Signal-to-noise ratios of seismic waves
Refer to given Exercise. Suppose the model from part c is fit, with the following result:
y^ = 1 + .05x1 + x2 + .05 x1 x2
Graph the estimated signal-to-noise ratio y^ as a function of the wavelet amplitude, x2, over the range x2 = 10 to x2 = 50 for frequencies of x1 = 1, 5, and 10. Do these functions agree (approximately) with the graphs you drew for given Exercise, part c?
Exercise
Signal-to-noise ratios of seismic waves. Chinese scientists have developed a method of boosting the signal-to-noise ratio of a seismic wave (Chinese Journal of Geophysics, Vol. 49, 2006). Suppose an exploration seismologist wants to develop a model to estimate the average signal-to-noise ratio of an earthquake's seismic wave, y, as a function of two independent variables:
x1 = Frequency (Cycles per second)
x2 = Amplitude of the wavelet
(a) Identify the independent variables as quantitative or qualitative.
(b) Write the first-order model for E(y).
(c) Write a model for E(y) that contains all first order and interaction terms. Sketch typical response curves showing E(y), the mean signal-to-noise ratio, versus x2, the amplitude of the wavelet, for different values of x1 (assume that x1 and x2 interact).
(d) Write the complete second-order model for E(y).