User:Bbenne1/sandbox

Deconvolution is a fundamental step of seismic data processing generally aimed at compressing seismic wavelets to spikes and suppressing multiple reflections (Menanno, G.M. et al. 2012). Deconvolution is a fundamental step of seismic data processing generally aimed at compressing seismic wavelets to spikes and suppressing multiple reflections (Menanno, G.M. et al. 2012).

Deconvolution
Deconvolution is a filtering technique used to increase resolution to obtain a more interpretable seismic section. This is achieved by compressing the basic wavelet in the recorded seismogram and by attenuating reverberations and short-period multiples (Sheriff, 2004). Deconvolution usually involves convolution with an inverse filter. The idea is that this will undo the effects of a previous filter, such as the earth or the recording system (Sheriff, 2004). By far the most widely used method of calculating the deconvolution filter is Wiener filtering (Robinson, 1957; Robinson et al. 1967, 1980).

Convolution
The simplest convolutional model regards the trace as the convolution of the effective seismic wavelet with the earth's reflection coefficients (Saggaf, M. et al. 2000). The model approximates the earth by a linear system and is used to explain how the seismic trace is formed (Al-Shuhail, 2012). In the convolutional model, the system’s earth’s response e(t) is a series of impulses corresponding, in time and amplitude, to the reflection coefficients at layers boundaries, also known as the reflectivity series (Al-Shuhail, 2012). The system’s input, w, is the source wavelet (Al-Shuhail, 2012). Reverberate multiples and propagation effects are also often included in this effective wavelet (Robinson, 1985). A noise component n, if present, is additive; hence, the seismic trace becomes: s = w * r + n (Al-Shuhail, 2012). Or more simply, if we assume noise is negligent and denote the trace by s, the seismic wavelet by w, and reflectivity by r, we have s = w*r, where * denotes the convolution operator (Saggaf, M. et al. 1999).

The basic convolutional model of the seismic trace assumes the following (Al-Shuhail, 2012): (1) The earth is made up of horizontal layers of constant velocity. (2) The source generates a P-wave, which is reflected on layer boundaries at normal incidence. (3) The source waveform is stationary. That is, it does not change its shape as it travels in the subsurface. (4) The noise component n is zero. (5) The Earth’s reflectivity r is random. (6) The seismic wavelet is a minimum-phase wavelet.

Application of Spiking Deconvolution
Often, one or more of the assumptions of convolution might not be satisfied; this is when deconvolution is used (Al-Shuhail, 2012). If the source wavelet in known, then the deconvolution is deterministic and we use inverse filtering, spiking deconvolution, to find the earth response (Al-Shuhail, 2012). Spiking deconvolution shortens the embedded wavelet and attempts to make it as close as possible to a spike (Sheriff, 2004). This is also called whitening deconvolution, because it attempts to achieve a flat, or white, spectrum. A process is said to be white noise if it consists of uncorrelated random variables (Saggaf,M. M. 1999). This kind of deconvolution may result in increased noise, particularly at high frequencies (Sheriff, 2004).

Objective of Seismic Deconvolution
The objective of seismic deconvolution is to design a filter that recovers the reflection coefficients from the trace, effectively removing (or compressing) the wavelet (Saggaf, M. et al. 2000). This means that we are eliminating w and leaving only r in s. That is, we want to find a filter f(t) such that: w * f = d (Al-Shuhail, 2012).

Spiking Filtering Process
This is a general schematic diagram of spiking filtering adapted from "Deconvolution" by Anton Ziolkowski.

Mathematical and Physical Explanations
Inverse filtering (spiking deconvolution) · The aim of this process is to compress the source wavelet w(t) into a zero-lag spike of zero width (i.e., d(t)). This means that we are eliminating w(t) and leaving only e(t) in s(t). · That is, we want to find a filter f(t) such that:
 * w(t) * f(t) = d(t) (3.3)

· f(t) is called the inverse filter for w(t).

· Taking the FT of equation (3.3):
 * W(f) F(f) = 1, (3.4)

where W(f), F(f), and 1 are the FTs of w(t), f(t), and d(t), respectively. · From equation (3.4):
 * F(f) = 1/W(f) = [1/|W(f)|] exp[-fw(f)],
 * |F(f)| = 1/|W(f)|, and
 * Φf(f) = -Φw(f),

where |W(f)| and fw(f) are the amplitude and phase spectra of w(t) and |F(f)| and ff(f) are the amplitude and phase spectra of f(t). · Therefore, the amplitude spectrum of the inverse filter is the reciprocal of that of the source wavelet whereas its phase spectrum is the negative of that of the wavelet. · The deconvolution is accomplished by convolving the inverse filter f(t) with the seismic trace s(t):
 * f(t) * s(t) = [f(t) * w(t)] * r(t) = δ(t) * r(t) = r(t), (3.5)

which is the earth’s response that we want to extract from the seismic trace. · The inverse filter of a minimum-phase wavelet has a minimum phase, too. · For small wavelets, the z-transform is used and equation (3.4) becomes:
 * W(z) F(z) = 1,

which means that: F(z) = 1 / W(z).

Example application
The objective of seismic deconvolution is to design a filter that recovers the reflection coefficients from the trace, effectively removing (or compressing) the wavelet (Saggaf, M. et al. 2000). The code shown below is taken from a lab exercise given by Professor Juan Lorenzo of Louisiana State University.

Conclusion
Deconvolution has been an essential part of seismic data processing since its introduction by Robinson 1954 (Margrave, G.F. et al. 2011). At present, the full capability of multicomponent seismology has not been exploited because the three-recorded components are usually processed separately (Menanno, G.M. 2012). Menanno et al., 2012 propose a method of multicomponent deconvolution based on quaternion algebra, an extension of complex numbers introduced by Hamilton (1844) that is able to jointly treat all the components, thus taking into account the vectorial nature of the data. (Menanno, G.M. 2012). Synthetic and real data examples from Menanno et al. 2012 show that quaternion deconvolution, either spiking or predictive, generally performs superiorly to the standard (scalar) deconvolution because it properly takes into account the vectorial nature of the wavefields. This provides a better wavelet estimation and thus an improved deconvolution performance, especially when noise affects differently the various components (Menanno, G.M. 2012).