# Poststack inversion methods

Poststack inversion methods generally refer to the various workflows used to transform stacked seismic data into quantitative rock physics parameters. From poststack inversion, the outcome is usually acoustic impedance, whereas prestack inversion may result in both acoustic impedance and shear impedance.  Poststack inversion methods include sparse spike inversion, model-based inversion, recursive inversion, and colored inversion. These methods are all grouped as deterministic methods as they do not involve probabilistic measures.

## Recursive inversion

This is the simplest and oldest form of inversion. It assumes that acoustic impedance can be calculated using the recursive equation:

${\textstyle Z_{i+1}=Z_{i}*{\frac {1+r_{i}}{1-r_{i}}}\ }$ where Z is the acoustic impedance and r is the reflection coefficient. By applying this recursive equation, seismic data can be directly inverted for acoustic impedance. However, these algorithms have various problems:  

• The result is band-limited (i.e. limited to seismic frequency ranges).
• The algorithms require zero-phase seismic
• Tuning effects are not removed as the wavelet in the seismic is ignored
• The recorded seismic trace does not directly correspond to a reflectivity series, rather it corresponds to the convolutional model of the wavelet and reflectivity

## Model-based inversion

In a model-based inversion, a simple initial acoustic impedance model is convolved with the wavelet to obtain a synthetic response that is compared with the actual seismic trace. The acoustic impedance model is altered iteratively until the difference between the inverted trace and the seismic trace is reduced to a threshold value. A model with a very small difference is accepted as a solution. An advantage of model-based inversion is that it gives satisfactory results, even with limited well control and poor quality seismic. The seismic dataset itself acts as the guide for inversion and a wavelet can be easily derived straight from the seismic.  The least-squares inversion method is a type of model-based inversion where the threshold value is the smallest least-squares error. Model-based inversion starts with an initial model that is updated iteratively until the residual trace is minimized to a threshold value. 

## Sparse spike inversion

Sparse spike inversion uses a minimum number of acoustic impedance interfaces to simulate a seismic trace that models subsurface reflectivity. It relies on the assumption that the reflection coefficient series associated with the acoustic impedance is sparse. That is, the seismic trace can be modeled with fewer reflection coefficients; only the large spikes (i.e. large impedance contrasts) are meaningful. Ideally, convolving the simulated seismic trace with the wavelet should reproduce the real seismic response. The goal is to obtain a high-resolution impedance profile from band-limited seismic data that directly relates to lithology or rock properties of the formation.  The inversion algorithm originally worked on a trace-by-trace basis, but now involves a multi-trace approach that incorporates low frequency variation from well logs and a priori geological information to constrain the solution and improve the inversion results.  There are two main types of sparse-spike inversion methods: linear programming and maximum likelihood.

### Linear Programming

This algorithm first extracts an estimate of the reflectivity by using frequency domain constraints to recover the high frequencies of the seismic spectrum. The reflectivity series is then integrated under the initial model, which then creates a sparse reflectivity that produces the best match between the synthetic and the seismic trace. Often, the L1 norm of the reflection is minimized to find the best model. 

### Maximum Likelihood

For this algorithm, the reflectivity series is perturbed on the basis of an initial model. A sparse reflectivity series is estimated by adding reflection coefficients until an optimal set has been found. The broadband reflectivity is then perturbed until it matches the seismic trace to a threshold value. This method relies on the assumption that the wavelet of the seismic is known and is the current wavelet. The top curve indicates the error. The middle curves are the synthetics derived from the model. The bottom curves are the true model (black) and the inverted model (red). This example used the linear programming method of inversion. 

## Colored inversion

Colored inversion compares the amplitude spectrum of the seismic dataset with that of well logs. An operator is designed that matches the amplitude spectrum of the seismic with the relative acoustic impedance data at the wells. This operator is then convolved with the seismic traces, resulting in an assumed acoustic impedance equivalent.  The advantages of this method are that it's simple, fast, and robust even in the presence of noise, making it ideal for quick and preliminary inversion. The disadvantages are that the output is rather imprecise, it's band-limited, and the input seismic must be zero phase as a wavelet is not used. Also, it assumes that the reflectivity spectra of one well is representative of the whole region. An inversion operator is derived from the amplitude spectrum and then applied to seismic so that the amplitudes are in agreement with well data. In this example, the lithological units are easily recognizable in the colored inversion.