Refer to Example 14.3 in which wavelet-based smoother exhibited notable difference from the standard smoother LOESS. Read the data earthquake. dat into MATLAB, select the wavelet filter, and apply the wavelet transform to the data.
(a) Estimate the size of the noise by estimating (T using MAD from page 276 and find the universal threshold λu.
(b) Show that finest level of detail contains coefficients exceeding the universal threshold.
(c) Threshold the wavelet coefficients using hard thresholding rule with λu that you have obtained in (b), and apply inverse wavelet transform. Comment. How do you explain oscillations at boundaries?
Example 14.3 A researcher was interested in predicting earthquakes by the level of water in nearby wells. She had a large (8192 = 213
measurements) data set of water levels taken every hour in a period of time of about one year in a California well. Here is the description of the problem:
The ability of water wells to act as strain meters has been observed for centuries. Lab studies indicate that a seismic slip occurs along a fault prior to rupture. Recent work has attempted to quantify this response, in an effort to use water wells as sensitive indicators of volumetric strain. If this is possible, water wells could aid in earthquake prediction by sensing precursory earthquake strain. We obtained water leveI records from a well in southern California, collected over a year time span. Several moderate size earthquakes (magnitude 4.0 - 6.0) occurred in close proximity to the well during this time interval. There is a a significant amount of noise in the water level record which must first be filtered out. Environmental factors such as earth tides and atmospheric pressure create noise with frequencies ranging from seasonal to semidiurnal. The amount of rainfall also affects the water level, as do surface loading, pumping, recharge (such as an increase in water level due to irrigation), and sonic booms, to name a few. Once the noise is subtracted from the signal, the record can be analysed for changes in water level, either an increase or a decrease depending upon whether the aquifer is experiencing a tensile or compressional volume strain. just prior to an earthquake.