## 1 Introduction
The \"dip\" of the wave refers to the slope of the wave in the matrix, or how much the row position of a wave changes as we move from one column to its adjacent column. Some of the problems encountered with local SVD were resolved with the development of structure-oriented singular value decomposition (SOSVD) [17]. By using plane wave destruction [18], SOSVD can identify several dominant slopes at each window location. While using plane wave destruction provides a noticeable improvement, numerical artifacts can still appear at the intersection of the waves with different slopes. Our method is inspired by the SOSVD and local SVD, and it also shifts the columns of the matrix before applying SVD. However, in order to avoid numerical artifacts, we use different processes to determine how columns should be shifted. Specifically, we do not use a moving window with a predetermined path. Instead, we use a geometric mean filter that adaptively chooses which elements to use in the geometric mean. We apply the geometric mean filter to seismic data in order to highlight areas which contain wave arrivals. The windows from the matrix to which we apply SMD depend on the results of the geometric mean filter. This allows us to use more singular vectors to describe areas with multiple wave arrivals, and fewer singular vectors to describe areas dominated by noise. Additionally, local SVD and SOSVD use SVD results solely to denoise seismic data. They use them to reconstruct denoised version of seismic data as soon as they obtain them and do not discuss the compression achieved by storing SVD results. In our work, instead of reconstructing the data immediately, we store the SVD results. The final product of our algorithm is the collection of SVD results, that can later be reconstructed into the denoised version of original data. By doing this, our algorithm can be used for data compression as well as noise suppression. We call our new method the shifted-matrix decomposition, or SMD for short. It should also be noted that DAS has seen increasing use outside geophysical exploration. Distributed acoustic sensing has been used to record signals from earthquakes and volcanic events [19]. Because of its unprecedented spatial and temporal resolutions, DAS is expected to see increasing use in earthquake monitoring, imaging of faults and many other geologic formations, and hazard assessment [20]. The growing potential of DAS application outside of geophysical exploration, adds importance to our method, which we believe will play an integral role in the processing of DAS data. We organize the paper as follows: First we give an overview of singular value decomposition and present two simple examples that show advantages and drawbacks of its application to seismic data. Next, we demonstrate the improvements achieved by shifting the traces before extracting singular vectors. In the following subsection, we describe in detail each step of the SMD algorithm. The SMD algorithm depends on several parameters. The optimal values of said parameters are determined in a machine learning stage following the algorithm description subsection. In the training stage we use seismic field data obtained from marine seismic gathers, which has a large amount of interference between coherent waves, and noise which can be difficult to differentiate from signal. After training on marine seismic gathers, SMD provides accurate results on other seismic data as well as marine seismic gathers, which allows us to skip the training stage in future applications. To confirm the accuracy of SMD, we reproduce synthetic data from [17], and compare results of SMD to results of local SVD and SOSVD. The SMD is then tested on real seismic data. While SMD is primarily developed for application to microseismic data recorded by DAS, we currently do not have access to such data. Instead, we apply SMD to field data obtained from marine seismic gathers [21].