**Supplemental material and extensive references for **

**"Making the Most Out of the Least (Squares Migration)"**

*YUNSONG HUANG,** **GAURAV DUTTA, **WEI DAI,** **XIN WANG, JIANHUA YU, and** GERARD SCHUSTER*

* *

Due to space limitation, very few references and only a brief introduction can be accommodated in our article in the special issue on Migration and Imaging, TLE, Sept. 2014. Here, more extensive references and background material are provided. Only the text that includes such citations, references, and background material are included in this webpage. We refer the readers to our TLE article for the rest of the text and the figures.

**Theory and
background**

Least-squares migration (Nemeth et al., 1999; Duquet et al., 2000) is an imaging algorithm (an excellent overview can be found in Schuster (2011)) equivalent to linearized waveform inversion (Lailly, 1984) that seeks to find the best reflectivity image given a fixed smooth background velocity model. This is achieved by minimizing an objective function

(1)

where
is the recorded reflection
data, represents the linearized
forward modeling operator, and the regularizer (Wang and Sacchi, 2007) encourages
the solution to adopt specified characteristics of .
The operator represents the solution of
the wave equation for a smooth background velocity model embedded with a sharp
reflectivity distribution represented by **m**.

The
Levenberg-Marquardt solution to equation 1 is formally given as, whereis a damping
parameter. Unfortunately, the Hessian inverse ** **is
too expensive to compute or store, so either an iterative solution in the data
domain (Nemeth et al., 1999) or an approximate inverse in the image domain (Hu
et al., 2000; Yu et al., 2006; Tang, 2009) is employed**. **In the image
domain, the Hessian inverse is
approximated with the migration deconvolution (MD) operator (Hu et al., 2000;
Yu et al., 2006). Then the least-squares migration (LSM) image is given by, where the contribution from the
regularizer is dropped for brevity. The benefits of the image domain approach
include: (1) once the approximate inverse Hessian is obtained, the LSM image is
computed at the cost of about one migration; and (2) MD can be computed in a
target-oriented manner where the computation is limited to a small target volume
of interest. The limitation is that the MD operator is an approximation that
assumes a local layered medium around the image point of interest.

In the data domain, on the other hand, the data misfit drives the iterative updates of the trial reflectivity model, such as

(2)

where represents the iteration index, is the step length, and is the preconditioner. If this preconditioner is inadequate, a conjugate gradient or quasi-Newton method is used instead for the iterative update. When and the initial model is assumed to be smooth (or a reflectivity model where only the direct wave is generated), then is the migration image computed by standard migration (Lailly, 1984; Mora, 1987). Further iterations will adjust the reflectivity model (not the migration velocity model) to enhance resolution and partly mitigate migration artifacts due to trace aliasing, finite aperture effects, geometric spreading, defocusing, and ringiness in the source wavelet. The aforementioned artifacts in the migration image are largely suppressed by LSM because such artifacts will not lead to minimization of the normed residual. The rest of this article focuses on the data domain implementation of LSM.

LSM can be implemented with Kirchhoff migration, one-way wave equation migration (Kuehl and Sacchi, 1999), or reverse time migration (Plessix and Mulder, 2004; Guitton et al., 2006; Dai et al., 2010; Yao and Jakubowicz, 2012). Typically, Kirchhoff migration and one-way wave equation migration produce fewer artifacts because they only smear received energy along the tentative reflectors coincident with the migration ellipse. In contrast, reverse time migration (RTM) automatically generates upgoing reflections from reflectors, and so the received energy is also smeared along the reflection wavepaths (known as the “rabbit ears”) to give rise to low frequency artifacts (Zhang and Sun, 2008). One way to suppress such artifacts is to smooth the migration velocity model. In the context of least squares reverse time migration (LSRTM), Born modeling is employed which assumes a weak scattering approximation.

...

**Curbing the
computational cost**

...

To reduce the cost of LSM, shot gathers can be encoded and blended together to form one supergather or several sub-supergathers (Romero et al., 2000; Tang and Biondi, 2009; Neelamani et al., 2010; Huang and Schuster, 2012; Dai and Schuster, 2013). To this end, there are at least five strategies for source-encoding: (1) random time series (Romero et al., 2000; Neelamani et al., 2010), (2) random polarity (Krebs et al., 2009), (3) random time shift (Dai and Schuster, 2009; Tang and Biondi, 2009), (4) random frequency division (Huang and Schuster, 2012), and (5) plane-wave encoding (Zhang et al., 2005; Etgen, 2005; Dai and Schuster, 2013; Wang et al., 2013).

...

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