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[[Inverse problem|Inverse modeling]] is a mathematical technique where the objective is to determine the phycical properties of the subsurface of an earth region that has produced a given [[seismogram]]. Cooke and Schneider (1983)<ref name="Cook1983">{{cite journal|last=Cooke|first=D. A.|coauthors= Schneider W. A.|title=Generalized linear inversion of reflection seismic data|journal=Geophysics|date=June 1983|volume=48|issue=6|pages=665–676|doi=10.1190/1.1441497}}</ref> defined it as calculation of the earth’s structure and physical [[parameters]] from some set of observed [[seismic]] 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]].<ref name="Pica1990">{{cite journal|last=Pica|first=A.|coauthors=Diet J. P., Tarantola A.|title=Nonlinear inversion of seismic reflection data in a laterally invariant medium|journal=Geophysics|date=March 1990|volume=55|issue=3|pages=284–292|doi=10.1190/1.1442836}}</ref> Some common earth properties that are inverted for include acoustic velocity, [[Formation (stratigraphy)|formation]] and fluid [[Density|densities]], [[impedance]], [[poisson's ratio]], formation compressibility, shear rigidity, [[porosity]], saturation etc.

The method has long been useful for geophysicists and can be categorized into two broad types:<ref name="Earthworks">{{cite journal|last=Francis|first=A.M.|title=Understanding Stochastic and Seismic Inversion|journal=First Break|date=November 2006|volume=24|issue=11|doi=10.3997/1365-2397.2006026}}</ref> [[Deterministic algorithm|Deterministic]] and [[stochastic]] inversion. [[Deterministic algorithm|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 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]] flow simulation, using geostatistical tools like [[kriging]]. As opposed to [[Deterministic algorithm|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 algorithm|deterministic]] models is the average of all the possible non-unique solutions of stochastic methods.<ref name="Earthworks"/> Since seismic[[Inverse problem|linear inversion]] is a [[Deterministic algorithm|deterministic]] inversion method, the [[stochastic]] method will not be discussed beyond this point.

[[File:Linear Seismic Inversion Flow Chart.jpg|thumb|upright=1.5|Figure 1: Linear Seismic Inversion Flow Chart]]

== Linear inversion ==

The [[Deterministic algorithm|deterministic]] nature of [[Inverse problem|linear inversion]] requires a [[Function (mathematics)|functional]] relationship which models, in terms of the earth model [[parameters]], 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]] and the field collected [[seismic trace]]. As in the field of optimization, this function to be minimized is called the [[objective function]] and in convectional[[Inverse problem|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 [[Inverse problem|linear inversion]] as a minimization problem, the different components that are necessary to invert seismic data.

=== Forward model ===

The centerpiece of seismic [[Inverse problem|linear inversion]] is the forward model which models the generation of the experimental data collected.<ref name="Cook1983"/> According to Wiggins (1972),<ref name="wiggns1972">{{cite journal|last=Wiggins|first=Ralph|title=The general linear inverse problem: Implication of surface waves and free oscillations for Earth structure|journal=Reviews of Geophysics|date=February 1972|volume=10|issue=1|pages=251–285|doi=10.1029/RG010i001p00251}}</ref> 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]]s for predicting [[particle displacement]] 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)<ref>{{cite journal|last=Tarantola|first=A.|title=Linearized and inversion of seismic reflection-data|journal=Geophsical Prospecting|year=1984|volume=32|issue=6|pages=908–1015|doi=10.1111/j.1365-2478.1984.tb00751.x}}</ref> 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)<ref name="kan">{{cite journal|last=Kanasewich|first=E. R.|coauthors=Chiu S. K. L|title=Least squares inversion of spatial seismic refraction data|journal=Bulletin of the Seimological Society of America|date=June 1985|volume=75|issue=3|pages=865–880|doi=10.1.1.122.1606}}</ref> used the brachistotrone model of John Bernoulli for travel time of a [[Ray (geometry)|ray]] along a path. In Cooke and Schneider (1983),<ref name="Cook1983"/> 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)<ref name="Cook1983"/> can only be used to invert CMP data since the model invariably assumes no spreading loss by mimicking the response of a laterally [[homogeneous]] earth to a plane-wave source

<ol start="1">
<li><math>t=\sum_{i=1}^n\frac{\big[(x_i-x_{i-1})^2+(y_i-y_{i-1})^2+(z_i-z_{i-1})^2\big]^{\frac{1}{2}}}{v_i}\!</math> </li>

where t is ray travel time, x, y, z are depth coordinates and vi is the constant velocity between interfaces i − 1 and i.

<li><math>\left[\frac{1}{K(\vec{r})}\frac{\partial^2}{\partial t^2}-\nabla\cdot\big(\frac{1}{\rho(\vec{r}\big)}\nabla)\right] U(\vec{r},t)= s(\vec{r},t)
\!</math></li>

where <math>K(\vec{r})\!</math> represent bulk modulus, <math>\rho(\vec{r})\!</math> density, <math>s(\vec{r},t)\!</math> the source of acoustic waves, and <math>U(\vec{r},t)\!</math> the pressure variation.

<li><math>s(t)=w(t)*R(t)\!</math></li>
</ol>

where ''s''(''t'') = synthetic trace, ''w''(''t'') = source wavelet, and ''R''(''t'') = reflectivity function.

===Objective function===

An important numerical process in [[Inverse problem|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 function]]s include [[sum of squares]]{{disambiguation needed|date=December 2013}} of difference between experimental and numerical data, as in the [[least squares]] methods, the sum of the [[Magnitude (mathematics)|magnitude]] 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 problem|inverse modeling]] procedure. These constraints, according to Francis 2006,<ref name="Earthworks"/> 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)<ref name="Cook1983"/> 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 problem|inverse modeling]] is to cast the generalized linear inverse problem into a simple[[Matrix (mathematics)|matrix]]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]] concepts, the forward model is[[Linearization|linearized]] by expansion using a [[Taylor series]] as carried out below. For more details see Wiggins (1972),<ref name="wiggns1972"/> Cooke and Schneider (1983).<ref name="Cook1983"/>

Consider a set of <math>m\!</math> seismic field observations <math>F_j\!</math>, for <math>j = 1, \ldots, m\!</math> and a set of <math>n\!</math>earth model [[parameters]] <math>p_i\!</math> to be inverted for, for <math>i=1,\ldots, n\!</math>. The field observations can be represented in either <math>\vec{F}\,(\vec{p})\!</math> or <math>F_j\,(p_i)\!</math> where <math>\vec{p}\!</math> and <math>\vec{F}\,(\vec{p})\!</math> are vectorial representations of model [[parameters]] and the field observations as a function of earth [[parameters]]. Similarly, for <math>q_i\!</math> representing guesses of model [[parameters]], <math>\vec{F}\,(\vec{q})\!</math> is the vector of numerical computed seismic data using the forward model of Sec. 1.3 Taylor's series expansion of <math>\vec{F}\,(\vec{p})\!</math> about <math>\vec{q}\!</math> is given below.

<ol start="4">
<li><math>\vec{F}\,(\vec{p}) = \vec{F}\,(\vec{q})+(\vec{p}-\vec{q})\frac{\partial \vec{F}\,(\vec{q})}{\partial \vec{p}}+(\vec{p}-\vec{q})^2\frac{\partial^2 \vec{F}\,(\vec{q})}{\partial \vec{p}^2} +O(\vec{p}-\vec{q})^3\!</math></li>

On [[linearization]] by dropping the non-linear terms (terms with (p⃗ − ⃗q) of order 2 and above), the equation becomes

<li><math>\vec{F}\,(\vec{p}) - \vec{F}\,(\vec{q})=(\vec{p}-\vec{q})\frac{\partial \vec{F}\,(\vec{q})}{\partial \vec{p}}\!</math></li>

Considering that <math>\vec{F}\!</math> has <math>m\!</math> components and <math>\vec{p}\!</math> and <math>\vec{q}\!</math> have <math>n\!</math>components, the discrete form of Eqn. 5 results in a system of <math>m\!</math> [[linear equation]]s in <math>n\!</math> variables whose [[Matrix (mathematics)|matrix]] form is shown below.

<li><math>\Delta \vec{F} = \mathbf{A}\,\Delta\vec{p}\!</math></li>

<li><math>\Delta\vec{F} = \begin{bmatrix}F_1(\vec{p})-F_1(\vec{q})\\\vdots\\F_m(\vec{p})-F_m(\vec{q})\end{bmatrix}\!</math></li>

<li><math>\Delta\vec{p} = \vec{p}-\vec{q} =
\begin{bmatrix}
p_1-q_1\\
\vdots \\
p_n-q_n
\end{bmatrix}\!</math></li>

<li><math>\mathbf{A} =
\begin{bmatrix}
\frac{\partial F_1(\vec{q})}{\partial p_1} & \frac{\partial F_1(\vec{q})}{\partial p_2} & \cdots & \frac{\partial F_1(\vec{q})}{\partial p_n} \\
\frac{\partial F_2(\vec{q})}{\partial p_1} & \cdots & \frac{\partial F_2(\vec{q})}{\partial p_{n-1}} & \frac{\partial F_2(\vec{q})}{\partial p_n} \\
\vdots & \frac{\partial F_j(\vec{q})}{\partial p_i} & \vdots & \vdots \\
\frac{\partial F_m(\vec{q})}{\partial p_1} & \frac{\partial F_m(\vec{q})}{\partial p_2} & \cdots & \frac{\partial F_m(\vec{q})}{\partial p_n} \\
\end{bmatrix}\!</math></li>
</ol>

<math>\Delta \vec{F}\!</math> is called the difference [[vector (mathematics and physics)|vector]] in (Cooke and Schneider 1983).<ref name="Cook1983"/> It has a size of<math>m\times 1\!</math> and its components are the difference between the observed trace and the numerically computed seismic data.<math>\Delta\vec{p}\!</math> is the corrector vector of size <math>n\times 1\!</math> while <math>\mathbf{A}\!</math> is called the sensitivity matrix. It has a size of <math>m\times n\!</math> and its comments are such that each column is the [[partial derivative]] 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===

<math>\vec{F}\,(\vec{q})\!</math> is computed from the forward model while <math>\vec{F}\,(\vec{p})\!</math> is the experimental data. Thus,<math>\Delta \vec{F}\!</math> is a known quality. On the other hand, <math>\Delta\vec{p}\!</math> is unknown and is obtained by solution of Eqn. \ref{eq_inv}. This equation is theoretically solvable only when <math>\mathbf{A}\!</math> is invertible, that is, if it is a square matrix so that the number of observations <math>m\!</math> is equal to the number <math>n\!</math> of unknown earth [[parameters]]. If this is the case, the unknown corrector vector <math>\Delta\vec{p}\!</math>, is solved for as shown below, using any of the classical direct or iterative solvers for solution of a set of [[linear equation]]s.

<ol start="10">
<li><math>\Delta \vec{p} = \mathbf{A}^{-1}\,\Delta\vec{F}\!</math></li>
</ol>

In most seismic inversion applications, there are more observations than the number of earth [[parameters]] to be inverted for i.e. <math>m>n\!</math>, leading to a system of equations that is mathematically over-determined. As a result, Eqn. \ref{eq_inv} is not theoretically solvable and an exact solution is not obtainable.<ref name="kan"/> An estimate of the corrector vector is obtained using the [[least squares]]procedure to find the corrector vector <math>\Delta \vec{p}\!</math> that minimizes <math>\vec{e}\,^T \vec{e}\!</math>, which is the sum of the squares of the error, <math>\vec{e}\!</math>.<ref name="kan"/>

The error<math>\vec{e}\!</math> is given by

<ol start="11">
<li><math>\vec{e} = \Delta\vec{F}-\mathbf{A}\,\Delta \vec{p}\!</math></li>
</ol>

In the [[least squares]] procedure, the corrector vector that minimizes <math>\vec{e}\,^T\vec{e}\!</math> is obtained as below.

<ol start="12">
<li><math>\begin{align}
\mathbf{A}\,\Delta \vec{p} &=\Delta\vec{F}\\
\mathbf{A}^T\mathbf{A}\,\Delta \vec{p} &= \mathbf{A}^T\Delta\vec{F}
\end{align}\!</math></li>
</ol>

Thus,

<ol start="13">
<li><math>
\Delta \vec{p} = (\mathbf{A}^T\mathbf{A})^{-1}\,\mathbf{A}^T\Delta\vec{F}\!</math></li>
</ol>

From the above discussions, the [[objective function]] is defined as either the <math>L_1\!</math> or <math>L_2\!</math> norm of<math>\Delta \vec{p}\!</math> given by
<math>\sum_{j=0}^n|\Delta p_j|\!</math> or <math>\sum_{j=0}^n|\Delta p_j|^2\!</math> or of <math>\Delta \vec{F}\!</math> given by<math>\sum_{i=0}^m|\Delta F_i|\!</math> or <math>\sum_{i=0}^m|\Delta F_i|^2\!</math>.

The generalized procedure for inverting any experimental seismic data for <math>m = n\!</math> or <math>m >n\!</math>, using the mathematical theory for [[Inverse problem|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]] <math>\vec{q}\!</math> is provided to initiate the inversion process.
# A synthetic seismic data <math>\vec{F}(\vec{q})\!</math> is computed by the forward model, using the model [[impedance]] above.
# The difference vector <math>\vec{F}(\vec{p})-\vec{F}(\vec{q})\!</math> is computed as the difference between experimental and synthetic seismic data. \item
# The sensitivity matrix <math>\mathbf{A}\!</math> is computed at this value of the [[impedance]] profile.
# Using <math>\mathbf{A}\!</math> and the difference vector from 3 above, the corrector vector <math>\Delta \vec{p}\,\!</math> is calculated. A new[[impedance]] profile is obtained as <ol start="14"><li><math>\vec{p}=\vec{q}+\Delta \vec{p}\!</math></li></ol>
# The <math>L_1\!</math> or <math>L_2\!</math> 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 $\vec{p}\!</math> from Eqn. \ref{corr_imp}. 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<ref>{{cite journal|last=Cooke|first=D|coauthors=Cant J.|title=Model-based seismic inversion: Comparing deterministic and probabilistic approaches|journal=CSEG Recorder|date=April 2010}}</ref> shows a typical example of impedance profile updating during successive iteration process. According to (Cooke and Schneider 1983),<ref name="Cook1983"/> use of the corrected guess from Eqn. \ref{corr_imp} 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]] represents the average value in the discrete interval.

Considering that [[Inverse problem|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)<ref name="Cook1983"/> 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 reduces, leading to greater accuracy in the results of the inversion [[algorithm]].

[[File:Amplitude Log.jpg|thumb|upright=1.5|Figure 8:Amplitude Log]]

[[File:Impedance Logs Inverted From Amplitude.jpg|thumb|upright=1.5|Figure 9a:Impedance Logs Inverted From Amplitude]]

[[File:Impedance Well Log.jpg|thumb|upright=1.5|Figure 9b: Impedance Well Log]]

==Inversion examples==

===Temperature inversion from Marescot (2010)<ref name="m"/>===

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 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
<ol start="15"><li>
<math>\vec{F}(\vec{q}) = \vec{T} = a+bz\!</math></li></ol>

where <math>\vec{q} = [a,b]\!</math>. Thus, the dimension of <math>\vec{q}\!</math> is 2 i.e the number of parameters inverted for is 2.

The objective of this inversion algorithm is to find <math>\vec{p}\!</math>, which is the value of <math>[a,b]\!</math> that minimizes the difference between the observed temperature distribution and those obtained using the forward model of Eqn. 15. Considering the dimension of the forward model or the number of temperature observations to be <math>n\!</math>, the components of the forward model is written as

<ol start="16"><li><math>\begin{align}
T_1&=a+bz_1 \\
T_2&=a+bz_2 \\
\vdots \\
T_{n-1}&=a+bz_{n-1} \\
T_n&=a+bz_n \\
\end{align}\!</math></li>

so that <math>\vec{F}(\vec{q}) = T\!</math>

<li><math>\mathbf{A} =
\begin{bmatrix}
1 & z_1 \\
1 & z_2 \\
\vdots & \vdots \\
1 & z_{n-1} \\
1 & z_n \\
\end{bmatrix}\!</math></li></ol>

We present results from Marescot (2010)<ref name="m">{{cite web|last=Marescot|first=Laurent|title=Introduction to Inversion in Geophysics|url=http://www.tomoquest.com|accessdate=3 May 2013}}</ref> for the case of <math>n = 2\!</math> for which the observed temperature values at depths were <math>T_1 = 19 ^{\circ}C\!</math> at <math>z=2m\!</math> and <math>T_2=22^{\circ}C\!</math> at <math>z=8m\!</math>. These experimental data were inverted to obtain earth [[parameter]] values of <math>a = 0.5\!</math> and <math>b=18^{\circ}C\!</math>. For a more general case with large number of temperature observations, Fig. 4 shows the final linear forward model obtained from using the inverted values of <math>a\!</math> and<math>b\!</math>. The figure shows a good match between experimental and numerical data.

===Wave travel time inversion from Marescot (2010)<ref name="m"/>===

This examples inverts for earth layer velocity from recorded seismic wave travel times. Fig. 5 shows the initial [[velocity]] guesses and the travel times recorded from the field, while Fig. 6a shows the inverted [[heterogeneous]] [[velocity]] model, which is the solution of the inversion [[algorithm]] obtained after 30 [[iteration]]s. As seen in Fig. 6b, there is good comparison between the final travel times obtained from the forward model using the [[Inverse problem|inverted]] [[velocity]] and the field recored travel times. Using these solutions, the ray path was reconstructed and is shown to be highly tortuous through the earth model as shown in Fig. 7.

===Seismic trace inversion from Cooke and Schneider (1983)===

This example, taken from Cooke and Schneider (1983),<ref name="Cook1983"/> shows inversion of a CMP seismic trace for earth model[[impedance]] (product of density and velocity) profile. The seismic trace inverted is shown in Fig. 8 while Fig. 9a shows the inverted[[impedance]] profile with the input initial [[impedance]] used for the inversion [[algorithm]]. Also recorded alongside the seismic trace is an impedance log of the earth region as shown in Fig. 9b. The figures show good comparison between the recorded[[impedance]] log and the numerical inverted [[impedance]]from the seismic trace.

==References==
{{Reflist}}

==Further reading==

*Backus, G. 1970. “Inference from inadequate and inaccurate data.” Proceedings of the National Academy of Sciences of the United States of America 65, no. 1.

*Backus, G., and F. Gilbert. 1968. “The Resolving Power of Gross Earth Data.” Geophysical Journal of the Royal Astronomical Society 16 (2): 169–205.

*Backus, G. E., and J. F. Gilbert. 1967. “Numerical applications of a formalism for geophysical inverse problems.” Geophysical J. of the Royal Astronomical Soc. 13 (1-3): 247.

*Bamberger, A., G. Chavent, C. Hemon, and P. Lailly. 1982. “Inversion of normal incidence seisomograms.” Geophysics 47 (5): 757–770.

*Clayton, R. W., and R. H. Stolt. 1981. “A Born-WKBJ inversion method for acoustic reflection data.” Geophysics 46 (11): 1559–1567.

*Franklin, J. N. 1970. “Well-posed stochastic extensions of ill-posed linear problems.” Journal of Mathematical Analysis and Applications 31 (3): 682.

*Parker, R. L. 1977. “Understanding inverse theory.” Annual Review of Earth and planetary sciences 5:35–64.

*Rawlinson, N. 2000. “Inversion of Seismic Dat for Layered Crustal Structure.” Ph.D. diss., Monash University.

*Wang, B., and L. W. braile. 1996. “Simultaneous inversion of reflection and refraction seis- mic data and application to field data from the northern Rio Grande rift.” Geophysical Journal International 125 (2): 443–458.

*Weglein, A. B., H. Y. Zhang, A. C. Ramirez, F. Liu, and J. E. M. Lira. 2009. “Clarifying the underlying and fundamental meaning of the approximate linear inversion of seismic dat.” Geophysics 74 (6): 6WCD1–WCD13.

[[Category:Mathematical modeling]]
[[Category:Geology]]

Burst error-correcting code - Revision history

en>Mark viking: expand first sentence