# 3D seismic survey design by maximizing the spectral gap

## Objectives/Scope (600 characters)

The massive cost of 3D acquisition calls for methods to reduce the number of receivers by designing optimal receiver sampling masks. Recent studies on 2D seismic showed that maximizing the spectral gap of the subsampling mask leads to better wavefield reconstruction results. We enrich the current study by proposing a simulation-free method to generate optimal 3D acquisition by maximizing the spectral gap of the subsampling mask via a simulated annealing algorithm. Numerical experiments confirm improvement of the proposed method over receiver sampling locations obtained by jittered sampling.

## Methods, Procedures, Process (1500 characters)

The spectral gap ratio is the ratio of the first and second singular values of a binary subsampling mask (binary matrix). It is a cheap-to-compute measure to predict wavefield reconstruction quality from acquisition geometry only. Motivated by recent success on 2D survey design methods driven by spectral gap ratio minimization, we consider 3D survey design where receivers are missing and sources are fully sampled. Because 3D wavefield reconstruction based on low-rank matrix completion relies on the non-canonical Source-X/Receiver-X (columns) Source-Y/Receiver-Y (rows) organization of the data into a matrix, we aim to minimize the spectral gap ratio of the subsampling mask in that domain. Fortunately, when sources are fully sampled, each single-receiver block of the global sampling matrix is either fully sampled or empty depending on whether that specific receiver is sampled. Consequently, the block structure of the global matrix leads to the exact same singular values as a single-source receiver sampling mask. We can therefore optimize a single-source mask to obtain the global optimized mask. The main computational cost is computing the first two singular values of the receiver sampling mask, which is negligible compared to approaches that require wave simulations. The resulting optimal mask with the lowest spectral gap ratio indicates the receiver sampling locations that favor 3D wavefield reconstruction via matrix completion in the non-cononical organization domain.

## Results, Observations, Conclusions (1500 characters)

To illustrate the efficacy of our method via a numerical experiment on a simulated 3D marine dataset over the compass model. The data volume consists of 501 time samples, 1681 sources and 10k receivers. The distance between the adjacent sources and receivers are 150m and 25m in each direction, respectively, with a time sampling interval of 0.01s. By removing 90% of receivers using jittered subsampling, we obtain a binary matrix with the spectral gap ratio 0.507 in the non-canonical domain. After applying simulated annealing algorithm, the spectral gap ratio of mask effectively decreases to 0.328. To validate the efficacy of our acquisition design method, we perform data reconstruction on a frequency slice at 16.8Hz via weighted matrix completion for the two subsampled datasets with jittered subsampling mask and the proposed mask, with results shown in Figure 2. The reconstruction signal-to-noise ratio from the observed data at proposed receiver locations is 12.27 dB, which is about 1.4 dB higher than the reconstruction signal-to-noise ratio 10.88 dB achieved from data observed at jittered sampled receiver locations. This confirms that the proposed optimized receiver sampling locations result in a superior seismic survey that leads to better wavefield reconstruction performance. More information is in slimgroup.github.io/IMAGE2023.

## Significance/Novelty (600 characters)

This is the first numerical case study that applies spectral gap ratio minimization techniques to seismic acquisition design that favors 3D wavefield reconstruction. Rather than requiring costly wave simulations, the proposed method only relies on a single binary matrix optimization which is computationally inexpensive. Experiments demonstrate that the proposed method generates an improved 3D seismic survey better suitable for 3D wavefield reconstruction.