Geoscience Reference
In-Depth Information
with
σ
y
the samplestandarddeviationofthey variable,andwheretheordinateis
n
N
∆
z
where
n
is the number of samples in each bin of width
z
out of the total popu-
lation
N
(equal to 200). The dashed curve in each case is the normal probability
distributionfunctionforavariablewithzeromeanandunitstandarddeviation:
∆
1
√
2
e
−
z
2
/
2
p
=
π
Afterseveralexamplescomparingthebootstrapsamplecdfs(obtainednumerically)
withthenormalcdfshowedveryminordifferences,wechosetomodifythemethod
slightlybyassumingthatyisnormallydistributed.Thusthe95%confidenceinterval
is givenby
−
1
.
96
<
z
<
1
.
96,orequivalently
CI
=
y
∓
1
.
96
σ
y
3.2.5 Averaging Time and the Spectral Gap
w
T
)
10
6
Heat flux values
with 95% confidence limits for each of the
realizations (Fig. 3.6) differ by considerably more than their respective error bars.
It is quite common to see large variation in covariance values from one 15-min
sample to the next even in a flow that is relatively steady. The reason for this is
implicit in the
wT
time series of Fig. 3.4, where the covariance is dominated by a
fewlargepositiveandnegativeeventsinwhichturbulenteddieswith timescalesof
minutestransfer heatup and down.If a sample includesan excess or deficit of just
a few of these events, it may affect covariance values appreciably. Thus sampling
strategy represents a tradeoff between remaining in a “spectral gap” where the re-
alization averaging time will capture most of the eddy events, but where changes
in the mean flow speed and direction will not adverselyaffect covariancestatistics.
Generally,ourapproachhasbeentodividetheflowregimeinto15-minrealizations
for which the streamline coordinatesadequatelyrepresentthe actual flow, and then
further average the covariance estimates for longer periods, typically 1-6h. Using
results fromthe bootstrapanalysis for theindividualrealizations, covarianceconfi-
dence intervals for the longer periods may be constructed by invoking the
central
limit theorem
(e.g.,BowkerandLieberman1959)whereweassumethatthecovari-
ancesdeterminedfromeachofthe15-minrealizationsarenormallydistributedwith
knownvariances.In that case, evenwith a small numberof samples (in the present
example,four valuesfor
(
4
×
w
T
)
the 95% confidenceinterval for the true mean,
µ
,
isrelatedtothesamplemeanby
σ
n
/
√
n
X
n
∓
CI
n
=
1
.
96
