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).