## Geophysical Curiosities

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**Geophysical curiosity 22**

April 24^{th} 2019

Steerable total variation regularization

In signal processing, total variation regularization (TV), is based on the principle that signals with excessive and possibly spurious detail have high total variation, that is, the integral of the absolute gradient of the signal is high. According to this principle, reducing the total variation of the signal, subject to staying close to the original signal, removes unwanted detail whilst preserving important details such as sharp contrasts.

This noise removal technique has advantages over simple techniques such as linear smoothing or median filtering (although also known to be edge preserving), which reduce noise but at the same time smooth away edges to a greater or lesser degree. By contrast, total variation regularization is remarkably effective at simultaneously preserving edges whilst smoothing away noise in “flat” regions, even at low signal-to-noise ratios.

Total Variation is used in FWI to regularize the output of the non-linear, ill-posed inversion.

A new development in TV is steerable total variation regularization. It allows steering the regularization of the FWI solution towards weak or strong smoothing based on prior geological information.

Hold-on, doesn’t that lead to a very pre-conceived result. That could indeed be the case when not done carefully.

However, there is a way in which it can be done by including that prior information in the regularization parameter rather than in the misfit term as done usually. A spatially variant regularization parameter based on prior information where sharp contrasts are expected can be used. It will first reconstruct the high contrasts (salt bodies) followed by filling in the milder contrasts (sediments) at later stages.

Ref: Qui et al, SEG2016, P1174-1177

**Geophysical curiosity 21**

April 24^{th} 2019

**Wasserstein distance**

A most curious metric

In Full-Waveform Inversion the subsurface model is updated by minimizing the difference between synthetic/modelled and observed data. This is usually done by a sample-by-sample comparison (trace-by-trace or globally) using the L2-norm. The advantage of the L2-norm is that it is an easily understood and fast comparison, the disadvantage is that the model update is prone to errors due to poorly updating the low wavenumber trends, also mentioned as cycle skipping (more than half-wavelength shifts). Several ways of overcoming this issue are available. The most common one is to use a multi-stage approach, in which we start with the lowest available frequency (longest wavelength=largest time difference for cycle skipping) and then using increasing frequencies (Hz) in subsequent stages (2-4, 2-6, 2-8, 2-10, etc).

Now recently, at least for geophysics, a new difference measure has been used with great success, the quadratic Wasserstein metric W_{2}. This metric appears to me as curious. It namely computes the cost of rearranging one distribution (pile of sand) into another (different pile of sand) with a quadratic cost function (cost depends for each grain on the distance it needs to be moved/transported). This explains the term “optimal transport” which is often mentioned.

The advantage of using the W_{2} metric is that the cost function is much more convex (a wider basin of attraction), mitigating cycle skipping and hence less sensitive to errors in the starting model.

However, some extra data pre-processing is needed to apply this method. The synthetic/modelled and observed data need to be represented as distributions, which means non-negative and sum-up to unity. This can easily be achieved.

Another interesting approach is not to use the samples themselves, but first map the data to another more “suitable” domain, for example the Radon domain.

And then there is another “unexpected” improvement by not using the samples themselves, but by using the matching filter between observed and modelled data. If the two are equal, the matching filter would be a delta function. Hence, the model update is done by minimizing the Wasserstein distance between the matching filter and a delta function (for stability a delta function with a standard deviation of 0.001 is used). And apparently, this cost function allows updating the subsurface model.

Great, isn’t it.

Ref: Yang et al., Geophysics, 2018, V83, N1, P.R43-62

**Geophysical curiosity 20**

April 2^{th} 2019

**Matching Pursuit**

How to explain matching pursuit?

I will use the above plot from Wikipedia to explain matching pursuit. Given a time signal, we like to decompose it into a minimum number of components. The number of possible components, contained in a dictionary, is large, not to say unlimited. So, I start by matching the signal first with the most important, best matching component, then I take the next one and I keep on doing this, I pursue to get a better match, till I am satisfied with the match, which means the signal has been sufficiently approximated using “only” N components.

In the example above, the axes are time vs frequency, the components are cosine functions in the frequency range 0-60 Hz and the results show that time signal can be well represented by the centres of the ellipses.

**Geophysical curiosity 19**

April 2^{th} 2019

**Inversion of seismic data**

The simplest definition of seismic data inversion I came across is:

“Going from interface to Formation information”

**Geophysical curiosity 18**

^{th}2019

**Do shear waves exist in an acoustic medium?**

This is a curious question as it is well known that shear waves do not exist in an acoustic medium, because that is the “definition” of an acoustic medium? In an acoustic medium \( \mu \) equals 0, hence V^{2}_{S}= \( \mu/ \rho=0 \), which means shear waves cannot propagate and hence cannot exist.

But another definition of an “acoustic” medium is also used (Alkhalifah, Geophysics, V63, N2, P623-631), namely defining it by V_{S0}=0, that is the vertical shear wave velocity equals 0. This definition of acoustic is used to simplify the equations for P-wave propagation. However, it defines the medium to be truly acoustic for an isotropic medium only. In case of VTI anisotropy the story is different. Namely, in that case a shear wave exists as demonstrated by the (phase velocity) expression below: Grechka et al, Geophysics Vol 69, No.2, P576-582.

Fig.1: Wavefronts at 0.5 s generated by a pressure source

in VTI medium with V_{P0}=2 km/s, V_{S0}=0 and \( \epsilon \) =0.4, \( \delta \) =0.0

From the formula we see that except for the vertical (\( \Theta \)=0°, 180°) and horizontal (+/-90°) directions a SV wave exists in case \( \epsilon \neq \delta \) (the VTI anisotropy is not “elliptical”). Wavefront modelling (Fig.1) indeed shows that a curious shear wavefront is generated in accordance with expression (1).

Hence, defining the medium to be “acoustic” by putting the vertical shear wave velocity (V_{S0}) to zero simplifies the P-wave equations, but do not make the medium truly acoustic. For anisotropic media, shear waves, are still excited in the medium. For modern high-resolution seismic modelling, the “noise” generated is not negligible.

Thus, putting V_{S0}=0 doesn’t make the medium truly acoustic in case of anisotropy.

P.S. To be fair to Alkhalifah he called it an acoustic approximation**.**

**Geophysical curiosity 17**

Feb. 20^{th} 2019

**Cloud computing in Geophysics**

Cloud computing is hardly used in geophysical processing. Whether that is due to security issues or the amount of data involved is not clear to me. Interestingly enough a recent publication by Savoretti et al, from Equinor in the first Break, V36, N10 gives some insight in the approach.

It is well known that cloud computing provides access to a scalable platform, servers, databases and storage options over the internet. The provider maintains and owns the network-connected hardware for these services.

It offers increased flexibility in access to compute power and data sets are easily transferred to the cloud during project work and removed afterwards. In cluster terms an allocated compute node participates in running workloads via a queue system, which keeps track of what can be done on which node most efficiently.

What I didn’t know is how it is commercially arranged. It appears that nodes can either be ordered permanently or from a “spot market” for lower prices. However, spot market nodes can be taken away suddenly in case there is a higher bidder. The work done so far on that node will then be lost.

To minimize the effect, the workflow has been adjusted to better tolerate loss of nodes in spot-market environment. Since then better use can be made of the cheaper spot-market nodes by automatic recovery and more fully automated mode ordering.

**Geophysical curiosity 16**

Dec. 24th 2018

**Machine learning**

Machine Learning, also called Artificial Intelligence or Algorithms, is today’s hot topic. It is also associated with “Big Data”, indicating that its use can mainly be found in case an overwhelming amount of data need to be explored.

Is Machine Learning also useful in geophysics? Before answering this question, I like to pose another question: Will Machine learning replace the Physics in Geophysics? By that I mean do we still need formulae and equations to explore for hydrocarbons, salt/fresh water, polluted/unpolluted subsurface, etc? To answer this question let’s look at “traditional” versus “new” geophysics.

In the traditional geophysics we try to interpret observations / data by applying two main methods: forward modelling and inversion. In forward modelling we assume a (best guess) model of the subsurface and use equations, like the visco-acoustic wave equation for seismic, to derive related observations (synthetic data). By comparing these synthetic data with the real observations, we derive an update of the assumed model using inversion. The success depends heavily on the accuracy (adequacy) of the forward modelling and the ability of solving the ill-posed inversion. So, it is essential to fully understand the physics of the problem.

In Machine Learning it is different. An understanding of the physics is useful, but not essential. It is now a matter of having a sufficiently large learning / labelled data set which (hopefully) contains a statistical relationship between the observations and the related subsurface. This relationship, which is nothing else than a “correlation”, will be derived by a clever algorithm that contains parameters, which are determined by tuning the algorithm using the learning (labelled) data set. This tuning will determine the parameter values resulting in a predictive capability of the algorithm with an appropriate accuracy. Among the numerous kinds of algorithms, the most popular ones are called Neural Nets.

Neural Nets consist of an input layer, hidden layers and an output layer, each containing nodes.

An input node is fed with a seismic sample or an attribute /feature from a point in the 3D data set. A node in a hidden layer would receive weighted inputs from all input nodes and depending on the sum provide input to the nodes in the output layer. Each node in the output layer represents a class, say lithology or fluid saturation of a reservoir interval. The sum of each output node would indicate whether the input belongs to its class. In the learning phase the weights will be determined. Once determined, the network would be able to predict the class /character of new input data (undrilled locations). It can also be used in an unsupervised way to determine clusters of data, which then still need to be interpreted. In general, more hidden layers are needed to be able to handle more difficult problems.

Now, if you like the physics of geophysics, that is formulae and equations give you a warm feeling as they are an unambiguous, short and clear language describing physical processes, then you will have a tendency to give preference to traditional geophysics, but if you need practical results to earn an income (say predictions of fluid saturations in carbonates), then you will have to rely on the benefits of applying machine learning in addition to an understanding of the underlying physics.

So, the answer to the question: “Which one will be the winner?” will be in my opinion: Neither of the two, it will be a combination of both.

**Geophysical curiosity 15**

**Quaternion **

Up to now I didn’t know of the existence of quaternions and that they can also be used in geophysics. What is a quaternion?

A complex number is defined as a+ib, a quaternion as q=a+bi+cj+dk, hence it is a hypercomplex number.

It
allows a Fourier domain representation of multi-component seismic data.
For example, so-called “Projection Onto Convex Sets” (POCS)
reconstructs all three seismic components at once.

Applications
are in image disparity methods (edge detection), interpolation,
time-lapse analysis, separation of up- and down-going waves (VSP) and
calculation of attributes.

However, so far I have encountered very few “real” applications*.*

**Geophysical curiosity 14**

**Migration or Wavefield reconstruction versus Imaging**

It is important to make a distinction between wavefield reconstruction and imaging.

Migration is a two-step process: 1) wavefield reconstruction and 2) imaging.

Wavefield
reconstruction is the determination of the wavefield over a volume of
interest. Imaging involves the making of a picture of the geometrical
distribution of the reflecting surfaces within the medium.

This
is best illustrated by the fact that we perform a downward
continuation, forward in time, of the wavefield generated by a source
and a downward continuation, but now backward in time, of the recorded
wavefield, followed by the application of an imaging condition to obtain
the reflection point in space and its reflection strength.

Again,
we have various methods for wavefield continuation, the more
sophisticated the more accurate (acoustic, elastic, visco-elastic) and
various so-called imaging conditions (cross-correlation, deconvolution,
least-squares deconvolution)*.*

**Geophysical curiosity 13**

**Question**: Is seismic a hologram?**Hologram**

An
optical hologram is not an image; it is a recording of a wavefield.
Remember how a hologram is made: A single frequency laser light is split
into two parts. One part is mirrored directly onto the recording plate;
the second one is reflected / diffracted by the object. What is
recorded on the plate is the interference pattern between the two.

When
we beam the laser onto the “developed” plate and look through we see
the object in 3D. Looking from different directions we see the object
from these different directions. Even more fascinating is that when we
break off a part of the plate, we can still see the total object from
all directions, although the smaller the remaining piece the poorer the
quality of the image.

**Seismogram**

Remember
that in seismic we can consider the subsurface to consist of a
collection of diffraction or scattering points. This would imply that a
single shot recorded by a single geophone would record diffraction
energy from any point of the subsurface. Hence, the single record
contains information about the entire subsurface and it would be
possible, in principle to reconstruct/image the complete subsurface.
This is like the concept that a single point on a hologram is enough to
observe the object in 3D. But again, we would prefer a larger piece of
the plate, i.e. more observations, to obtain a better-quality image.**Lesson learned**

If we would apply holographic seismic imaging, we could do with a much sparser and therefore cheaper survey.**Ref.:**

“Extended resolution: Neidell is right”, E.A. Robinson, TLE 2018V37, N1

‘holistic migration”, E.A. Robinson, TLE 1998, V17, N3**Added statements** by Öz Yilmaz

“Spatial under sampling can be compensated by temporal oversampling”

“Temporal oversampling enables the extension of the spectral bandwidth beyond the Nyquist frequency”

**Geophysical curiosity 12**

What is the difference between updating a velocity model in the data or in the image domain? Are we using the same data?

Updating
of velocities means that we have an estimated velocity-depth model and
look for diagnostics and ways which will allow improving the velocities.**Data domain**

We
have observed the arrival times at all receivers for all shots. We know
the shot and receiver locations at the surface and a rough/first
estimate of the velocity model, which will be represented by a grid of
cells, with for each cell \( (i) \) a velocity \( (v_i) \). Then we
apply raytracing from each shot to the receivers. That means that the
ray will go through a subset of all cells. For each ray we can calculate
the arrival time by adding all the \( \Delta t_i \) calculated from
the pathlength \( (l_i) \) in each cell divided by the cell velocity \(
v_i \). We will find a difference between the calculated/modelled and
the observed travel time. This difference could come from anywhere along
the ray. So, we simply divide the difference over the total ray path,
leading to a small velocity update in each cell traversed. Having many
shot-receiver combinations, means that (hopefully) each cell will be
traversed by many rays. Each traverse will give a small contribution to
the update. If they agree the sum will be a significant update, if they
disagree the sum will be “zero”, that means no update. That’s all there
is to it in the data domain. So we don’t do any migration/imaging.**Image domain**

We
migrate the recorded data using the first estimate of the
velocity-depth model. The results are Common Image Gathers (CIG). They
will show residual move-out (diagnostic), which will be used to update
the velocity model.

Although
in both cases we use the same data, the advantage of using the image
domain is that the signal-to-noise ratio will usually be much better.
The disadvantage is that it needs more computer time.

**Geophysical curiosity 11**

What happens with reflections beyond the critical angle?

Let’s
look at AVA, that means the reflection coefficient with angle of
incidence (offset). This can for a flat interface between two
homogeneous layers be calculated exactly using the Zoeppritz equation,
but also approximated quite accurately to first order in the velocity
and density contrasts using an expression formulated in terms of
horizontal slowness p (Aki & Richard, Quantitative Seismology, eq.
5.44):

\(
R(p) = \frac{1}{2}(1 - 4 \beta^2p^2) \frac{ \Delta \rho }{ \rho } +
\frac{1}{2(1 - \alpha^2p^2)} \frac{ \Delta \alpha }{ \alpha } - 4
\beta^2p^2 \frac{\Delta \beta}{\beta } \)

Aki
& Richards mention that the linearized expression is only valid
when none of the angles are near 90°. When the P-wave velocity below the
interface is larger than above, the angle of transmission \( \Theta
\)_{t} can be 90° or “larger”.

However, the expression can also be given in terms of the average angle \( \Theta=( \Theta_1+ \Theta_t)/2 \)

\(
R( \Theta)= \frac{1}{2} (1-4 \frac{ \beta^2 }{ \alpha^2 } sin^2 \Theta)
\frac{ \Delta \rho }{ \rho } + \frac{1}{2cos^2 \Theta } \frac{ \Delta
\alpha }{ \alpha } -4 \frac{ \beta^2 }{ \alpha^2 } sin^2 \Theta \frac{
\Delta \beta }{ \beta } \) ,

but now the transmitted angle can become complex:

\(
\Theta_t= \frac{ \pi }{2} -i\ cosh^{-1} ( \frac{ \alpha_2 }{ \alpha_1
} sin \ \Theta_r) \) , \( \frac{ \alpha_2 }{ \alpha_1 } sin\ \Theta_r
\geq1 \)

As a consequence, the reflection coefficient becomes complex. This means that the amplitudes (trace samples) become complex.

This
makes us realize that for AVA, we calculate at an angle of incidence \(
( \Theta_i) \) the reflection amplitude for each sample. That means
looking at a sequence of samples, forming a wavelet, the same wavelet,
but scaled with the reflection coefficient, will be returned. But only
in case the reflection coefficient is real. If complex, all samples will
have the same phase change and the reflected wavelet will have a
different appearance. As the phase change is angle (offset) dependent,
the wavelet change will be angle (offset) dependent, complicating the
analysis somewhat.

Ref: Downton & Ursenbach, Geophysics, 2006, V71, N4.

**Geophysical curiosity 10**

Systematically
organised fractures or cracks play an important role in seismic wave
propagation and in fluid flow of reservoirs. If of a size significantly
below seismic wavelength their effect on seismic wave propagation can be
described by anisotropy.

For
a VTI medium we estimate the Thomsen weak anisotropy parameters \(
\epsilon, \delta, \gamma, \eta \) from travel time information of P and
PS or S wave.

For a HTI medium, caused by the presence of a single set of cracks, we estimate the Thomsen-type parameters \( \epsilon \)^{(v)}, \( \delta \)^{(v)}, \( \gamma \)^{(v)}, \( \eta \)^{(v)}
from either the azimuthal NMO velocities or AVA gradients of P and PS
waves or from shear wave splitting (birefringence), in which case the
fracture infill (brine, oil, gas, dry) has no influence. Note that for P
or PS waves the infill has an influence.

There
are several crack models, based on isolated infinite fractures with
linear slip boundary conditions (Schoenberg), isolated penny-shaped
cracks (Hudson) and hydraulically connected cracks and matrix pore space
(Thomsen). Fortunately, seismic data are not sensitive to the **shape **of the fractures.

However, seismic is sensitive to pore fill & **saturation **and whether the pore spaces are **hydraulically connected**,
which depends on the pore throat micro-geometry and the fluid
properties. At seismic (low) frequencies, connection (pressure
equilibrium) between fracture and medium pore space can be assumed.
“Squirt flow” occurs at frequencies between sonic and ultrasonic bands.
In that case Gassmann theory should be replaced by Biot theory. At high
frequencies (ultrasonic) no fluid interaction occurs between fracture
and matrix pore space. At very high frequencies Rayleigh scattering
becomes important. Note that at intermediate frequencies, dispersion
will occur due to the fluid flow between fracture and matrix pore space.

An
empty crack is compliant because of its 2D shape, equant (non-flat)
pores are stiffer. If a crack is isolated or connected to other cracks
of similar shape and orientation than a pore fluid will stiffen the
crack(s). If connected with equant matrix pore space, the fluid will
“squirt” into the stiffer matrix pore space. Hence, the pores effect the
compliance, that is the anisotropy.

There
several ways of describing the presence of cracks or fractures. One way
is to describe the influence of fractures on the wave propagation in
terms of the parameters \( \epsilon \)^{(v)}, \( \delta \)^{(v)}, \( \gamma \)^{(v)}, \( \eta \)^{(v)}
(penny shaped cracks). The other way is to consider the fracture
properties or excess fracture compliance parameters, thus the additional
compliances due to fractures. Two sets are used. One uses K_{N} (excess normal fracture compliance) and K_{T} (excess tangential compliance). The other set uses similar, but dimensionless and normalised parameters, \( \Delta \)_{N} (normal weakness) and \( \Delta \)_{T} (tangential weakness) together with the Lamé parameters \( \lambda \) and \( \mu \) (linear slip theory). \( \Delta \)_{T} gives estimate of the crack density and \( \Delta \)_{T}** /**\( \Delta \)_{N} is sensitive to fluid saturation. Obviously, the sets are related.

Crack
density can be derived from the seismic observations of P, PS and/or
AVA gradients (azimuths parallel/strike or normal to the fractures) or
from shear wave splitting.

The
main difference, except for shear waves, is between pore space filled
with fluids (brine, oil) or filled with gas (dry). The dry-crack model
is close to elliptical (\( \epsilon \)^{(v)}= \( \delta \)^{(v)}), for fluid-filled cracks \( \delta \)^{(v)} is close to the crack density for typical background V_{s}/V_{p} ratios (around 0.5). In the (\( \epsilon \)^{(v)}, \( \delta \)^{(v)}) plane all possible pairs of \( \epsilon \)^{(v)}, and \( \delta \)^{(v)} are confined to a narrow area.

An
important aspect is the frequency of the seismic wave. Low frequent
(seismic) means that there is enough time for the pressure to equalize
in fractures and matrix (equant) porosity, so the entire pore volume is
involved. For moderately high frequencies (ultra-sonic) only the
fracture volume is involved.

A
complete characterization of fractures can be based on wide-azimuth 3D
P-wave data or a combination of P and PS data or S data.

**Geophysical curiosity 9**

I thought I knew a reasonable number of statistical distributions, until I went to Wikipedia (https://en.wikipedia.org/wiki/List_of_probability_distributions) and realised I only knew a few. (So, it is true that you don’t know what you don’t know).

More interesting, however, is that I recently came across a distribution called logistic distribution. What the name means I am lost, as logistics refers to moving goods from one place to another. However, I found it a very useful distribution for geophysics, as it can handle limited interval distributions of geophysical parameters of different magnitudes. Again, a description can be found in Wikipedia (https://en.wikipedia.org/wiki/Logistic_distribution)

An interesting application can be found in an article by J. Eidsvik, D. Bhattacharjya and T. Mukerji: Value of information of seismic amplitude and CSEM resistivity. (Geophysics, 2008, Vol73, N4)

**Geophysical Curiosity 8**

The local angle domain. (LAD, Koren & Ravve, Geophysics, 2011, V76, N01).

When we consider fractures, we look at the source-receiver azimuth as an important parameter. Using either the NMO velocity as a function of azimuth (low resolution) or the AVA as a function of azimuth (high resolution) we determine the orientation of the fracture system(s). But is this correct?

It might be, but not always. Namely, what is important is the local azimuth of the “incident ray” and the “reflected ray” at the reflection point, because they are directly influenced by the local fracture orientation at both sides of the interface. So, in case the “local azimuth” is significantly different from the surface source-receiver azimuth, it is important to consider it.

But how often does this occur? I think seldom, but yes in cases of complicated geology and stress fields. Now those occur often in places where significant hydrocarbons can be found and where drilling is very expensive: deep water with reservoirs below salt or basalt.

So, there are cases when considering the local angle domain (LAD) is a must for obtaining a correct image of the subsurface.

**Geophysical Curiosity 7**

A
P-wave reflected by an interface will change in amplitude according to
the reflection coefficient. Hence, when the acoustic impedance (ρvp)
increases across the boundary, a positive amplitude (meaning the
wavefront starts with a compression) will remain a positive amplitude
upon reflection.

But how about a
S-wave? What will happen to the amplitude when reflected by an
interface where the shear impedance (ρvs) increases?

The first thought would be that the reflection coefficient will be positive, so the amplitude will keep the same polarity.

But, no, no, that is not the case. How confusing. Why can that be?

The answer lies in the fact that we use in the industry a moving reference frame.

So, for P waves, the positive direction is downwards on the way down and upwards on the way up.

What
does this mean for S waves, say SH? It means on the way down the
positive shear direction is to the right and on the way up the positive
shear is of course also to the right. But if the, say SH wave front
start with a movement to the right and the shear impedance increases
across the interface, on the way up the movement will be in the same
direction. However, now that direction is negative due to the choice of
the frame we are using. So, the reflection coefficient is negative for
an increase in shear impedance.

I hope it is no longer confusing.

**Geophysical Curiosity 6**

But do we still do stacking? Yes, we still do stacking, but in a different way. The stacking we do takes place in the imaging part of processing. Namely, in a modern processing scheme the data is prepared for imaging and after each offset is imaged, then the results are stacked. This can be most easily seen in case image gathers are built and displayed. After inspection and possibly re-imaging, these are the traces that are stacked.

**Geophysical Curiosity 5**

We
are all aware of gravity, especially when you fall of your bike after
too many drinks. Especially then it looks that gravity is a powerful
force, although strictly speaking it is an acceleration. Interesting
enough, gravity is the weakest force in nature. How can that be? You
would agree when you realize that it needs the whole earth, that is a
mass of 6 10^{24} kg, to
make me feel it. Also, according to Newton, if I meet another person, I
should feel the “attraction” between us, but I don’t. This shows how
weak gravity is.

**Geophysical Curiosity 4**

*How confusing can you make it?*” Take the description of anisotropy. The confusion is mainly due to using misleading symbols and different notations for the same quantities. Two examples: 1) using the symbol C for

**in**compressibility and 2) the use of \( \delta \)

^{(2)}in one set of publications and \( \delta \)

_{1}in other publications, both describing the same anisotropy parameter for propagation in the (x

_{1},x

_{3}) plane. So, when reading publications make sure you are aware of the definitions.

In describing wave propagation, the elastic and anisotropy parameters occur in the so-called stress-strain relationship which needs to be combined with Newton’s first law of motion to obtain the wave equation. In principle (“worst case”: triclinic symmetry) there are 21 different (C

_{ij}) parameters needed to describe the medium. Luckily in practice we can get away with fewer parameters and in the “best case” with only 2. For almost all practical applications we have the following cases: orthorhombic (7 C

_{ij}parameters), Transverse Isotropic (5 C

_{ij}parameters) and Isotropic (2 C

_{ij}parameters). Orthorhombic describes a finely layered medium with vertical fractures, Transverse Isotropic describes either fine layering or a set of vertical fractures only, whereas Isotropic describes a medium without any sub-seismic scale layering nor fractures.

For weak anisotropy, we use the following parameters:

Orthorhombic:

**Κ**, \( \mu \), \( \epsilon \)

^{(1)}, \( \epsilon \)

^{(2)}, \( \delta \)

^{(1)}, \( \delta \)

^{(2)}, \( \delta \)

^{(3)}, \( \gamma \)

^{(1)}, \( \gamma \)

^{(2)},

Transverse Isotropy: VTI: (

**Κ**, \( \mu \), \( \epsilon \), \( \delta \), \( \gamma \)), HTI: (

**K**, \( \mu \), \( \epsilon \)

^{(v)}, \( \delta \)

^{(v)}, \( \gamma \)

^{(v)})

Isotropy:

**K**, \( \mu \)

Unhappily, all rocks should be considered triclinic, but in reality, we can “ignore” that. We wouldn’t be able to derive the needed 21 parameters from seismic observations anyway. So, let’s forget it and restrict ourselves to at most orthorhombic or tilted orthorhombic in case of dipping structures.

One quantity we can observe quite well with modern wide- or multi-azimuth acquisition and that is the orientation of fracture sets. Namely, the NMO velocity of waves travelling perpendicular to fractures is lower than waves travelling parallel to fractures. In addition, if S-waves can be observed (multi-component land or OBC acquisition) shear-wave splitting also indicates the orientation of fractures..

**Geophysical Curiosity 3**

“*Keep it simple, you stupid*”
This slogan is of course based on the well-known political slogan “It’s
the economy, you stupid” and here refers to “why introducing anisotropy
parameters” in the description of wave propagation, given that a full
description of wave propagation is already complicated enough. The
reason is that in general we don’t honour the real Earth when we try to
model geophysical observations. In 1992, I gave an “oration” when I
became Professor at the University of Utrecht, called “Do we do justice
to the Earth” with a wink to my wife’s occupation as a judge. It must be
clear, that the title refers to all the approximation we make to make
geophysical problems tractable. If we want to model exactly the seismic
response of real rocks, that means including small-scale
inhomogeneities, we need complex mathematical algorithms and still can
only do the modelling in detail for specific, well-described cases. And
even then, you might wonder whether in practice we know or even can use
such detail.

It
is exactly for that reason that effective media are used. In these
media, the detail is “summarised/replaced” by “simple” anisotropy
parameters. It might appear “making it more complicated”, but after
having familiarised yourself with it, you realize that it makes the
description of wave propagation more realistic and especially more
applicable. Nice examples are shown in the AVA exercise in the
“Quantitative Reservoir Characterization” course.

**Geophysical Curiosity 2**

We
know that seismic energy can be propagated in the earth in two ways:
P-waves and S-waves. Now in an isotropic elastic medium the P-wave
particle displacement is in-line with the propagation direction and in
case of an S-wave the particle displacement is perpendicular to the
propagation direction. This particle displacement can be in any
direction as long as it is perpendicular to the propagation direction.

In anisotropic media, the situation is quite different.

For
P-waves the particle displacement is deviating slightly from the
propagation direction (ray direction), as the displacement is normal to
the wave-front. The ray vector is namely no longer perpendicular to the
wave-front.

For S-waves it is
even more complicated. It was noticed in field experiments that the
vertical S-wave reflection time from a horizontal interface differed
depending on the polarization direction. Hence, the velocity did depend
on the polarization. Therefore, it was concluded that two S-wave modes
exist, one with the particle displacement parallel (S-fast), the other
with the particle displacement perpendicular (S-slow) to fracture
orientation (or fine layering). In addition, the S-wave can, and that is
quite odd, **only **propagate with these two polarizations. So, if a source generates an S-wave with polarization in between, this S-wave will **not **propagate
at all, but will split into the two modes discussed before and each
mode will propagate with its own velocity. This is called “shear-wave
splitting” or “birefringence”. And again, the polarizations are not
exactly perpendicular to the propagation direction. Instead, they are
perpendicular to the P-wave polarization, which as mentioned before, is
neither in the propagation direction.

**Geophysical Curiosity 1**

When
you look at the formula for the P-wave velocity, namely \( \sqrt[]{
\frac{K+4/3 \mu }{\rho}} \), with K the elastic modulus, which I rather
call the stiffness or resistance to pure compression (only change in
volume) and μ the resistance to deformation (distortion without a change
in volume), you realize that the P-wave is not a pure compressional
wave. Pure compressional waves occur in media which have a shear modulus
μ=0, such as water and air.

*Please feel free to comment or add points of view and indicate whether you would like it to be mentioned on the website or not (email: j.c.mondt@planet.nl). *

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