Page 108 - MATLAB Recipes for Earth Sciences

5.5 Interpolating and Analyzing Unevenly-Spaced Data 101

[Pxy,f] = cpsd(y1,y2,[],0,512,10);
phase=angle(Pxy);
plot(f,phase)

The phase shift at a frequency of f=0.2 (period τ=5) is

interp1(f,phase,0.2)
which produces the output of

ans =
1.2568

The phase spectrum is normalized to one full period τ=2›, therefore a phase
shift of 1.2568 equals (1.2568*5)/(2*›) = 1.0001, which is again the phase
shift of one that we introduced at the beginning.

5.5 Interpolating and Analyzing Unevenly-Spaced Data

Now we use our experience of evenly-spaced data to run a spectral analysis
on unevenly-spaced data. Such data are very common in earth sciences. For

example, in the field of paleoceanography, the deep-sea cores are typically
sampled at constant depth intervals. Transforming evenly-spaced length-pa-
rameter data to time-parameter data in an environment with changing length-
time ratios results in unevenly-spaced time series. Numerous methods exist
for interpolating unevenly-spaced sequences of data or time series. The aim
of these interpolation techniques for tx data is to estimate the x-values for an
equally-spaced t vector from the actual irregular-spaced tx measurements.
Linear interpolation is relatively simple and straightforward method for ex-
trapolating between two equally spaced data points. It predicts the x-values
by effectively drawing out a straight line between two neighboring measure-
ments and by calculating the appropriate point along that line. However,
the method also has its limitations. It assumes linear transitions in the data,
which introduces a number of artifacts, including the loss of high-frequency
components of the signal and limiting the data range to that of the original
measurements.
Cubic-spline interpolation is another method for interpolating data that
are unevenly spaced. Cubic splines are piecewise continuous curves, pass-
ing through at least four data points for each step. The method has the ad-
vantage that it preserves the high-frequency information contained in the
data. However, steep gradients in the data sequence could cause spurious