than impedance, Cooke and Schneider (1983)
gives them to include source wavelet and scale factor.
Finally, by treating constraints as known impedance values in some layers or discrete intervals, the number of unknown impedance values to be solved for are reduced, leading to greater accuracy in the results of the inversion algorithm.
Inversion examples
Temperature inversion from Marescot (2010)
We start with an example to invert for earth parameter values from temperature depth distribution in a given earth region. Although this example does not directly relate to
seismic inversion In geophysics (primarily in oil-and-gas exploration/development), seismic inversion is the process of transforming seismic reflection data into a quantitative rock-property description of a reservoir. Seismic inversion may be pre- or post- stack, d ...
since no traveling acoustic waves are involved, it nonetheless introduces practical application of the inversion technique in a manner easy to comprehend, before moving on to seismic applications. In this example, the temperature of the earth is measured at discrete locations in a well bore by placing temperature sensors in the target depths. By assuming a forward model of linear distribution of temperature with depth, two parameters are inverted for from the temperature depth measurements.
The forward model is given by
-
where