Greedy Automatic Signal Decomposition and Its Application to Daily GPS Time Series

The recognition of transient motion in terrestrial continuous Global Positioning System (GPS) time series implies the knowledge of certain time functions that we assume to be ever present in the time series. By assuming that the permanent time functions are the long-term secular velocity of the Earth and the seasonal oscillations, we define the total remaining signal as transient motion. Here we adopt the multitransient as a versatile function for modeling transient motion over a range of time scales. We define the multitransient as the sum of two or more transient decaying functions with different characteristic time scales and identical onset times. We then demonstrate the greedy approach to fitting the time series by using a minimum number of multitransients (sparse functions) in addition to the permanent time functions in a linear regression. The Greedy Automatic Signal Decomposition algorithm decomposes the signal into three parts: (1) background seasonal motion, (2) secular and transient motion, and (3) a residual (noise). We describe the greedy algorithm with synthetic examples before demonstrating its application to time series of daily GPS solutions. The implementation of the multitransient allows for a more realistic plate-trajectory model, whereby a full range of transient signal time scales, from short-duration slow slip to longer-duration processes such as postseismic slab accelerations or postseismic decay, can all be estimated with the same function. Since Greedy Automatic Signal Decomposition algorithm automatically estimates trend, its application to a GPS network allows for the common mode filter to be applied seamlessly. Plain Language Summary Over the past few decades, earthquake scientists have been increasing the deployment of continuous GPS stations. This is because high precision time series of surface motions are great for testing the hypotheses of earthquake physics. There is now a wealth of stations (over 15,000 worldwide) but unfortunately no quick way of separating the expected, background plate motion from the unexpected, unusual transient signal. In this study, we present a method aimed at solving this very problem. Adapting an algorithm used in the fields of statistics and electrical engineering called Greedy Optimization we show how, by assuming that transient signals are rare occurrences in the time series, it is possible to separate them from the expected signals in a computationally efficient way. We believe that this method, if developed to run even faster (for example with parallelization), will pave the way for rapid analysis of unexpected signals on a spatial scale that has previously been impossible. This opens the door to many new exciting discoveries into how the earth is moving in response to plate-tectonics and weather. Furthermore, we conclude that this method has the potential to be applied to a wide variety of time series data.