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.

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                ~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

-------
 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.

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 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 806—514—4

-------
     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	Hi—I
                                                                    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	1—H	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	It—I	h—I
                                                                   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.

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 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 8C6—514—5

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