Colored inversion

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Colored inversion
Article  in  The Leading Edge · October 2017
DOI: 10.1190/tle36100858.1
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Martin Blouin
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858 THE LEADING EDGE October 2017
G e o p h y s i c a l T u t o r i a l — C o o r d i n a t e d b y M a tt H a l l
Colored inversion
Whether it is deterministic, band-limited, or stochastic, seismic
inversion can bear many names depending on the algorithm
used to produce it. Broadly, inversion converts reflectivity data to
physical properties of the earth, such as acoustic impedance (AI),
the product of seismic velocity and bulk density. This is crucial
because, while reflectivity informs us about boundaries, impedance
can be converted to useful earth properties such as porosity and
fluid content via known petrophysical relationships.
Lancaster and Whitcombe (2000) published a fast method
for band-limited inversion of seismic data known as colored
inversion (CI) that generated widespread interest among inter-
preters. Recognizing that the popular sparse-spike inversion
process could be approximated by a single operator, yielding
relative impedance via simple convolution with the reflectivity
data, the authors showed that this operator can be derived from
well logs. Like other inversions, CI can help remove the smearing
effects of the seismic wavelet and enhance features such as thin
beds and discontinuities. What’s more, since CI is directly linked
to seismic data, the relative impedance it produces can be used
as a base for comparison with other inversion to see what kind
of information is introduced by numerical constraints or the
low-frequency model.
In this tutorial, we follow the steps presented by Lancaster
and Whitcombe in their 2000 expanded abstract to achieve the
so-called “fast-track colored inversion.” Using open-source algo-
rithms, we describe all the steps to go from reflectivity data to
inverted cubes:
1) Fit a function to the log spectrum(s).
2) Get a difference spectrum by substracting the seismic
spectrum.
3) Convert the difference spectrum to
an operator.
4) Convolve the operator with the
stacked seismic.
5) As a quality control (QC) step,
check the residuals by comparing
log and AI section spectrums.
The idea of this tutorial is to achieve
a CI without any external software, just
Python code with the NumPy and
SciPy libraries. You can read and run
the code for yourself from the repository
at github.com/seg/tutorials-2017. This
tutorial focuses on presenting the whole
workflow in the simplest fashion rather
than trying to optimize parameters to
recover interpretable features. Prior to
Martin Blouin1
and Erwan Gloaguen2
running the workflow on your data, noise attenuation should be
performed to ensure that the inversion process recovers frequencies
associated only with the earth.
The data set
For our demonstration, we use one inline from the 1987 Dutch
F3 volume (Figure 1) plus the AI log from the F02-1 well. Good
descriptions of the geologic setting of this data set can be found
in Sorensen et al. (1997) and Overeem et al. (2001). We use the
dip-steered median filter stacked data set to get reduced noise on
our input.
Fit a function to the log spectrum
Walden and Hosken (1985) observed that the reflectivity
sequences in sedimentary basins display a logarithmic decay in
amplitude as frequency increases. For the first step of our inversion,
we look at the reflectivity spectrum and make sure it behaves as
predicted. We load the F02-1 well, convert it to the time domain,
and calculate the spectrum (Figure 2).
n_log = AI_f021.shape[0]
k_log = np.arange(n_log-1)
Fs_log = 1 / np.diff(time_f021/1000)
T_log = n_log / Fs_log
freq_log = k_log / T_log
freq_log = freq_log[range(n_log//2)]
spec_log = np.fft.fft(AI_f021) / n_log
spec_log = spec_log[range(n_log//2)]
We would like to simplify the spectrum, so we proceed to a
regression. To make the process more robust, CI packages offer
1
GeoLEARN Solutions.
2
INRS-ETE.
https://doi.org/10.1190/tle36100858.1.
Figure 1. F3 dip-steered median filtered stacked seismic data at inline 362 with well location and bounding
horizons of the inverted region.
Downloaded10/02/17to198.168.46.102.RedistributionsubjecttoSEGlicenseorcopyright;seeTermsofUseathttp://library.seg.org/

October 2017 THE LEADING EDGE 859
one for the error function. We then use scipy.optimize to find
the best fit, shown in Figure 3.
def linearize(p, x):
return p[0] * x**p[1]
def error(p, x, y):
return np.log10(y) - np.log10(linearize(p, x))
args = (freq_log[1:2000], np.abs(spec_log[1:2000]))
qout, success = optimize.leastsq(error,
(1e5, -0.8),
args=args,
maxfev=3000)
Compute the difference
Now we have a continuous function that mathematically
approximates the well-log spectrum. The next major step of the
workflow is to compute an operator that is representative of the
difference between the log spectrum and the seismic spectrum.
We first define some boundaries to our modeled spectrum
in the frequency domain. This is a critical part that will have
great influence on the end result. We define a simple function to
generate a 50-points Hanning-shaped taper at both ends of the
spectrum; the result is shown in Figure 4. Then we take seismic
traces at the well location, average them, and repeat the procedure
we did on the well to get the spectrum. Figure 4 shows the resulting
modeled spectrum. In this example, taper windows of 0–5 Hz
and 100–120 Hz have been defined. It is important to note that
our end result will be influenced greatly by this choice, and care
should be applied at this stage.
After this step is completed, we simply substract the seismic
spectrum from our modeled spectrum (Figure 5).
Convert to an operator
Now, continuing with Lancaster and Whitcombe’s methodol-
ogy, we can derive an inversion operator. We move back to the
time domain with an inverse Fourier transform, then shift zero
time to the center of the time window. Finally, we rotate the phase
by taking the quadrature of the signal, represented by the imagi-
nary component and shown in Figure 5.
gap = spec_log_model - spec_seismic_model
operator = np.fft.ifft(gap)
operator = np.fft.fftshift(operator)
operator = operator.imag
Convolve operator with seismic
Once the operator is calculated a simple trace-by-trace convolu-
tion with the reflectivity data is needed to perform CI. NumPy’s
apply_along_axis() applies any function to all columns of an
array, so we can pass it the convolution:
def convolve(t):
return np.convolve(t, operator, mode='same')
ci = np.apply_along_axis(convolve, axis=0, arr=seis)
Figure 2. F02-1 well data. (a) AI log and (b) calculated power spectrum.
Figure 3. F02-1 well spectrum approximated by a linear function in log space.
Figure 4. Comparison of well log, seismic, and modeled frequency power spectrum.
you the option of averaging this spectrum over all the available
wells, but we can skip that as we are working with only a single
well. To proceed to a linear regression, we define two simple
functions, one that serves as the linearization of the problem and

860 THE LEADING EDGE October 2017
Since the relative impedance result
contains no low frequencies, it does not
inform us about trends, so it makes more
sense to look at the result in a single
formation. We will focus on an area of
interest of about 300 ms in thickness
between two horizons; Figure 6 presents
the resulting relative impedance section
scaled (0–1) in this area.
Check the residuals
To make sure we did a good job
defining our operator, we now look at
the spectrum from the relative impedance
result at the well location to see if we
achieved a good fit with the well-log
spectrum (Figure 7). It is to be noted
that this fit was achieved with some
arbitrary decisions, and the workflow
could be iterated upon to try to minimize
the difference between the two spectrums.
Conclusion
We have presented a straightforward application of CI using
only Python with the NumPy and SciPy libraries. This open-
source workflow is documented at github.com/seg. It is called
a “robust” process in the literature, but it is somewhat sensitive
to the chosen frequency range. There are also choices to be made
about the window of application and the number of traces used
to compute the seismic spectrum. Notwithstanding all this, the
process is simple and fast and yields informative images to
interpreters.
Acknowledgment
Thanks to Matt Hall for useful comments and suggestions
on the workflow and manuscript.
Figure 5. (a) Resulting difference spectrum. (b) Associated operator.
Figure 6. Relative impedance result in a single stratigraphic unit.
Figure 7. QC of the CI workflow by comparison between input and output spectrums.

October 2017 THE LEADING EDGE 861
Corresponding author: martin.blouin@geolearn.ca
References
Lancaster, S., and D. Whitcombe, 2000, Fast-track ‘coloured’ inver-
sion: 70th
Annual International Meeting, SEG, Expanded
Abstracts, 1572–1575, https://doi.org/10.1190/1.1815711.
Overeem, I, G. J. Weltje, C. Bishop-Kay, and S. B. Kroonenberg,
2001, The Late Cenozoic Eridanos delta system in the Southern
North Sea Basin: A climate signal in sediment supply?: Basin
Research, 13, no. 3, 293–312, https://doi.org/10.1046/j.1365-2117.
2001.00151.x.
Sørensen, J. C., U. Gregersen, M. Breiner and O. Michelsen, 1997,
High frequency sequence stratigraphy of upper Cenozoic deposits:
Marine and Petroleum Geology, 14, no. 2, 99–123, https://doi.
org/10.1016/S0264-8172(96)00052-9.
Walden,A.T.,andJ.W.J.Hosken,1985,Aninvestigationofthespectral
propertiesofprimaryreflectioncoefficients:GeophysicalProspecting,
33, no. 3, 400–435, https://doi.org/10.1111/j.1365-2478.1985.
tb00443.x.
Suggestions for further reading
dGB Earth Sciences and TNO, 1987, F3 data set. Data set accessed
31 August 2017 at http://opendtect.org/osr/.
Francis, A., 2014, A simple guide to seismic inversion: GEOExPro,
10, no. 2, 46–50.
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