GG413 Geological Data Analysis
Homework #11: Autocorrelation and Cross-Correlation
Reading: Sections 5.5 & 5.6, Due Tue Nov. 18.
As usual please be as complete and concise in explaining your thought process in solving each problem below.
1) The geothermal gradient can be determined by measuring the temperature in an observation well that has
been left undisturbed sufficiently long so that the hole has come to thermal equilibrium with the enclosing
rocks. Temperature measurements typically are made using a string of thermistors lowered into the hole.
However, repeated measurements made over a period of time may exhibit temperature fluctuations that
reflect the movement of fluids in convection cells that develop inside the bore hole. The durations of any
such fluctuations are important to determine because they reflect upon the geothermal gradient, among a
few other factors (e.g. fluid viscosity, bore-hole diameter).
File “TEMPER.TXT” contains temperature measurements (in C) taken at 5-min intervals at a depth of
180 m in the Foster No. 6 well in San Jacinto County, Texas. Your goal is to determine whether there is
any cyclicity in the record, and hence any active fluid convection.
(a) First plot the time series
(b) Compute and plot the autocorrelation function of this record.
(c) Identify any autocorrelation peaks that reflect cyclicity and that are significant at the 5% level. What
is(are) the time period(s) associated with the inferred convectively driven temperature oscillation(s)?
(2) In the Chesapeake Bay of Maryland, as in many coastal estuaries, there are complex changes in salinity
caused by the mingling of freshwater with seawater during the diurnal tidal cycle. Fresh Chesapeake
River water floats across the denser, more saline, brine in the Bay; during periods of low tide, the fresher
surface water moves farther seaward, down the estuary. However, there is a counter-flow along the
bottom that carries dense marine water up the Bay during the sinking tide. Salinity measurements have
been made at 1.5 hr intervals over a 24-hr period, for both surface water and bottom water (11-m depth
average) at a collection station offshore from Annapolis, Maryland. The salinities (in ppt) for the top and
bottom waters are given in the 1st and 2rd columns of “Chesapeake_salinity.txt”, respectively (the first
record was taken at 2:30 p.m. the second at 4:00 p.m. etc).
Your objective is to determine whether the differential motion between the top and bottom waters is
expressed as time-shifts between salinity highs and lows between the two records.
(a) Plot the two records adjacent to one another so they share the same axes.
(b) Compute and plot the cross-correlation function. For simplicity, it is ok to examine lags in only one
direction (e.g., only positive).
(c) Identify any cross-correlations that are significant at the 5% level. What is(are) the time(s) of the
significant lags?
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