1D convolution has been one of the key concepts of seismic acquisition, processing, inversion and quantitative interpretation (QI) developments. Convolutional modelling states that the migrated image is linearly related to the reflection coefficients which are then approximately related to small changes in subsurface elastic properties. 1D convolution, as a special case, assumes horizontal layering and in return comes with a great computational efficiency i.e. at each location a 1D wavelet is convolved with the reflectivity series to obtain a synthetic (or modelled) seismic trace. This is why 1D convolution is at the heart of seismic amplitude versus offset (or angle) inversion algorithms. So whilst 1D convolution correctly expresses the vertical resolution for 1D earth, it would fail to do so when dips are present when compared with migrated seismic image. Migration stretch effect on dipping structures was discussed in details in Tygel et al. (1994). 1D convolution also ignores the resolution limits of migration operator as a function of offset, depth and acquisition geometry (the so called “illumination” effects).
Overcoming the resolution limits of migration has been explored in the imaging literature by incorporating 3D filters referred to as the Green’s function for migration, the migration impulse response or Point Spread Functions (PSF). As an example, the migration deconvolution method as proposed in Yu et al. (2006) and the least-squares migration approach in Fletcher et al. (2016) both aim to enhance the resolution of the migrated image and compensate for the structural and illumination effects by using the PSFs to approximate the Hessian operator. Also, Lecomte et al. (2015) discussed in detail the application of PSFs as 3D convolutional operators to model PSDM images more accurately. A much simpler approach was introduced by Cherrett (2013) to model the effect of migration stretch on dipping structures using a constant velocity approximation. Cherrett’s constant velocity approach can be considered as a 3D convolution with a single PSF (as opposed to a grid of temporally- and spatially-varying PSFs). Lecomte et al. (2015) referred to such approximation as a PSF with “perfect illumination”.
Seismic inversion algorithms based on AVO/AVA are commonly (if not always) based on 1D convolution to forward model the synthetic, comparison with the seismic and the use of the difference to update the provided initial models of acoustic and/or elastic properties until reaching minimal difference. Consequently, given the inconsistency between the migrated seismic and synthetic due to dip and illumination effects mentioned above, a possible solution is to replace the 1D convolution forward modelling in AVO/AVA inversion and/or other QI applications with a form of 3D convolution. That means the forward modeling engine then produces a synthetic that resembles the seismic including the imaging footprints, and in return helps to invert the seismic more accurately. An alternative possibility, proposed in this paper, is to remove those effects from the input seismic. If such effects, such as migration stretch, are not accounted for, the inconsistency between the seismic and the synthetic would be injected into the inverted model so that the difference between the seismic and synthetic is minimized (after all that is the cost function for the inversion).
Finally, there are some inversion algorithms based on spectral shaping, like colored Inversion (Lancaster and Whitcombe, 2000), which are commonly used in QI. Although, in colored inversion, there is no convolution as in the form of convolving a 1D reflectivity with a wavelet, convolution is still present in the form of convolving a filter with the seismic so that the output frequency spectrum matches the earth’s impedance spectrum. This filter though assumes a homogenous horizontally stratified model which means when applied to seismic with structural dip, it would bias the result due to the presence of migration stretch. This is effectively the same inconsistency as what we had for seismic inversion with a difference that it acts in the spectral domain. In this paper, I propose a first order deconvolution method that removes the migration stretch of dipping events. This is a post-stack correction and can be considered as a “conditioning” step before the inversion whether that is model-based or colored inversion.
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