# Vertical Seismic-Profile Data Local Full-Waveform Inversion Based on Marchenko Redatuming
## 1 Introduction  
The inversion of seismic data is one of the most effective methods to characterize reservoirs. In general, the conventional input for wave impedance inversion is a post-stack seismic profile or seismic volume, which assumes that all diffracted energy has been focused on the diffracted object by the migration process [1]. In practice, inaccuracies in the velocity model are common, leading to conflicts with that assumption. The seismic responses corresponding to small-scale reservoirs (like caves, faults, and fractures) with high heterogeneous patterns are mainly generated from diffraction [2, 3]. These lead to the inability of conventional inversion methods to correctly characterize these small-scale reservoirs. How to make sensible use of seismic full waveform information is the key to inversion of these complex reservoirs. [PERSON] (1984) proposed a generalized least squares-based full waveform inversion (FWI) method in the time domain [4, 5]. The method reconstructs the model parameters by analyzing the complete seismic waveforms (including amplitudes and travel times). With the high accuracy of the inverted model reconstructed by FWI and the enhanced computational power, the FWI method has become a focal area in current geophysical research. [PERSON] (1999) proposed the frequency-domain FWI method which directly performs inversion for various frequency components. Compared with time-domain inversion, frequency-domain FWI makes multiscale inversion more convenient. Generally, in the frequency domain, the multiscale approach is implemented by the successive inversion of single frequencies or groups of selected frequencies from low to high [6, 7, 8]. High-frequency FWI has the potentialto recover high-resolution subsurface model parameters [8; 9]. The results of this method can help in identifying fine reservoir characteristics and in interpreting complex subsurface structures. However, high-frequency FWI requires tremendous computer resources because of its small scale spatial and time sampling to ensure the stability of the wave propagating simulation [10]. However, for reservoir characterization, we are usually interested in a small range. Therefore, the inversion of the entire model is not needed. Moreover, it is difficult to perform FWI on deeply buried reservoirs under complex overburdens because the strong scattering from the overburden tends to obscure the reflections from the region of interest [11]. Target-oriented local inversion methods were developed to focus the inversion on a target area [12; 13; 14; 15]. [PERSON] et al. developed a time-lapse seismic local FWI method based on the wavefield injection technique [12; 13; 16]. However, this method requires accurate model parameters of overburden to simulate [PERSON]'s function on the subsurface virtual receivers [12; 13]. [PERSON] and [PERSON] obtained virtual data at a subsurface datum level by introducing a robust waveform-based redatuming approach [17]. Using the retrieved virtual data, high-resolution local FWI and time-lapse FWI are performed without overburden. [PERSON] and [PERSON] extended the redatuming method to be adapted into elastic media [18]. [PERSON] et al. proposed the multidimensional Marchenko equations based on the one-dimensional Marchenko representation by introducing the focusing function [19; 20; 21; 22]. The multidimensional Marchenko equations can be used for redatuming [PERSON]'s functions of the virtual receivers. This method requires only a smoothed velocity model with correct kinematic information and does not need physical sources or receivers in the subsurface medium [19; 20; 21; 23]. By using the Marchenko redatuming method, we can obtain subsurface receiver redatuming along the boundary enclosing the zone in which we are interested. However, the standard Marchenko method is insensitive to lateral propagation of the wavefield under highly complex geological structures [24].