Theory of JMI

Joint Migration Inversion (JMI) was proposed as one of the methods to overcome the above-mentioned limitations in FWI. In Figure 1 the flow diagram of the JMI process is shown. The main engine of the JMI method is a forward-modeling process, called Full Wavefield Modeling (FWMod) based on the parameters reflectivity and velocity, which is described in Berkhout (2012), Davydenko and Verschuur (2012), and Berkhout (2014a). With this recursive and iterative two-way modeling process, from the current estimate of the reflectivities and the velocity model, the seismic reflection responses are being generated. In this modeling process, multiples and transmission effects are included. Next, the modeled responses are compared to the measured ones and the resulting difference data, being the residual of the inversion, is back-projected into the parameter space via reverse extrapolation of the residual into the medium and a subsequent transformation of this residual energy into updates of reflectivity and velocity. The parameters are updated, from which new seismic responses are modeled, yielding the next version of the residual data. In this way, the residual is slowly driven to zero (Berkhout, 2014c; Staal and Verschuur, 2013; Berkhout, 2012; Staal and Verschuur, 2012). We can treat the whole procedure as minimizing the following objective function:

$$J_{JMI} = \sum\limits_{\omega} \vert\vert \mathbf{D}^-\left(z_0\r... ..._{mod} \left( z_0,\mathbf{r}, \mathbf{v} \right) \vert\vert _2 ^2 + constraint,$$ (2)

where the $\vert\vert . \vert\vert^2$ describes the sum of the squares of the values (i.e., the energy), $\mathbf{D}^-\left(z_0\right)$ is the collection of all recorded surface seismic shot records in the $\left( x,\omega \right)$ domain, and $\mathbf{P}^-_{mod} \left( z_0,\mathbf{r}, \mathbf{v} \right)$ describes the modeled surface shot records as a function of reflectivity $\mathbf{r}$ and velocity $\mathbf{v}$. Note that by using the reflectivity and propagation velocity as parameters, density variations are implicitly included in $\mathbf{r}$. Even though JMI has a reduced non-linearity, the velocity update still suffers from local minima. With a proper constraint, JMI can lead to a more accurate inverted velocity, and therefore a better inverted reflectivity.

jmi_scheme
Figure 1. JMI flow chart.