Seismic full waveform inversion (FWI) is a powerful method for providing a quantitative description of the subsurface properties by iteratively minimizing an objective function that measures the misfit between observed and predicted data in the least-squares sense (Tarantola, 1984). However, FWI is a non-linear and ill-posed inverse problem and its objective function may suffer from local minima that are not informative about the true parameters (Virieux and Operto, 2009; Fu and Symes, 2017b; Chen et al., 2016; Fu and Symes, 2017a). Using regularization methods is an effective way to mitigate this ill-posedness and non-uniqueness of FWI.

Joint Migration Inversion (JMI) was proposed as one of the methods to overcome the above-mentioned limitations in FWI (Berkhout, 2014b; Verschuur et al., 2016; Staal, 2015). It is an inverse algorithm to automatically derive both velocity and reflectivity based on the full wavefield modeling (FWMod) process (Berkhout, 2014a) that takes into account transmission effects and surface/internal multiples. In the FWMod procedure, the velocity only affects the kinematics without any scattering effect in the modeling operators and the reflectivity only deals with scattering effects. These characteristics lead to a reduced non-linearity in the inversion process. Even though not as severe as FWI, the velocity update may still suffer from being trapped in local minima. With the help of regularization, JMI can get a more accurate inverted velocity, and thus achieve a better inverted reflectivity (Qu and Verschuur, 2016b,2017b).

The popular regularization methods include: quadratic L2-norm-based regularization, such as Tikhonov regularization (Hu et al., 2009), and laplacian smoothing (Guitton et al., 2012; Qu and Verschuur, 2017a,2016a), which tend to produce models with blurred discontinuities; non-quadratic L1-norm-based regularization, such as total variation (TV) (Qu and Verschuur, 2016b; Anagaw and Sacchi, 2011), smooths the model by enhancing the sparsity of the spatial gradient of the velocity, thereby preserving its edges. However, regular TV regularization only tends to reduce the horizontal and vertical gradients of each gridpoint in the model regardless of their structural direction. Thus, TV is not suitable where the local geologic structure has a dominant structural direction. Unlike general digital images, the spatial changes of the seismic model always have some specific geological structures, like tilted layers, faults, or edges of a salt body (Wu and Bai, 2018; Bai et al., 2018; Chen et al., 2020,2017). Bayram and Kamasak (2012) proposed a directional TV method and applied it to digital image denoising. However, they only consider one single dominant direction for all pixels, which is obviously ineffective for complex-textured geologies. Therefore, we propose a directional TV constraint based on a rough estimate of the subsurface image.

The paper is organized as follows: we first briefly introduce the basics of FWI and JMI. Next, we formulate the conventional TV and the proposed directional TV. Finally, using two complex Marmousi-model-based examples, we show that the proposed method is more effective than the alternative methods, when the model contains tilted layers and steep faults. At the end, using the JMI-based example, we also show that the L1 directional TV works better than the L2 directional Laplacian smoothing regarding the preservation of edges and the steering of the update away from the local minimum. Note that this paper is an extended version of work published in Qu et al. (2017).