U.S. DEPARTMENT OF HEALTH, EDUCATION, AND WELFARE
Public Health Service
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APPLICATION OF SPECTRAL ANALYSIS
TO
STREAM AND ESTUARY FIELD SURVEYS
I. Individual Power Spectra
T. A. Wastler
Technical Services Branch
Robert A. Taft Sanitary Engineering Center
U.S. DEPARTMENT OF HEALTH, EDUCATION, AND WELFARE
Public Health Service
Division of Water Supply and Pollution Control
Cincinnati, Ohio
November 1963
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Public Health Service Publication No. 999-WP-7
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CONTENTS
Page
Abstract v
Introduction 1
The Concept of Spectral Analysis 2
The Technique of Spectral Analysis 6
Interpretation of Spectra 18
Design Criteria for Spectral Analysis 26
Conclusion 29
Acknowledgments 30
Bibliography 31
in
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ABSTRACT
The application of spectral analysis techniques to sanitary
engineering stream and estuary studies is discussed from a
practical operational viewpoint. Techniques of interpretation
and the data requirements are emphasized rather than the math-
ematical basis and details of the technique. The usefulness of
spectral analysis in analyzing records obtained from continuous,
automatic monitoring stations is pointed out. Spectral analyses
applied to tidal height records and dissolved oxygen records ob-
tained in a field study of the Potomac Estuary are discussed.
The discussion is limited to the application of individual power
spectra computation to sanitary engineering investigations.
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APPLICATION OF SPECTRAL ANALYSIS
TO
STREAM AND ESTUARY FIELD SURVEYS
I. Individual Power Spectra
INTRODUCTION
The expanding population in the coastal areas of this conti-
nent has stimulated interest in the sanitary engineering study of
estuarine systems. The speed with which conditions change in
estuaries and the complex interactions that occur require new
techniques of estuarine sampling and data analysis.
_ Spectral analysis is a data analysis tool that shows consider-
able promise for both estuary and stream study. This technique
emerged from the work of Fourier and Laplace near the begin-
ning of the nineteenth century, and much of the mathematical
theory underlying its practical application was developed in the
work of G. I. Taylor, Norbert Wiener, and S. O. Rice. It was
not until the advent of automatic sampling devices and high-speed
digital computers that its practical application to many scientific
and engineering problems became feasible. Since the 1940's
spectral analysis has been used with considerable success in
such diverse fields as communications engineering, aerodyna-
mics, oceanography, and meteorology.
The purpose of this discussion is to present the computation
and interpretation of individual power spectra, the basic opera-
tion of spectral analysis, as an engineering tool. The mathema- -
tical and statistical basis of spectral analysis, which is essen-
tially a mathematical and statistical technique, has been almost
entirely ignored. It is hoped that the references given in the
bibliography may satisfy any need for fuller explanation of these
aspects of spectral analysis.
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APPLICATION OF SPECTRAL ANALYSIS
THE CONCEPT OF SPECTRAL ANALYSIS
One of the classic experiments in the study of optics is to
pass a beam of sunlight through a triangular prism and observe
the rainbow of colors into which the beam of white light is split.
This experiment serves as a useful physical analogy to the oper-
ation of spectral analysis on a body of data.
In Figure la is presented a schematic diagram of the results
of passing a beam of sunlight through a triangular prism. The
beam of sunlight is split into a spectrum of colors ordered ac-
cording to their respective wave lengths (or frequencies). If
this spectrum of colors is allowed to strike a battery of light-
sensitive cells, the intensity of the light of each color can be
measured; these results can then be plotted as a "light inten-
sity spectrum, " which might look like that in Figure la if each
of the six principal colors had the same intensity.
If a similar experiment is performed with a light beam that
is composed of only three of these colors (present at different
intensities), the resulting light frequency spectrum and light in-
tensity spectrum might resemble those in Figure Ib.
This experiment demonstrates the resolution of a complex
physical phenomenon into a group of simpler phenomena that may
be easier to examine from both theoretical and practical view-
points.
The effect of using spectral analysis on a record of observed
field data is directly analogous to the effect of the prism on the
light beam. This analogous effect is presented schematically in
Figure Ic. The actual technique of spectral analysis is dis-
cussed later; at this time the significance of the result of the
computation is of concern.
In the prism experiment the relative intensities of the re-
solved light frequencies can be observed and studied. Spectral
analysis of a record of observations results in a sorting of the
total "variance" of the record into its component frequencies.
The variance of a data record is therfore analogous to the inten-
sity of the light beam.
The variance is defined as the sum of the squares of the de-
viations from the mean divided by one less than the number of
observations. This is the definition of variance as it is normally
used as a descriptive statistic of a body of data. Conceptually,
the variance is a measure of the dispersion of observations about
the mean value. In the statistical interpretation of data, it is
ordinarily regarded that this dispersion of values about the mean
is due to chance.
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INTENSITY OF LIGHT
LIGHT
BEAM
a. PRISMATIC RESOLUTION
OF A BEAM OF SUNLIGHT
LIGHT
FREQUENCY
SPECTRUM
LIGHT
INTENSITY
SPECTRUM
LIGHT
BEAM
b. PRISMATIC RESOLUTION
OF A BEAM OF LIGHT
INTENSITY OF LIGHT
0-2
LIGHT
FREQUENCY
SPECTRUM
LIGHT
INTENSITY
SPECTRUM
DATA RECORD
c. RESOLUTION OF A DATA RECORD
BY SPECTRAL ANALYSIS
MAGNITUDE OF VARIANCE
"POWER SPECTRUM"
OR
"VARIANCE SPECTRUM1
Figure I. Physical analogy to spectral analysis.
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4 APPLICATION OF SPECTRAL ANALYSIS
The statistical manipulation of time-series data by spectral
analysis results in the computation of those parts of the variance
of a record that recur at constant time intervals as well as the
part that is random (nonrecurring) in character. As the light
beam may be resolved by the prism into its component colors of
different intensity, so is the variance of a time-series record
resolved into its component parts by spectral analysis.
The estimates of variance for each frequency resolved in the
spectral analysis form the "power spectrum" of the record from
which the computations are made. (The term "variance spec-
trum" would be more accurate; but in the pioneering work in
spectral techniques done in communications engineering the term
"power, " which is closely related to record variance in that
frame of reference, became common usage. ) The power spec-
trum computation is the fundamental operation of data reduction
and interpretation by means of spectral analysis; the many other
computations that can be made in spectral analysis are all based
firmly upon the concept and calculation of individual power spec-
tra. It may be stated that the computation of individual power
spectra bears about the same relationship to spectral analysis as
differentiation and integration bear to the Calculus.
The interpretation of variance as a statistic descriptive of
both the random and nonrandom characteristics of a time-series
data record is most important in understanding the significance
of spectral results. In the usual type of statistical analysis,
variance is conceptually regarded as a measure of the random
dispersion of the observations from their mean value. In many
cases this is true; but that this is not a necessary condition for
the existence of a variance can be demonstrated with the aid of
Figure 2, in which -segments of three hypothetical records and
the corresponding power spectra are presented.
Figure 2a shows a record that has a constant value, i. e. ,
all values are equal to the mean. Since there are no deviations
from the mean, the variance is zero and the power spectrum is
zero at all frequencies.
Figure 2b shows a -record that forms a sloping straight line.
The segment of the record shown in this figure has a mean of
3. 55 and a variance of 0. 69. It is apparent that none of this
variance results from a "random" dispersion about the mean,
but is the result of a secular (time-dependent) trend in the re-
cord. If spectral analysis -were done on the record of which
this segment is a part, all of the variance (or "power") would be
concentrated in the zero-frequency spectral estimate as shown in
Figure 2b. The zero-frequency spectral estimate includes all of
the record variance that does not recur during the length of the
record used in the analysis. It therefore includes (1) any truly
random fluctuations in the record, (2) any linear trends in the
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TO STREAM AND ESTUARY FIELD SURVEYS
record, and (3) any periodic components in the record that are
of so low a frequency that they appear as linear trends in the
record. For example, spectral analysis of the segment "A" in
Figure 2c would result in a power spectrum similar to that in
Figure 2b, simply because the record length is not great enough
to resolve the periodic fluctuation exhibited in Figure 2c.
SEGMENT OF RECORD
2 PLOT OF POWER SPECTRUM
7 -
6 -
5 -
4 -
3 -
2 -
1 -
0 -
TIME'
a. STRAIGHT LINE OF SLOPE =0
/I fz f3 /4 fs
FREQUENCY
7 -
6 -
5 -
4 -
3 -
2 -
I -
0 -
S«
TIME »
b. SLOPING STRAIGHT LINE
FREQUENCY
WAVE LENGTH
CORRESPONDING TO 'z
* *
7 - .
6 -
TIME *
c. SINE WAVE
FREQUENCY
Figure 2. Typical spectra obtained from several types of curves.
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APPLICATION OF SPECTRAL ANALYSIS
The record of which Figure 2c is a small part also exhibits
a variance. The mean of this record is 4. 0, and the variance is
entirely the result of the sinusoidal fluctuation about the mean.
Spectral analysis of this record, which should be about 10 times
longer than the segment presented in Figure 2c, would result in
a power spectrum in which all of the variance is concentrated at
the frequency designated £9, which corresponds to the wave
length of the sine wave in the data record.
If the data record were a combination of Figures 2b and 2c,
the power spectrum would be a. combination of the spectra in
these figures, i. e. , there would be components at the zero fre-
quency and at i-^. It is this characteristic that makes spectral
analysis such a useful tool in analyzing records that represent
complex phenomena in natural systems. For example, the di-
urnal effects of photosynthetic activity could be separated from
the longer-period effects of waste loads and river discharges by
spectral analysis of the stream dissolved oxygen (DO) record.
The power spectra in Figures 1 and 2 are presented in
bar graphs to emphasize two characteristics of spectral results:
(1) estimates of the variance at several discrete frequencies
are obtained from the analysis and (2) each variance estimate
represents the variance concentrated in a band around the
nominal frequency of each variance. This bar graph represen-
tation is not the usual way in which spectra are presented; in the
remaining figures discussed in this paper, the conventional
point-and-line representation is used.
Like any other method of data analysis, this method has its
limitations and disadvantages. Three major requirements for
the record may be regarded as limiting. First, the record must
be fairly long -- generally having over 100 sequential measure-
ments. Second, there must be no missing data -- if measure-
ments are missing, suitable values must be interpolated before
spectral analysis is attempted. Third, the mathematical pro-
cedures require so much computation that the use of a high-
speed digital computer is essential for most analyses. These
restrictions are discussed in more detail.
THE TECHNIQUE OF SPECTRAL ANALYSIS
The approach used here is to present the technique of
spectral analysis by following through the steps in the actual
spectral analysis of a record. For the benefit of those who wish
to explore the mathematical basis for spectral analysis compu-
tations, pertinent references are presented in the bibliography.
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TO STREAM AND ESTUARY FIELD SURVEYS 7
The record chosen for detailed examination here is a record
of water level obtained from a U. S. Geological Survey station
in the Potomac Estuary near Washington, D. C. This record
was chosen because it exhibits a simple periodicity with very
little random interference ("noise"), and because the data were
obtained as a continuous recording so that a wide choice of
sampling intervals was possible.
A portion of this record is presented in Figure 3. A visual
examination of this record shows that there is a dominant period
of about 12 hours and that there is some long-period change.
Although the entire computation can be carried out on high-
speed digital computers, with available programs, the individual
steps are presented here to illustrate the technique. Only the
initial steps in the data preparation need be carried out by hand
or on semi-automatic equipment.
Step 1. A sampling interval of 4 hours was chosen for this par-
ticular analysis, and the record was read at this interval for a
total of 145 readings. (Considerations governing the number of
points read and the sampling interval are discussed later in
detail. ) The starting point was arbitrary. The number obtained,
in order, are
1. 30 = value 1
2. 57 2
3.79 3
1.49 4
2.30 5
4.73 6
3. 10 140
1.46 141
3.16 142
3.30 143 . -
1.41 144
2.35 145
Mean = 2.43
Square of Mean = 5. 90
Step 2. The autocorrelation function of these numbers is then
formed. This is a very large name for a very simple, but very
useful, process. Each number in the record is multiplied by
another number in the record, and from the mean of the sum
is subtracted the square of the arithmetic mean of the entire
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0.0
DATE (AUGUST, 1959)
Figure 3. Portion of tide height record in the Potomac Estuary.
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TO STREAM AND ESTUARY FIELD SURVEYS 9
record. If the individual products are formed by multiplying
each number by itself, the result is called the "autocorrelation
at lag 0, " which is purely and simply the "variance" as ordinarily
defined in statistics. If the individual products are formed by
multiplying each number by the number that follows it in the
sequence, the result is called the "autocorrelation at lag 1. "
The autocorrelations computed to 12 lags for the record being
analyzed here are
CQ = Autocorrelation at lag 0 Cj = Autocorrelation at lag 1
1.30x1.30= 1.69 1.30xZ.57= 3.34
2.57x2.57= 6.60 2.57x3.79= 9.74
3.79x3.79=14.36 3.79x1.49= 5.65
1.49x1.49= 2.22 1.49x2.30= 3.43
2.30x2.30= 5.29 2.30x4.73=10.88
4.73 x 4.73 = 22.37 4.73
3.10x3.10= 9.61 3.10x1.46= 4.53
1.46x1.46= 2.13 1.46x3.16= 4.61
3.16x3.16= 9.99 3.16x3.30=10.43
3.30x3.30-10.89 3.30x1.41= 4.65
1.41x1.41= 1.99 1.41x2.35= 3.31
2. 35 x 2.35 = 5.82 2.35
Sum = 1046. 9 Sum = 804. 67
1046'9=7.22 804'67 =5.588
145 -S^O 144 -5.90
C0 = 1.32 C1 = -0. 312
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10 APPLICATION OF SPECTRAL ANALYSIS
C2= Autocorrelation at lag 2 Cj = Autocorrelation at lag 3
1.30x3.79= 4.93 1.30x1.49= 1.94
2.57x1.49= 3.83- 2.57x2.30= 5.91
3.79x2.30= 8.72 3.79x4.73= 17.93
1.49 x 4.73 = 7.05 1.49 x .
2. 30 x . . 2. 30 x .
4.73 x . . 4.73 x .
3.10x3.16= 9.80 3.10x3.30= 10.23
1.46x3.30= 4.82 1.46x1.41= 2.06
3.16x1.41= 4.46 3.16x2.35= 7.43
3.30 x 2.35 = 7.76 3.30
1.41 1.41
2.35 2.35
Sum = 764. 62 Sum = T015. 3
764.62 _ c ,._ 1015.3
1.20
The remaining autocorrelations are computed similarly and have
the values
C4 = 0. 154 C? = 0.0476 C10 = 0.252
C5 = 0.735 C8 = -0.917 Cn= -0.998
C6 = 1- 10 Cg = 0.921 C12 = 0.798
This operation may be expressed mathematically as
n-r n _2
Cr =_L
n-r 1
where
Cr = autocorrelation at lag r,
xj = record value at t,
t = 0, 1, 2 ... n = sequential index of values,
r = 0, 1, 2 ... m = lag number,
m = total number of lags.
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TO STREAM AND ESTUARY FIELD SURVEYS 11
Step 3. The Fourier cosine transform for each autocorrelation
is next computed. This serves to smooth out some of the wide
fluctuations in the autocorrelations and consists of applying a
cosinusoidal weighting factor to each autocorrelation calculated
in the preceding step. This operation can be expressed mathe-
matically as
m-1
r m I 0 m
q = 1
where
Vr = Fourier cosine transform of the autocorrelation at
lag r,
q = lag number, having values between 1 and m-1
k = a constant, k = 1 for r = 1,2 ... m-1,
k - 1/2 for r = 0,
r = m,
and the other letters have the definitions previously given.
The Fourier cosine transforms calculated for each auto-
correlation of the tide height record are
V0 = ... * . 1.32 + 2^ c >cos q(0) * + 0.798 cos(O) v\
(Z/ll^l I f n m I
L qTT q J
= fl.32 + 2(1) (-.312 - .553 + 1.20 - 0. 154 - 0.735
24 L
+ 1. 10 + 0.0476 - 0.917 + 0.921 + 0.252 - 0.998)
+ 0.798 (1)1
= 0.0765
V = -L [ 1.32 + (0.798) cos(l) *] + £- [-(0.
X I ^ L Jl^L
-(0.553)cos<2)154)
i. Lt \. L*
i L*
+ (0.0476)cos(7)(i1^ * - (0.917)coa(8)(1)ir + (0.921)
cos
i21iiUL+
.
i. Lt 1 Lt
Vj = 0. 101
Similarly, the Fourier cosine transforms for the remaining auto-
correlations can be computed:
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12 APPLICATION OF SPECTRAL ANALYSIS
V2 = - 0.0332 V8 = 0.995
V3 = 0.0570 Vg = -0.224
V4 = -0.0443 V10 = 0.146
V5 = 0.0943 Vu = -0. 118
V6 = -0. 136 V12 = 0.0561
V? = 0.353
Step 4. The final step in the spectral analysis of a single record
is another weighting operation that counteracts some distortion
of the spectrum resulting from the small sample size.
This step can be expressed mathematically as
u0 =0.54 [YO + V!].
Ur =0.23 Vr_1 + 0.54 Vr + 0.23 Vr+1 ,
for r = 1, 2, 3 m-1
Um = 0.54 Vm.i + 0.54 Vm ,
where UQ, Ur, Um are the power spectrum estimates corres-
ponding to the respective lags, and the remaining symbols have
the meanings previously assigned.
The power spectrum estimates for the tide gage record are
U0 = (0. 54){0. 0765) ,- (0. 54)(0. 101) = 0. 0959
Uj = 0.0643
U2 = 0.0183
U3 = 0.0130 U8 = 0.567
U4 = 0.0109 U9 = 0. 142
U5 = 0.00947 U10 = 0.000446
U6 = 0.0295 Un = -0.0173
U7 = 0.388 U12 = -0.0334
Each of these spectral estimates represents the part of the
total record variance that is estimated to occur with a certain
periodicity. The period corresponding to each lag is determined
from the lag number and the sampling interval by this relation:
Tr = 2m AT
r
where Tr = period corresponding to lag r,
A T = sampling interval.
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TO STREAM AND ESTUARY FIELD SURVEYS
For the record analyzed here, m = 12 and AT = 4 hours.
The periods corresponding to the lag numbers are these:
13
Lag Number
0
1
2
3
4
5
6
7
8
9
10
11
12
Period (hours)
CD
96
48
32
24
19.2
16
13.7
12
10.7
9.6
8.7
8
The spectral estimates are plotted as functions of period in
Figure 4.
0.6
0.5
0.4-
0.3-
o
N
O
z
<
at
<
O.2
O.I r
1 1 I 1 T~
TOTAL NO. OF POINTS =145
SAMPLING INTERVAL =4 hr
80% CONFIDENCE BAND
96 48 32
24 19.2 16
PERIOD, hr
Figure 4. Spectrum of e tidal height record.
137 12 10.7 9.6 8.7
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14 APPLICATION OF SPECTRAL ANALYSIS
From these results it can be seen that about 79 percent of
the observed variance in -water level can be attributed to periodi-
cities of around 12 hours, that there are very small diurnal
effects in the tide at this point, and that long-period changes
account for about 12 percent of the observed variance.
The effects of using different sampling intervals and com-
puting to different numbers of lags may be examined by consider-
ing the results of further spectral computation with this record.
Figures 4, 5, and 6 illustrate spectra obtained from the same
record, with variations only in the sampling interval and in the
number of lags used in calculation. The total record length in
each case was 576 hours. The spectral estimates from which
Figure 4 was plotted were obtained by computation to 12 lags
from values read at 4-hour intervals (145 points). Figure 5
was obtained from computation to 12 lags from values read at
hourly intervals (577 points). Figure 6 was obtained from com-
putation to 24 lags from values read at hourly intervals. The
abscissal scale in Figure 6b is 50 times greater than that in
Figure 6a; this shows short-period effects more clearly.
While each of these figures shows the dominance of approxi-
mately semidiurnal periodicities in this record, the effects of
the different sampling intervals and the different numbers of
lags can be seen also.
Comparison of Figures 4 and 5 shows the effects of a change
in sampling interval on the spectrum of the tide height record.
First, the use of a smaller sampling interval for a given record
length increases the number of measurements used in the analy-
sis and therefore increases the number of degrees of freedom
upon which each estimate is based. This results in the smaller
confidence band shown in Figure 5. Second, the change of sam-
pling interval from 4 hours to 1 hour permits the resolution of
components with periods as short as 2 hours in the latter case
instead the 8 hours possible in the former. However, the use of
the same number of lags with the shorter sampling interval does
not permit as high a degree of resolution of long-period pheno-
mena as was obtained with the longer sampling interval. In fact,
the spectrum calculation leading to Figure 5 does not permit an
estimate of the diurnal and longer-period effects because these
are effectively masked by the dominant 12-hour component.
-------
~l 1 1 1 1 1 1 1 T
TOTAL NO. OF POINTS «577
1 SAMPLING INTERVAL = i hr
80% CONFIDENCE BAND
l l i i l i i l i l i
00 24 12 8 6 4.8 4 34 3 27 2.4 2.2 2.0
PERIOD, hr
5a
80% CONFIDENCE BAND
00 24 12 8 6 48 4 3.4 3 27 2.4 22 2.0
-0002
5b
Figure 5. Spectrum of a tidal height record.
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CP
'5
o
CM
O
<
n:
06r
0.5
0.3
0.2
O.I
TOTAL NO. OF POINTS »577
SAMPLING INTERVAL I hr
80% CONFIDENCE BAND
I I I I I I I I I I I
00 24 12 8 6 48 4.0 33 3.0 2.7 2.4 Z.2 2.0
PERIOD, hr
6a
0.0
80% CONFIDENCE BAND
00 24 12 8 6 48 40 34 3.0 Z.7 ^4
-0.002
6b
Figure 6. Spectrum of a tidal height record.
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TO STREAM AND ESTUARY FIELD SURVEYS 17
Comparison of Figure 5 with Figure 6 shows the effect of
increasing the number of lags without changing the sampling in-
terval. The resolving power is increased, i.e., estimates are
made for a greater number of spectral bands, but with a slight
loss of confidence in the estimates, as is indicated by the wider
confidence band in Figure 6 than in Figure 5. In this particular
case, however, the increased resolving power demonstrates the
existence of a small but statistically significant overtide (a har-
monic of the semidiurnal tidal component) with a period of about
6 hours. This overtide is not shown in Figure 5b because the
dominating semidiurnal band has overlapped the 6-hour-period
band sufficiently to mask the very small overtide. In Figure 6b
the resolving power of the 24 lags used in computation produces
band-widths sufficiently narrow to prevent overlap of the 12-
and 6-hour periods. The existence of higher harmonics is also
shown in both Figure 5b and Figure 6b. The former figure shows
significant fourth and fifth harmonic overtides, -whereas the
latter shows these and several other short-period components.
The results of the spectral analysis of this record may be
used to point out several characteristics of this technique.
First, each of the spectral values obtained represents an
estimate of the variance over a range of periods in the vicinity
of the nominal value. The range of periods for which each esti-
mate is computed is determined by the sampling interval and the
number of lags used in computation. This range is called the
"equivalent width" (Wg) of the spectral band and can be calcu-
lated in terms of frequency by
we = i
m AT
The spectral estimate reported is an average value for all
periods in the band over the range Wg. For example, for the
results presented in Figure 4, the spectral estimate reported
as corresponding to a 12-hour period is actually an average for
the range of periods from 10. 7 hours to 13. 7 hours. The over-
lapping of spectral estimates is illustrated in Figure 5, and
Figure 6 demonstrates how this overlapping can be reduced by
increasing the number of lags, thereby reducing the equivalent
width of the band for each spectral estimate.
Second, the precision of each estimate is a function of the
total number of samples and the number of lags used in compu-
tation. A method for estimating the number of degrees of free-
dom for each estimate and for establishing the confidence in-
tervals has been presented by Blackman and Tukey . In this
method, the process being measured is regarded as Gaussian
and the degrees of freedom and confidence intervals are based
on a Chi-square distribution. The 80 percent confidence bands
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18 APPLICATION OF SPECTRAL, ANALYSIS
indicated on Figures 4, 5, and 6 have been determined by this
method. It has been previously noted that a decrease in the total
number of samples or an increase in the number of lags causes
a widening of the confidence band, which represents a loss of
precision in each spectral estimate.
Third, if the sampling interval is not small enough to per-
mit resolution of the shortest periods that contribute signifi-
cant variance to the record, the short-period variance in the
record is not lost but is reported at harmonics of the true period.
For example, if this record, as analyzed in Figure 4, contained
significant variance with a period of 4 hours, this would be
shown in the computations as part of the spectral estimate for
the 8-hour period. This occurrence is known as "aliasing" or
"folding. " In Figure 4, the appearance of a significantly large
estimate at the shortest period computed in the analysis indi-
cates that there is probably some aliasing of the short-period
variance. The use of a smaller sampling interval, as shown in
Figure 5, eliminated the aliasing in this spectrum.
Fourth, the appearance of an occasional negative spectral
estimate is an artifact of the computation and results from the
use of a finite record length for spectral computations. If the
record were infinitely long, there would be no negative spectral
estimates; but, when an estimate is close to zero in magnitude,
it may appear with either a positive or negative sign. This
result is interpreted only as a very small quantity of variance,
and no physical significance is attributed to the negative sign.
INTERPRETATION OF SPECTRA
In order to illustrate how the computed spectra of a set of
observations can be used to gain an insight into the structure of
a river or estuarine system, some data obtained in a field sur-
vey of the Potomac Estuary near Washington, D.C. , are pre-
sented here as a basis for discussion. Since the purpose of
this discussion is to present a conceptual picture of how spec-
tral results may be interpreted, the quantitative results obtained
from this particular study are not presented at this time.
In Figures 7 and 8 are presented time series of DO and
5-day biochemical oxygen demand (BOD) measurements ob-
tained at six stations in this estuary. In Figures 9 and 10 are
the spectra computed from these records. Each record con-
tained 145 points obtained at 4-hour intervals; computation was
carried out to 12 lags. Figure 11 is a schematic representation
of the station locations.
-------
TO STREAM AND ESTUARY FIELD SURVEYS 19
Each spectrum shown in Figures 9 and 10 consists of three
major effects: long-period, diurnal (24-hour), and semidiurnal
(12-hour). The estimates adjacent to the long-period, 24-hour,
and 12-hour periods are affected by the strength of the dominat-
ing period and the overlapping of spectral bands, as was dis-
cussed earlier in relation to the tide gage record. The signifi-
cance of each of these effects may be considered separately.
The effects reported in the computation as "long-period" or
of "infinite" period include all components whose periods are
too long to be resolved in the computation. Since in these com-
putations there is overlapping of the 96-hour and "infinite"
periods, this means that all components with periods longer than
48 hours are regarded as long-period components. The use of
the infinity symbol in the figures is merely a convenience in
plotting and should be interpreted as meaning "long-period. "
The components reported as long-period may have one or
more of these physical interpretations:
1. The record may be affected by the existence of
periodic components that are too long to be re-
solved by the length of record available. For
example, a record of air temperature for a
length of several months might show a long-
period effect that, from a record of several
years duration, would appear as an annual cycle.
2. There may be a secular trend in the record that
is fundamentally aperiodic in nature. For ex-
ample, a record of world population over several
centuries might show such a trend.
3. Random sampling, reading, and analytical errors
appear as long-period effects. A constant bias
in the data affects the mean of the record only,
and since spectra are computed from deviations
from the mean, such a bias would not affect the
spectral results.
F.rom the computed spectra alone, it is not possible to
differentiate among random errors, aperiodic effects, and long-
period effects. In a practical sense it is often quite adequate to
regard the long-period spectral estimates as secular trends for
the available record length and to regard any random error as
constant for all records of the same measurement.
The DO spectra presented in Figure 9 exhibit long-period
effects that generally decrease from the upstream to The down-
stream stations. If a constant random error is assumed, the
existence of an effect that is a function of distance from the
head of tidewater may be postulated, and it might be deduced
that this effect is related to the river discharge entering the
estuary above Station 3. Since the parameter being measured
GPQ 8065144
-------
000
6.00'
STATION 3
1 1 1 1 1 1-
~\ 1 1 1 h
E
Q.
D-
D
O
CD
0
000
6.00
5.00
4.00
3.00
2.00
1.00
0.00
STATION 5
-\ 1 1 1 1 1 1
H - 1 - 1
1 - 1
IO 20 30 40 50 60 70 80 9O IOO 110 120 130 I4O
SAMPLE NUMBER
STATION 4
I 1 1 1 HiI
STATION 8
I 1 1_ 1 1 1 1 1 1 J 1 -1.-U I _ ...LJ
10 20 3O 40 50 60 7O 80 9O IOO 110 I2O I3O 140
SAMPLE NUMBER
Figure 7. Dissolved oxygen records obtained in the Potomac Estuary, August 1959.
-------
OL
Q.
Q
O
m
i
6,00
5,00
4.00
300
2.00
1.00
O.OO
800
7.00
6.0O
5.00
400
300
2.00
1.00
8.00
7.00
6.00
5.00
4.00
3.00
1.00
"I P
STATION 3
H 1 1 1 1 1 1 1 ( 1 1 1 1 h
STATION 5
1 1 1H 1 1-
'0 10 20 30 40 50 60' 70 80 90 IOO 110 120 130 140
SAMPLE NUMBER
1 STATION 6
I 1 1 h ItI hI
20 30 40 50 60 70 80 90 100 110 120 130 140
SAMPLE NUMBER
Figure 8. Biochemical oxygen demand records obtained in the Potomac Estuary, August 1959.
-------
0.500
STATION 3
MEAN = 4.60 ppm
I 1 I I I J _ 1 I 1 I I I I
0.500,
48 24 16 12 96 8
PERIOD, hr
STATION 4
M£AN=3.29 ppm
o.ooo
I I I ! I I I I I I
OO 48 24 16 12 9l6
PERIOD, hr
0.400
STATION 5
MEAN = 2.70 ppm
0.400 r-
0.00'
0.300
24 16 12
PERIOD, hr
STATION 7
MEAN =2.53 ppm
O.Oi
24 16 12
PERIOD, hr
STATION 6
MEAN = 2.68 ppm
48
0.300|-
M
ct 0.200
O
Z
<
ec. 0.100
0.000
24 16 12 9.6
PERIOD, hr
STATION 8
MEAN = 2.83 ppm
00 48
24 16 12
PERIOD, hr
9.6 8
Figure 9. Dissolved oxygen spectra in the Potomac Estuary.
-------
O.300
N
§1 O.2OO
o
<
a: 0.100
<
o.ooo
STATION 3
MEAN = 2.44 ppm
00 48 24 16 12 9.6 8.0
PERIOD, hr
0.300
N
a.
OL o.200:
LU
O
<
oi 0.100
0000
STATION 4
MEAN = 3.44 ppm
00 48 24 16 12
PERIOD, hr
9.6 8.0
0.400i
0.000 «-
STATION 5
MEAN=373ppm
0.400'
00 48 24 16 12 9.6 BO
PERIOD, hr
STATION b
MEAN = 4.10 ppm
O.OOO1
OO 48 24 16 12 9l6 8.0
PERIOD, hr
O.SOOi
STATION 7
MEAN = 4.80 ppm
3°° 00 48 24 16 12 96 80
PERIOD, hr
0500
0.400
STATION 8
MEAN = 5.51 ppm
0000
00 48 24 16 12 9.6 80
PERIOD, hr
Figure 10. Biochemical oxygen demand spectra in the Potomac Estuary.
-------
24
APPLICATION OF SPECTRAL ANALYSIS
is DO this long-period effect might be interpreted as a measure
of the amount of DO advected to the system in the river dis-
charge or as an effect of river discharge on the reaeration
characteristics of the system, or both. Considerable caution
is in order if these results are to be interpreted in this manner;
the change in the long-period estimate from Station 3 to Station 8
is barely significant at the 80 percent level, and the differences
Ul
I
o
O
<
10
a
0-
10.000-
20000-
30,000-
40,000-
50,000-
60,000-
HEAD OF
TIDEWATER
STA 3
STA 4
STA 5
STA 6
STA 7
STA 8
POTOMAC
Jt RIVER
MAJOR
WASTE
LOAD
TIDAL
EXCURSION
FROM
MAJOR WASTE
OUTFALL
(Volume
Displacement Basis)
TO
CHESAPEAKE
BAY
Figure 11. Schematic representation of Potomac Estuary sampling station locations.
in this estimate between successive stations are not significant
at the 80 percent level. The existence of a definite trend does
indicate, however, that the interpretation given is reasonable
and affords a basis for the examination of other results in terms
of this hypothesis.
The BOD spectra in Figure 10 present an aspect somewhat
different from the DO spectra at the same stations. The long-
period effects at four of the six stations have very similar values.
-------
TO STREAM AND ESTUARY FIELD SURVEYS 25
Station 7 exhibits a long-period component that is somewhat
higher than these but still lies within the 80 percent confidence
band of the four stations. Station 3 exhibits a long-period effect
that is significantly less than the corresponding effects at the
other five stations. These components at the five downstream
stations can be interpreted as the result of a secular trend in the
major waste discharge to the system as well as random error in
the determination of BOD.
Comparison of the long-period effects at Stations 3 and 4 with
those of stations in the immediate vicinity of the waste outfall
suggests that the limits of a tidal excursion upstream from the
outfall may actually lie between Stations 3 and 4 instead of down-
stream of Station 4, as the calculations based on the tidal prism
volume displacement indicate. The long-period effects at
Station 3 might then be regarded as resulting from the longi-
tudinal mixing of the waste discharge, whereas the similar effects
at the other stations may be regarded as reflecting a combination
of advective and diffusive processes.
The diurnal components of the DO and BOD spectra may be
interpreted as expressing the effects of diurnal variations in
waste discharge and in photosynthetic activity of the planktonic
population. Diurnal variations in waste discharge would affect
stations within a tidal excursion of the outfall more strongly than
those beyond this distance, whereas photosynthetic activity in
the system would be more pronounced at those stations exhibit-
ing the higher nutrient concentrations, generally reflected in
higher mean stream BOD's. Interpreted on this basis, the di-
urnal spectral components suggest that the extent of the up-
stream tidal excursion is between Stations 3 and 4, whereas
Stations 6, 7, and 8 are subject to considerable photosynthetic
activity in addition to the diurnal waste load variations. The
diurnal components of the DO spectra at each station correspond
in size to the respective BOD components; this is a result that
might be expected from the theory of DO-BOD relationships in
streams.
The semidiurnal component of these spectra reflects the ad-
vective motion of DO and stream waste load due to tidal action.
The magnitude of this component at each station is a measure of
the longitudinal concentration gradient of each parameter within
a tidal excursion of the station. The DO spectra show large
semidiurnal effects at Stations 3, 4, and 5, and relatively smaller
ones at Stations 6, 7, and 8; these results indicate that there is
a large DO gradient in the upper part of the system and a small
one in the lower part. Examination of the mean DO values at each
station suggests the existence of an oxygen-sag regime in which
Stations 3 and 4 represent a zone of rapid degradation and
Stations 5, 6, 7, and 8 a zone of critical DO and the beginning of
-------
Z6 APPLICATION OF SPECTRAL ANALYSIS
recovery, a situation that is in close agreement with the spec-
tral results. From this point of view the region of critical
DO is the location that has the smallest semidiurnal spectral
component, in this case Station 7.
The semidiurnal BOD spectral components present a picture
somewhat different than the DO spectra. At the two upstream
stations this component of the BOD spectrum is small. There is
a significant increase in this component at Station 5 and again at
Station 6, whereas the semidiurnal effects at Stations 7 and 8
are of the same magnitude as that at Station 6. These results
suggest a low BOD gradient in the upper reaches of the system
and a high gradient in the lower reaches, and Station 5 repre-
sents a region of transition. The strong gradients near and
below the waste outfall suggest that the semidiuranl spectral
components are affected by variations in the initial mixing of
the waste load throughout the volume of water passing the out-
fall in a tidal excursion. It is apparent that the mean BOD's
do not show such changes in gradient along the estuary. Com-
parison of the BOD gradient regime shown by the spectral analy-
sis (with that shown by the mean BOD values at each station
illustrates the sensitivity of spectral analysis) as a tool in
estuarine engineering.
It has been the purpose of this discussion to demonstrate
how spectral results can be interpreted in terms of familiar
sanitary engineering concepts. It is not intended to suggest that
the spectra can supply no inofrmation in addition to that dis-
cussed. Spectral analysis is purely and simply a tool for the
analysis of time-series data. It provides by itself no theoreti-
can insight into natural processes, but it does permit one to
examine individual periodic components of the data, with a
minimum of interference from other components. As with any
other statistical technique, the final interpretation of these
results must be based on an understanding of the natural pro-
cess, not on some magic numbers produced by the manipula-
tion of data.
DESIGN CRITERIA FOR SPECTRAL ANALYSIS
In the design of any spectral analysis program the require-
ments for precision in each spectral estimate, for resolution of
sufficient spectral bands, and for eliminating aliasing, or folding,
must all be considered and balanced against each other before
the sampling program and data analysis are begun.
An acceptable balance among these requirements can be es-
tablished by careful choice of the sampling interval, the record
length, and the number of lags used in computation.
-------
TO STREAM AND ESTUARY FIELD SURVEYS 27
The choice of appropriate sampling and computational fac-
tors should be based on reasonable assumptions of what the
dominant periodicities in the system are. The shortest period
it is necessary to resolve determines the sampling interval re-
quired, whereas the longest period necessary determines the
total record length. In most sanitary engineering applications,
it may be assumed that diurnal fluctuations in waste discharges
and in photosynthetic activity will be of considerable importance.
In tidal systems there will be significant semidiurnal periodi-
cities; there may also be some effects of this short a period in
waste discharges and in the biological regime for a non tidal
system. As a general basis for experimental design, it may
be assumed that resolution of 24-hour periods will be required
in a non tidal system and that resolution of 12-hour periods will
be required in a tidal system.
The shortest period it is theoretically possible to resolve
with a given sampling interval is the period that is twice the
sampling interval. That is, with a sampling interval of 6 hours,
it is theoretically possible to resolve a 12-hour period. From a
practical standpoint it is not possible to do this, since the 12-hour
estimate would be the shortest period computed, and there would
arise the question whether this is a valid estimate of a 12-hour
period or whether it merely represents the aliasing of periods
shorter than 12 hours. It is advisable that the sampling interval
chosen be small enough to provide spectral estimates for
several periods shorter than the expected dominant shortest
period, so that this period is minimally affected by any aliasing
that might occur.
As a rule of thumb, it is suggested that a maximum sam-
pling interval of 8 hours is required to resolve 24-hour periods
and an interval of 4 hours to resolve 12-hour periods. In gen-
eral, a sampling interval of not more than one-third the length
of the shortest significant period is recommended.
There are no clear criteria for determining the longest
period that can be resolved from a given record. The longest
period resolved in any particular analysis (other than the "in-
finite" period estimate) is determined by the number of lags
used in computation and by the sampling interval. In general,
computation to a number of lags greater than 10 percent of the
total number of measurements in the record is not recommended,
i. e. , for a record of 140 measurements, computation to no more
than 14 lags is recommended. Each additional lag used_in com-
putation reduces the precision of all spectral estimates com-
puted, and it is generally regarded that the 10 percent value
affords an optimum balance between precision of individual es-
timates and resolution of spectral components. As a guide in
determining the required record length for design purposes, it
may be assumed that a record length at least 10 times as long as
-------
28 APPLICATION OF SPECTRAL ANALYSIS
the longest significant period to be resolved will be required.
For many sanitary engineering field surveys, the 24-hour period
is about the longest significant period it is necessary to resolve
from field survey data; in such cases a minimum record length
of 240 hours would be required.
As a basis for selecting the number of lags to be used in
computation 10 percent of the number of measurements in the
record is used as an upper limit. The number of lags finally
chosen will probably be based on the resolution believed to be
required. For example, if it is desired to separate diurnal and
semidiurnal components, the number of lags chosen must be
large enough to include several estimates between the 24-hour
and 12-hour estimates, so that there is a negligible amount of
interference between the major components. An acceptable
degree of precision is then obtained by increasing the record
length if the numbers of degrees of freedom on which each
estimate is based give a confidence band that is too large to give
usable results.
Any type of time-series record can be subjected to spectral
analysis if it represents sampling at uniform time intervals and
if there are no missing points; if a few points are missing, how-
ever, a limited amount of interpolation may be done. Interpola-
tion of the mean value of all the measurements or linear inter-
polation of a missing point between two measurements is the
usual approach. There are no definite criteria that serve as
an indication of how much interpolation can be done in any par-
ticular case; a general consideration of the process suggests"
however, that if the missing points are widely scattered up to
possibly 5 per cent of the data may be interpolated without ser-
ious effects on the computed spectra.
The obtaining of data suitable for spectral analysis is
most simply and cheaply accomplished by means of automatic
sampling and recording equipment. Conversely, spectral analy-
sis offers the most effective means of analyzing and correlating
the large quantities of data produced by such instruments. This
does not mean, of course, that the computation of spectra is
limited to time-series data obtained from automatic devices.
The DO and BOD data discussed here were obtained by conven-
tional manual sampling procedures, whereas the tide height
record was obtained from an automatic recording device. If
the requirements of uniform sampling interval and record
length are met, the means by which the data are obtained is
immaterial.
The computation of spectra is most cheaply and efficiently
accomplished by high-speed digital computer. The spectra pre-
sented here were computed on an IBM 704; the total cost, in-
cluding preparation of the data for the computer, was estimated
-------
TO STREAM AND ESTUARY FIELD SURVEYS 29
at less than 10 dollars per spectrum. Computation by hand for
all but very short record lengths is prohibitive from the stand-
points of time, money, and accuracy.
CONCLUSION
It has been the purpose of this paper to present the technique
of spectral analysis as a statistical tool that can be used in a
wide variety of sanitary engineering applications. A strictly
operational viewpoint has been maintained, and the theoretical
basis of generalized harmonic analysis and spectral computa-
tion has been ignored.
It is unfortunate that there is no single work that can be
offered as a primary reference on the theory of spectral
analysis. Pertinent references on the theory and practice of
spectral analysis are given in the bibliography. Of these,
Panofsky and Briar present an introduction to the subject with
emphasis on the meteorological uses of the technique, Bendat
present some of the more practical aspects of the measurement
of power spectra. The other references are concerned with the
application of spectral techniques to the fields of meteorology,
oceanography, and aeronautical engineering, and with more de-
tailed discussions of the mathematical basis of spectral analysis.
Although a knowledge of the theory underlying the technique of
spectral analysis is certainly desirable, a lack of appreciation
of the finer points of the mathematical development need not pre-
vent the successful use of spectral analysis as a useful statisti-
cal tool in the solving of engineering problems.
The discussion here has been limited to the computation and
interpretation of the spectrum of an individual time-series
record. When two different records (perhaps a DO record and a
BOD record) are analyzed together in spectral computation, the
result, called the "cross-sprectrum, " is a much more powerful
tool than are the individual spectra. Cross-spectra, when com-
bined with other spectral calculations, produce among other
things quantitative information on the response of one record to
another (the change of DO as a function of the change in BOD,
for example) and on the time lag with which the response occurs.
Although this type of information is certainly of considerable im-
portance in engineering problems, this discussion has been
directed toward presenting the foundation upon which the more
esoteric spectral calculation rest.
-------
30 APPLICATION OF SPECTRAL ANALYSIS
ACKNOWLEDGMENTS
The author is deeply indebted to Dr. Blair Kinsman of the
Chesapeake Bay Institute for supervising his initiation into the
mysteries of spectral analysis.
The data discussed here were obtained by the Public Health
Service during a field survey of the Potomac Estuary. This sur-
vey was conducted at the request of the U.S. Army Corps of
Engineers as part of a comprehensive study of the water re-
sources of the Potomac River Basin. The permission of the
Corps of Engineers to use these data is gratefully acknowledged.
GPQ 806-514-3
-------
BIBLIOGRAPHY
1. Barber, N. F. Experimental Correlograms and Fourier
Transforms. Pergamon Press, New York, N. Y. , 1961.
2. Bendat, J. S. Principles and Applications of Random
Noise Theory. Wiley, New York, N. Y. , 1958.
3. Blackman, R. B. andj. W. Tukey. The Measurement of
Power Spectra. Dover, New York, N.Y., 1958.
4. Grenander, U. and M. Rosenblatt. Statistical Analysis of
Stationary Time Series. Wiley, New York, N. Y. , 1957.
5. Hannan, E. J. Time Series Analysis. Wiley, New York,
N.Y., 1960.
6. Kinsman, B. Surface.Waves at Short Fetches and Low Wind
Speeds --A Field Study. Chesapeake Bay Inst. , Johns
Hopkins Univ., Tech. Rep. XIX, May I960.
7. Marks, W. and W. J. Pierson. The power spectrum
analysis of ocean-wave records. Trans. Am. Geophys.
Union. 33:834-44. 1952.
8. Munk, W. H. , F. E. Snodgrass, and M. J. Tucker.
Spectra of low-frequency ocean waves. Bui. Scripps Inst.
Oceanog. Univ. Calif. 7(4): 283-362. 1959.
9. Panofsky, H. A. Meteorological applications of power
spectrum analysis. Bull. Am. Meteorol. Soc. 36:163-66.
1955.
10. Panofsky, H. A. and G. W. Brier. Some Applications of
Statistics to Meteorology. Pennsylvania State University.
1958.
11. Press, H. and J. W. Tukey. Power Spectral Methods of
Analysis and Their Application to Problems in Airplane
Dynamics. Flight Test Manual, NATO, Advisory Group
for Research and Development, IV-C, 1-41. June 1956.
12. Rice, S. O. Mathematical analysis of random noise. Bell
System Tech. 23:282-332. July 1944; 24:46-156. Jan.
1956. Reprinted in: Selected Paper on Noise and Stochastic
Processes. N. Wax, ed. Dover, New York, N. Y. , 1954.
13. Taylor, G. I. Statistical theory of turbulence. Proc.
Roy. Soc. (London). A151:421-78. 1935.
31
CPQ 806-514-2
-------
BIBLIOGRAPHIC: Wastler, T.A. Application of
spectral analysis to stream and estuary field
surveys. I. Individual power spectra.
PHS Publ. No. 999-WP-7. 1963. 31 pp.
ABSTRACT: The application of spectral analysis
techniques to sanitary engineering stream and
estuary studies is discussed from a practical
operational viewpoint. Techniques of interpre-
tation and the data requirements are emphasized
rather than the mathematical basis and details
of the technique. The usefulness of spectral
analysis in analyzing records obtained from
continuous, automatic monitoring stations is
pointed out. Spectral analyses applied to tidal
height records and dissolved oxygen records
obtained in a field study of the Potomac Estuary
are discussed. The discussion is limited to the
application of individual power spectra computa-
tion to sanitary engineering investigations.
ACCESSION NO.
KEY WORDS:
Spectral Analysis
Stream Data
Estuarine Data
Statistics
Power Spectra
Potomac Estuary
BIBLIOGRAPHIC: Wastler, T.A. Application of
spectral analysis to stream and estuary field
surveys. I. Individual power spectra.
PHS Publ. No. 999-WP-7. J963. 31 pp.
ABSTRACT: The application of spectral analysis
techniques to sanitary engineering stream and
estuary studies is discussed from a practical
operational viewpoint. Techniques of interpre-
tation and the data requirements are emphasized
rather than the mathematical basis and details
of the technique. The usefulness of spectral
analysis in analyzing records obtained from
continuous, automatic monitoring stations is
pointed out. Spectral analyses applied to tidal
height records and dissolved oxygen records
obtained in a field study of the Potomac Estuary
are discussed. The discussion is limited to the
application of individual power spectra computa-
tion to sanitary engineering investigations.
ACCESSION NO.
KEY WORDS:
Spectral Analysis
Stream Data
Estuarine Data
Statistics
Power Spectra
Potomac Estuary
BIBLIOGRAPHIC: Wastler, T.A. Application of
spectral analysis to stream and estuary field
surveys. I. Individual power apectra.
PHS Publ. No. 999-WP-7. 1963. 31 pp.
ABSTRACT: The application of spectral analysis
techniques to sanitary engineering stream and
estuary studies is discussed from a practical
operational viewpoint. Techniques of interpre-
tation and the data requirements are emphasized
rather than the mathematical basis and details
of the technique. The usefulness of spectral
analysis in analyzing records obtained from
continuous, automatic monitoring stations is
pointed out. Spectral analyses applied to tidal
height records and dissolved oxygen records
obtained in a field study of the Potomac Estuary
are discussed. The discussion is limited to the
application of individual power spectra computa-
tion to sanitary engineering investigations.
ACCESSION NO.
KEY WORDS:
Spectral Analysis
Stream Data
Estuarine Data
Statistics
Power Spectra
Potomac Estuary
GPQ 8C65145
------- |