# Seismic inversion (post-stacking vs pre-stacking)

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Seismic inversion, most of the time used in oil and gas industry, is the process used to reconstruct earth properties. It combines seismic and well data to predict rock properties (lithology, fluid content, porosity) across a survey. These rock properties can be used to identify hydrocarbon and reservoir.

Seismic Inversion Workflow. Credit: Farfour [1]

## Description

### Importance of impedance attribute

Many of these rocks properties can be identified by using well log data (gamma ray, water saturation, shale volume) or seismic data (more difficult to obtain). However, with seismic inversion we can obtain other rocks properties such as impedance and its attributes (P-impedance, S-impedance, Poisson’s Ratio, Vp/Vs, Lambda*Rho, Mu*Rho). Those properties are linked to fluid content, porosity, lithology. For example, for a given lithology, if we know the P-impedance, we can predict its porosity. From this relationship, by combining impedance at the well and the impedance calculated from seismic data, we can predict hydrocarbon across the survey.

### How to calculate impedance

P-impedance= density *P-velocity

S-impedance= density*S-velocity

Note that all other attributes can be calculated from the impedance.

From well log data we used density and velocity data to get the impedance and the Poisson’s ratio. From Seismic data, we obtain impedance by using seismic inversion which converts seismic from a boundary property to a layer property. Indeed, well data show us properties of the rock’s layer and seismic data show us information about the boundary between rock layers. In seismic inversion, the seismic data is changed into impedance.

### How seismic inversion works?

Seismic Inversion removes the imprints of the wavelet in the seismic data by deconvolution and then by converting the result into impedance.

Mechanism of Seismic Inversion. Source: Youtube Screenshot

## Post-stacking inversion

Seismic inversion can be divided into two techniques which are post-stacking and pre-stacking inversion.

The first technique, Post-Stacking inversion is the most common approach used for inversion. This technique transforms a single seismic information volume into an acoustic impedance volume by using seismic data, well data, and basics knowledge in stratigraphy for interpretation. By removing the wavelet from seismic data, we assist at the creation of a high resolution image of the subsurface.

Post-stacking Inversion Workflow. Credit: Petrologic [2]

The post-stacking methods are colored inversion, model-based inversion, sparse spike inversion and Band-limited impedance inversion.

### Colored Inversion

Colored Inversion (CI) is a process to approximately match or convolve the amplitude spectrum of a seismic with the acoustic impedance spectra issued from well data. It calculates its own inverted operator based on the frequency of seismic and well data. This method is very easy to use, fast, and strong even if there is noise. It can also be applied with or without well data and background model. The inconvenient is that the seismic input must be at zero-phase and it considers the reflectivity data of one well as the data of all the region.

Color inversion example. An operator is derived from the spectrum amplitude and then applied to the seismic. The lithology is easy to recognize. Credit: Veeken [3]

### Model-based inversion

In this method, the seismic trace (initial model) is convoluted with a wavelet to get a synthetic seismic trace. Then the impedance is confronted to many iterations until the difference between the inverted trace and the initial trace is reduced to a limit value.

Seismic trace = (wavelet*Reflectivity) +noise

[[File:Mbi.png|thumb|center|Model-based inversion example. It starts with an initial model and ends with a residual model. The initial model is process by iteration. Credit: Veeken [3]

### Sparse Spike Inversion

Sparse spike inversion assumes the seismic trace models the subsurface reflectivity by using small number of acoustic impedance interface. It presents two techniques: linear programming and maximum likelihood. Note that it final purpose is to obtain high resolution impedance.

#### Linear programming

It is an algorithm which extracts an estimate of the reflectivity by using frequency domain constraints to recover the high frequencies of the seismic spectrum. This reflectivity is by the end incorporated under the initial model, where the sparse reflectivity is created. This method is to restore the impedance by minimizing error between the modeled and the initial trace.[4]

#### Maximum Likelihood

The particularity of this algorithm is to disturb the reflectivity from the seismic trace. Assuming the wavelet is known, its job will be to add reflections coefficients until a best match is found.[4]

### Band-Limited Impedance inversion

Band-limited impedance inversion (BLI) changes post stack seismic data into impedance, density and P-wave velocity. This method is defined by the relationship between the seismic trace and seismic impedance.[4]

## Pre-stacking inversion

To obtain multiple impedance attributes we do pre-stacking. Pre-stacking inversion changes seismic into P-impedance, S-impedance, and density by integrating well and seismic data.

Pre-stacking Inversion Workflow. Credit: Petrologic [5]

The two techniques used on pre-stacking are: simultaneous and elastic inversion. Note that those technics required global wavelets and background model.

### Simultaneous inversion

Simultaneous pre-stacking inversion is defined by [4]:

### Elastic inversion

Recall:

Aki-Richards equation [4]

Elastic Impedance [4]

Elastic inversion model:

Elastic Inversion Model. Source: Youtube Screenshot

## Differences and Similarities

 Similarities Differences The stratigraphic interpretation is easier on impedance data because P-impedance is a layer property. High resolution of layers by reducing wavelet effects, side lobes, and tuning. P-impedance can be directly computed and compared to well data. Relation between porosity and acoustic impedance. Color inversion can be used with or without a well data and a background model. Post-stacking is cheap and easier to process Pre-stacking is more efficient to identify lithology and fluid content Pre-stacking identifies both acoustic impedance and shear impedance Pre-stacking is the best method to estimate Vp/Vs ratio. Post-stacking uses a single seismic trace while Pre-stacking uses a linear model of AVO.

## References

1. Farfour, M., Yoon, W. and Kim, J. (2015). Seismic attributes and acoustic impedance inversion in interpretation of complex hydrocarbon reservoirs. Journal of Applied Geophysics, 114, pp.68-80.
2. Petrologic.dmt.de. (2018). Post-Stack Inversion: Petrologic Geophysical Services GmbH. [online] Available at: http://petrologic.dmt.de/services/post-stack-inversion.html [Accessed 24 Apr. 2018].
3. Veeken, P. C. H., and Da Silva, M., 2004, Seismic inversion methods and some of their constraints: First Break, 22, 47-70, accessed October 28, 2017; https://www.researchgate.net/publication/277392423_Seismic_Inversion_Methods_and_some_of_their_constraints
4. Maurya, S.P., and Sarkar, P., 2016, Comparison of Post stack Seismic Inversion Methods:A case study from Blackfoot Field, Canada: International Journal of Scientific & Engineering Research, 7, 1091-1101, accessed October 28, 2017; https://www.ijser.org/onlineResearchPaperViewer.aspx?Comparison-of-Post-stack-Seismic-Inversion-Methods-A-case-study-from-Blackfoot-Field,-Canada.pdf
5. Petrologic.dmt.de. (2018). Pre-Stack Inversion: Petrologic Geophysical Services GmbH. [online] Available at: http://petrologic.dmt.de/services/pre-stack-inversion.html [Accessed 24 Apr. 2018].