Analysis and prediction of streamflow and precipitation data
by Alfred Benjamin Cunningham
A thesis submitted to the Graduate Faculty in partial fulfillment of the requirements for the degree of
MASTER OF SCIENCE in Civil Engineering
Montana State University
© Copyright by Alfred Benjamin Cunningham (1971)
Abstract:
The double mass analysis and a related method of data synthesis are used to develop a computerized
data analysis and generation model. Total monthly volumes of precipitation and streamflow are the
types of data for which the model is designed. The double mass analysis is used to check the
consistency of the data at a particular station. Then, using the equation of the double mass curve,
periods of missing record are synthesized for the station in question. A comparison can be made to
determine if cyclic variations exist between the synthetic and actual data. In the event that cyclic
variations do occur, correction factors can be applied to improve the accuracy of the synthetic data.
Although the data analysis and generation model was developed for use on the streamflow and
precipitation data in Montana, its general structure does not restrict its use to any one geographic
region. Statement of Permission.to Copy
In presenting this thesis in partial fulfillment of the
requirements for an advanced degree at Montana State University,
I agree that the Library shall make it freely available for
inspection. 'I further agree that permission for extensive copying
of this thesis for scholarly purposes may be granted by my major
professor, or, in his absence, by the Director of Libraries.
It
is understood that any copying or publication of this thesis for
financial gain shall not be allowed without my written permission.
Signature
Date
ANALYSIS AND PREDICTION OF STREAMFLOW AND PRECIPITATION DATA
by
ALFRED BENJAMIN CUNNINGHAM
A thesis submitted to the Graduate Faculty in partial
fulfillment of the requirements for the degree
of
MASTER OF SCIENCE
Civil Engineering
-yappro;
d:
Head, Major Deparment
Chairman, Examining Committee
Graduate Dean
/
MONTANA STATE UNIVERSITY
Bozeman, Montana
December, 1971
• ACKNOWLEDGMENTS
The author wishes to extend his thanks to the faculty of the
Civil Engineering and Engineering Mechanics department of Montana
State University for their willing assistance, and especially to
Professor T. T. Williams for Ais help and guidance in preparing this
thesis.
Thanks are also extended to the author's wife who, along with
Mrs. Neta Eckenweiler prepared and typed this thesis.
XV
TABLE OF CONTENTS
Page Number
VITA.................... ............................ .
ACKNOWLEDGEMENTS. . . . . .
I
..................
ii
....
iii
LIST OF FIGURES AND TABLES............................
vi
ABSTRACT. . .............................
ix
INTRODUCTION.......... ............................
I
STATE WATER PLANNING MODEL........ ....................
I
SUMMARY OF PROBLEMS WITH EXISTING DATA. . . . . . . . .
4
STATEMENT OF OBJECTIVES ..............................
4
LITERATURE REVIEW........................................
6
MODELING TECHNIQUES FOR HYDROLOGIC STUDIES............
6
Mechanical Analog.......................... ..
6
Electric Analog....................................
6
Digital Computer ......................
..........
7
STREAMFLOW SYNTHESIS.......... ........................
7
PRECIPITATION SYNTHESIS . . ............'..............
8
CONSISTENCY CHECK FOR STREAMFLOW AND PRECIPITATION DATA
8
DEVELOPMENT OF TECHNIQUE FOR DATA ANALYSIS AND GENERATION.
10
\
DATA A N A L Y S I S ........ ■...............................
10
Consistency Check..................................
10
Correction of Inconsistent Data........ ..
11
V
DATA GENERATION
Page Number
15
Criteria Necessary for SyntheticData Generation
^
Double Mass Extension..........
^
Multiple Regression...........................
MODEL STRUCTURE..................................
DATA ANALYSIS SUBROUTINE..... ....................
22
22
Discontinuity Detection.............. .........
Evaluation of the Data Analysis Subroutine...,.
2b
DATA CORRECTION AND GENERATION SUBROUTINE.........
27
Data Correction...............................
27
Data Generation...............
27
Evaluation of the Data Correction and
Generation Subroutine.........................
28
PRESENTATION OF RESULTS.................................
30
'
DESCRIPTION OF RESULTS.... .........................
30
SIGNIFICANT PARAMETERS.............................
30
DISCUSSION OF RESULTS............
50
ACCURACY VS. CORRELATION...........
50
EFFECTIVENESS OF DOUBLE MASS ANALYSIS.....
52
CYCLIC VARIATIONS..................................
54
OPERATION PROCEDURE FOR DATA ANALYSIS
AND GENERATION MODEL................
56
vi
Page Number
SUMMARY AND RECOMMENDATIONS FOR FURTHER RESEARCH...........
APPENDIX A. LITERATURE CONSULTED...........................
APPENDIX B . PROGRAM LISTING................................
58
'
61
62
.
vii
LIST OE FIGURES AND TABLES
Figure
I
Title .
Page Number
ADJUSTMENT OF INCONSISTENT DOUBLE MASS POINTS. . . .
12
I
2
DOUBLE MASS EXTENSION.............
18
.
3
MULTIPLE LINEAR REGRESSION .........................
20
1
4
DETECTION OF DISCONTINUITIES IN DOUBLE MASS ANALYSIS
23
5
GENERATION OF SYNTHETIC DATA BY DOUBLE MASS EXTENSION
29
6-A
DOUBLE MASS DIAGRAM - HYALITE CREEK AT RANGER
STATION VS. GALLATIN RIVER AT GALLATIN GATEWAY . .
35
COMPARISON OF DATA FOR HYALITE CREEK - ACTUAL DATA
VS. SYNTHETIC DATA ............................. .
36
6-B
7
PER CENT ERROR VS. CORRELATION COEFFICIENT . . . . .
37
8-A
DOUBLE MASS DIAGRAM - POWDER RIVER AT LOCATE VS.
YELLOWSTONE RIVER AT SIDNEY. . . . . . . . . . . .
38
8-B
9-A
DATA COMPARISON FOR POWDER RIVER - ACTUAL DATA VS.
SYNTHETIC D A T A ..............................
39
DOUBLE MASS DIAGRAM - GALLATIN RIVER AT GALLATIN
GATEWAY VS. MADISON RIVER AT WEST YELLOWSTONE. . .
40
!.
9-B
IO-A
10-B
11-A
DATA COMPARISON FOR GALLATIN RIVER - ACTUAL DATA VS.
SYNTHETIC AND ADJUSTED SYNTHETIC DATA........
41
DOUBLE MASS DIAGRAM -YELLOWSTONE RIVER AT
YELLOWSTONE LAKE VS. MADISON RIVER AT WEST
YELLOWSTONE............ ..........................
42
DATA COMPARISON FOR YELLOWSTONE RIVER - ACTUAL DATA
VS. SYNTHETIC AND ADJUSTED SYNTHETIC DATA.
....
43
DOUBLE MASS DIAGRAM - PRECIPITATION AT WYOLA VS.
PRECIPITATION AT SIDNEY........ ..
44
viii
Figure
11-B
12-A
12-B
13-A
13-B
Table
I
!H i
. i
Title
Page Number
DATA COMPARISON FOR WYOLA PRECIPITATION - ACTUAL
DATA VS. SYNTHETIC DATA. .........................
45
DOUBLE MASS DIAGRAM - PRECIPITATION AT MONTANA
STATE UNIVERSITY VS. PRECIPITATION AT TRIDENT. . .
46
DATA COMPARISON FOR MSU PRECIPITATION - ACTUAL
DATA VS. SYNTHETIC DATA..........................
47
DOUBLE MASS DIAGRAM - PRECIPITATION AT PRYOR VS.
PRECIPITATION AT BILLINGS........................
48
DATA COMPARISON FOR PRYOR PRECIPITATION - ACTUAL
DATA VS. SYNTHETIC DATA. . . . . . ..............
49
Title
Page Number
OCCURRENCES WHICH CAUSE DISCONTINUITIES IN DOUBLE
MASS ANALYSIS.....................................
14
DATA FOR THE PRECIPITATION AND STREAMFLOW STATIONS
WHICH WERE STUDIED . . . .........................
33
ix
ABSTRACT
The double mass analysis and a related method of data
synthesis are used to develop a computerized data analysis
and generation model.
Total monthly volumes of precipitation
and streamflow are the types of data for which the model is
designed.
The. double mass analysis is used to check the con­
sistency of the data at a particular station.
Then, using
the equation of the double mass curve, periods of missing record
are synthesized for the station in question.
A comparison can
be made to determine if cyclic variations exist between the
synthetic and actual data.
In the event that cyclic variations
do occur, correction factors can be applied to improve the
accuracy of the synthetic data.
Although the data analysis and generation model was de­
veloped for use on the streamflow and precipitation data in
Montana, its general structure does not restrict its use to
any one geographic region.
Chapter I
INTRODUCTION
In the last several years, the proper development and management
of water resources has become an issue almost everywhere.
Previously
the greatest concern had been to maintain an adequate water supply
for large urban areas; however, it is now becoming apparent that
proper development and management is needed everywhere if the optimum
use of available water resources is to be achieved.
The state of
Montana, even with its abundant water resources and sparse population,
is no exception.
In fact, because so many other areas of the country
depend on the water originating in Montana, the need for management
in Montana is indeed great.
Until recently, there has not been a concentrated effort to put
water resource planning and management techniques to use in this
state.
However, with the organization of the Montana Water Resources
Board out of the old Water Conservation Board in 1967, and the
subsequent increase in public support, water resource development
and management has become a top priority issue.
STATE WATER PLANNING MODEL
In 1968 the Departments of Civil Engineering & Engineering
Mechanics and Industrial & Management Engineering at Montana State
University contracted with •'the, Montana Water Resources Board to
-2develop a statewide water planning model.
Although there are several
types of modeling techniques in use, the approach taken was to develop
a mathematical model for use on the digital computer.
This is accom­
plished by deriving mathematical expressions to represent the inter­
action of certain hydrologic parameters and programming these ex­
pressions for a computer solution.
Once these expressions have been
derived, it is possible to determine the distribution of the ground
and surface water throughout the state for any specified time interval.
Having this capability, it will then be possible to evaluate the
effects of proposed projects on the state's water resource system.
. To accomplish this objective, the state water model will utilize
large quantities of hydrologic and geologic data - most of which have
never been collected on a regular basis.
It is apparent, therefore,
that a key to a successful model lies in developing accurate methods
of estimating missing data.
Two types of data which are vital to the model are.(I) records
of the precipitation which falls on the state and (2) records of the
streamflow which occurs within the state.
Although these data are
collected for Montana by the National Weather Service and the U.S.
Geological Survey, the records at many locations are presently
inadequate.
This is primarily because precipitation and streamflow
records in Montana range in length from one or two years up to fifty
or more years.
Also, the data for some locations nave not been
-3continuously collected throughout the lifetime of the station.
In
. fact an examination revealed that missing record periods are present
in approximately 80% of the available precipitation and streamflow
records for Montana.
Thus if a method could be developed whereby
periods of missing record could be synthesized accurately, a signifi­
cant improvement to the State Water Planning Model would be made.
"
Another inadequacy in the data that must be recognized is that
some periods of record are "inconsistent
It is important here to
realize what is meant and not confuse the term "erroneous" with the
term "inconsistent."
The record for a particular station is "erro­
neous" if the gage inaccurately measures the true quantity of pre.cipitation falling or streamflow passing.
The record is "inconsistent"
if some, natural or man-caused occurrence (such as the physical fe:location of a gage to a different site) causes a shift or discontinuity
'in the data.
v
For example, consider a stream gaging station at which data
.’have been collected for a long period of years.
If this station
"were then to be moved upstream past a major tributary .to the river, '
■the streamflow record after the move would be inconsistent with the
i•
! record before the move. That is, the recorded flow would then be less
I
•
j
compared to what it would have been if the gage had not been moved.
.
Notice in this example that the gaging station is assumed to record
'the true flows at each location.
But if the record from this station
is assumed to consist of the two segments of data collected from the
-4two sites without adjusting the data from one, then the record would
be said to be "erroneous."
It is easily seen that if the presence of inconsistent records
is not detected, any attempt to use these records in a watershed model
or to synthesize missing data could result in serious error.
Therefore
a necessary prerequisite for data synthesis is that existing data
which is to be used, first be tested for consistency.
I
SUMMARY >OF PROBLEMS WITH EXISTING DATA
The problems associated with the streamflow and precipitation
data for the State Water Planning Model can be summarized in the
following way.
The primary problem with the data is that periods of data are
missing from about 80% of the existing records.
Therefore a way must
be found to generate synthetic data in order to have complete records
at all desired locations.
The second problem which is less significant (though still
important) is that some existing records are inconsistent and must be
corrected before being used for data generation or any other purpose.
Therefore a way must be found to test existing streamflow and pre­
cipitation data for consistency.
STATEMENT OF OBJECTIVES
Based on the foregoing statements concerning the need for the
development of a "Data Analysis and Generation Model,"' the following
objectives are stated.
-5(1) .
Construct a computerized "Data Analysis Model" which
will check the consistency of existing precipitation
and streamflow data.
(2) .
Construct a computerized "Data Generation and Correction
Model" capable of synthesizing missing periods of monthly
streamflow and precipitation volumes - as well as
correcting existing data which are inconsistent.
(5).
Describe the operation procedures for these models
which will yield the best possible results.
i
,'i
■$>
Chapter 2
LITERATURE REVIEW
A survey of pertinent literature was conducted to analyze the
I various hydrologic modeling techniques as well as the various methods
I of data analysis and generation.
MODELING TECHNIQUES FOR HYDROLOGIC STUDIES
Three types of modeling techniques which are currently being
used in hydrologic studies are given by M'ount, (1965).
These three
types are (I) the mechanical-analog model, (2) the electire-analog
model, and (3) the digital computer model.
Mechanical Analog
JS
An example of applying a mechanical-analog model to a hydro-
i logic study would be to use a stretched membrane to represent the
; effect of pumping water from a well.
As the well is pumped, the water
table assumes the shape of an inverted cone with the apex at the well.
; The same shape occurs in a membrane when it is pressed with a sharp
instrument.
In this case the elastic properties of the membrane are
: analogous to the water transmitting properties of the aquifer and
. the magnitude o'f the point force applied to the membrane is analogous
to the rate of withdrawal from the aquifer.
Electric Analog
Electrie-analog models, however, appear to have greater utility
than mechanical-analog models because electric properties have
-7analogs in many- physical systems.
However, in hydrologic studies,
electrical systems can be used to model certain aspects of the .
hydrologic cycle.
For example, an electrie-analog model could be
constructed to depict the changes in ground and surface water condi­
tions for a given area.
Here, the ground and surface storage would be
''represented by capacitance
streamflow, precipitation, and ground
water movement would be represented by current; and soil.transmissibility would be represented by resistance.
Digital Computer
Although the electrie-analog and the mechanical analog models
are useful in many cases, the digital computer model is more versatile
and therefore used more often.
The application of digital computer
modeling to hydrology involves determining mathematical relations
which represent the interaction of hydrologic parameters.
For example,
consider the Stanford Watershed Model which is described by Linsley
. and Crawford, (1966) .
In this model virtually every aspect of the
hydrologic cycle has been represented mathematically and programed
for a computer solution.
Once the required hydrologic data have been
obtained for a particular watershed, the Stanford Watershed Model can
be used to predict the outflow hydrograph resulting from any theoreti­
cal storm which could occur on the watershed.
STREAMFLOW SYNTHESIS
A computer solution for estimating periods of missing streamflow record was developed by Beard, (1967).
The procedure was to
-8make the existing data at a particular gaging station a function of
the data from a group of surrounding stations.
The method of least
squares is used as the basis for this analysis and results in the
development of a linear relation between the data for the particular
station and the data for the surrounding stations.
Once this equation
is determined, Beard's program can then fill in any periods of missing
record for the particular station.
PRECIPITATION SYNTHESIS
'■
.
■Linsley, Kohler, Paulhus, (1950) presented a method which
graphically determines the distribution of rainfall from a particular
storm over a given area.
This is done by first plotting the recorded
,3 rainfall at each of the gaging stations for the area in question.
>. Then, isohyets or "rainfall contours" are drawn to indicate how the
rainfall was distributed over the area.
Thus the rainfall could be
\
estimated for any station which was not operating at the time of the
storm.
Though it would be . quite time consuming, an entire period of
, record at a given station could be estimated by repeating this proi cedure for each storm that -occurred" during the time the gage was not
operating.
• CONSISTENCY CHECK FOR STREAMFLOW AND PRECIPITATION DATA
A survey of available literature revealed that the standard
method for testing the consistency of hydrologic data is the "double
mass analysis" (DMA).
literature.
In fact no other method was found in the
A development of the DMA for use in testing the
consistency of precipitation data is presented by Linsley, Kohler, &
Paulhus, (1958).
Although the DMA is primarily used for precipitation
data, it is pointed by Linsley, Kohler, Paulhus, (1949), that the DMA
can be used to test the consistency of streamflow data as well.
It is because of this capability for testing the consistency of
both types of data that the double mass analysis was chosen as the basis
'for the data analysis and generation model.
R. Singh (1968) developed a computer solution for the double mass
analysis.
However, his program was deemed inappropriate for this study
because it could not detect the presence of more than one period of
inconsistent data.
V
Chapter 3
DEVELOPMENT OF TECHNIQUES FOR DATA ANALYSIS AND GENERATION
Based on the results of the literature survey, techniques were
developed for both the analysis of existing data and generation of
synthetic data.
DATA ANALYSIS
As previously stated, the double mass analysis is used by hy­
drologists to test the consistency of both precipitation and streamflow data.
Consider now how the DMA is applied to precipitation data1.
Consistency Check
The consistency of precipitation data is checked by comparing
the accumulated precipitation at a given station with the accumulated
precipitation for a group of surrounding stations.
.The procedure is
to plot the accumulated rainfall values at the particular station as
the dependent variable and the concurrent accumulated rainfall for
the surrounding stations as the independent variable.
If the records
from the stations are well correlated, as they should be if the
stations are in the same hydrologic unit, the points should plot
approximately on a straight line.
However if a slope change is
necessary to fit line segments through the points, it is highly
probable that an inconsistency exists in the data for the dependent
station.
If a slope change is detected, it is important to check
the history of the dependent station to confirm the existence of the
-11data inconsistency.
If the gage history reveals the occurrence of art
event which could cause a slope change in the data, then it is almost
certain that an inconsistency in the dependent station record exists.
If the slope change cannot be substantiated from the gage history, it
is a judgment decision as to whether the slope change does in fact
indicate a discontinuity in the data.
Care must be taken in this case
not to interpret the natural scatter of the points as an inconsistency
in the data.
The possible reasons for discontinuites in both precipi­
tation and streamflow data are discussed later in this chapter.
The above discussion is an explanation of the procedure involved
in performing the double mass analysis.
Although this procedure was
discussed with respect to precipitation data, it is important to note
that streamflow data can be analyzed in exactly the same manner.
The
only difference is that these two types of data will have different
units.
The streamflow data used in this study will have the units of
acre-ft/month while the precipitation data will be in inches/month". Correction of Inconsistent Data
Once the DMA has detected a discontinuity in the record of the
dependent station, correction of the erroneous data is easily done.
The procedure is to multiply each erroneous double mass ordinate by
the ratio of the slopes of the two line segments constructed through
the double mass points..
To illustrate, a double mass curve is shown in Figure I.
individual data values are shown along with the resulting double
The
-12-
L
-
1 .0
2 .5
1 .0
.5
2 .0
.5
3 -0
1 .0
2 .0
2 .0
CUMULATIVE
D E P E N D E N T ^ STA T | O N
DATA
25
X
X
2.0
2 .0
7 .0
9 .0
1 0 .0
i4 .o
1 5 .0
1 6 .5
1 7 .0
1 8 .0
1 9 .0
Y
-
1 .0
3 .5
4 .0
6 .0
6 .5
9 .5
1 0 .5
1 2 .5
1 4 .5
1 6 .5
5 .0
2 .0
1 .0
4 .0
1 .0
1 .5
.5
1 .0
1 .0
y = Dependent station data.
Y = Ordinates on double
mass diagram.
x = Surrounding station data. X = Abscissa values
for double mass
diagram.
Slope
segment 2
10
^ Adjusted data points
S
Segment I
lope = .5
1940
5
CUMULATIVE
FIGURE I.
10
SURROUNDING
15
STATI ON
20
DATA
ADJUSTMENT OF INCONSISTENT DOUBLE MASS POINTS
25
-13mass coordinates to demonstrate how the double mass lines were con­
structed.
Plotting the double mass coordinates (Y vs. X), shows that
a slope change occurs approximately in 1950.
Assuming that the slope
change does in fact represent a discontinuity in the dependent station
record, the record from 1950 to 1960 can be corrected by multiplying
the segment of the ordinates which lie above the discontinuity by the
ratio .5/2.0 so as to be consistent with the points on segment I.
Thus far it has been assumed that if a discontinuity is found by
a double mass analysis the record for the dependent station is auto­
matically in error.
However, in the event that only one surrounding
station is used as the independent station, it is equally possible that
a discontinuity could be due to erroneous data for the independent
station.
In this case the record for both the independent station and
the dependent station must be examined if a slope change is found by
the double mass analysis.
The most common causes for the occurence of
discontinuities in streamflow and precipitation data are listed in
Table I.
Where precipitation data is concerned, it has been observed
by the National Weather Service, according to Linsley, Kohler, and
Paulhus (1958) , that a change in gage location can be a significant
factor in causing the data to be inconsistent.
In some cases a change
in location of less than five miles can cause significant error in the
gage record.
Changes in the exposure of a gage can also be a signifi­
cant factor in affecting the accuracy of the gage record.
This happens
in areas where trees or vegetation are allowed to grow up around a
gage, thus affecting the catch.
-14TABLE I
OCCURENCES WHICH CAUSE DISCONTINUITIES
IN THE DOUBLE MASS ANALYSIS
PRECIPITATION DATA
1.
Changes in gage location
2.
Changes in the gage exposure
3.
Changes in instrumentation
4.
Changes in observation techniques
STREAMFLOW DATA
1.
Construction of hydraulic structures
2.
Changes in diversion practices
3.
Changes in gage location
4.
Changes in observation procedure
5.
Changes in instrumentation
6 . Erosion or sedimentation in vicinity of gage
Variations in observation procedure or changes in instrumenta­
tion can also affect the accuracy of raingage data.
For example
changing from a simple volumetric recorder to a continuous recorder
could change the accuracy of a monthly gage record considerably.
Also if the observation procedure for a simple volumetric recorder is
-15changed so that the gage is checked every month instead of daily, a
decrease in the accuracy of the monthly precipitation values could be
observed.
It is these types of "accuracy changes" which can also
cause inconsistencies in rain gage data.
In the case of monthly streamflow data, changes in instrumenta­
tion gage location and observation procedure could affect the accuracy
of the record in much the same way as they affect precipitation records.
However other possible causes for discontinuities in streamflow data
are changes in diversion practices and construction of hydraulic
structures.
The occurrence of either event above a stream gaging
station will almost certainly cause a discontinuity in the record.
DATA GENERATION
.
Once all existing periods of data for a particular station
have been checked'for consistency, the next task is to fill in the
missing record periods for which there are concurrent records from
surrounding stations.
Before the various alternatives are explained
it is necessary to define the criteria needed to permit missing data
to be synthesized.
These criteria are presented below.
Criteria Necessary For Synthetic Data Generation
(I)
The data used to obtain a relationship between the de­
pendent variable
( the data for the dependent station)
and the independent variable
( the data for the surround­
ing stations) must be tested and found to be consistent.
-16(2)
The independent variable data which is used to fill in
missing periods of dependent variable data, must be
consistent.
In the light of these criteria two alternate methods for data
generation were explored.
Double Mass Extension
The first method considered was an extension of the double mass
curve.
The procedure is to substitute known "X" (double mass
de­
pendent variable values) values for the period of missing record,
into the established double mass curve equation of the form:
Y = MX -f b
where
% “ The independent variable
Y = The dependent variable.
M = The slope of double mass
line.
b = The intercept of double
mass line.
The result is the generation of synthetic values of "Y" for
the missing record period.
The points on the double mass curve are
determined by the relationship:
2 Y'i
i=1
Yn=
and
- Jh
where
Yn" The ordinate of the n ^
double mass point.
y^= The individual base
station data values.
Xn= The abscissa of the n ^
double mass point.
Xi= The individual surround­
ing station data values.
-17The actual synthetic missing data values are found by sub­
tracting each "Y" from the "Y" value immediately succeeding it.
That is :
y^= The ith synthetic data
value.
?!= Yi- Yi-1
Yi and i-1 = The synthetic
double mass ordinates
used to obtain y^,
Thus the necessary missing record for the dependent station
is generated in this manner.
Multiple Regression
The second method for data generation which was explored was
the method of "multiple regression," which is a direct application
of the Theory of Least Squares.
Multiple regression in this case would involve using a period
of known record to make the dependent station data a function of the
data from surrounding stations.
This is accomplished by fitting the
following type of polynomial to the existing data.
y = C0 +
C1X1 + C2 x2... Cn Xn
where
y = data for dependent station
X1
= Concurrent data for surI)
rounding stations.
Cq
= coefficients which are
determined by the least
square fit of the poly­
nomial to the data.
The least squares fit of the polynomial to the data results
- 18-
in the determination of the coefficients C0 ,
C
n.
Once these are
known, periods of missing dependent station record (y values) can be
determined merely by knowing the concurrent "x" values for the
surrounding stations.
At first glance little similarity is apparent between the two
methods for data generation.
However, further examination shows
that there is in fact, a great deal of theoretical similarity.
It
can be shown for a special case that theoretically double mass ex­
tension and multiple regression will produce the same synthetic data
values.
This special case occurs when the data at the dependent
station is made a function of the data from only one neighboring station.
The proof is as follows :
Consider first how synthetic data are generated using an
extension of the line of best fit through the double mass points.
Y
YrmX ♦ b
Figure 2.
DOUBLE MASS EXTENSION
=
xi . (i
Ii
The individual data
values for the dependent
station
Xj^ =
.The individual data
values for the surround­
ing stations
x U
•H
-- .
y±
h-»
Yn =
. Ii
In the previous diagram,
1=>
-19-
I'j••« jn)
To estimate any synthetic data value yn , the following pro­
cedure is used.
yn = Yn - Yn -1
=
(mXn + b > “ <^n-l + b>
This can be rewritten as:
yn =
m(Xn - Xn^ 1) + b - b
Since (Xn - Xn_1) = xn
yn =
mxn
where m = slope = y n
Xn
Now consider how synthetic data are calculated from multiple
linear regression.
Using concurrent data from the dependent station and the
surrounding station, a linear relation is obtained.
yn = co + c lxn
where
yn = Individual data
values for the
dependent sta.
xn = Individual data
values for the
surrounding sta.
This relation is represented graphically as follows:
qj
-20-
Figure 3.
MULTIPLE LINEAR REGRESSION
To obtain an estimate of the synthetic data value yn the
only information needed is the known value for xn .
Analysis of the above diagram leads to the following equation.
since
* •
c
^n
=
y -c
slope = _n____o
xn - 0
(^n"
cq )
But E . I can also be written:
0r:
xn + Cq
(2)
yn = (yn + C q - Cq) xn
yn = (yn - cO ) xn + co
xn
Since E q. 2 and the final form of Eq. I are identical, it
has therefore been shown that the synthesis techniques of "double
mass extension" and "multiple linear regression" will produce the
same synthetic data values - provided that the dependent station
is made a function of only one surrounding station.
(3 )
-21This proof provided the basis for the decision to use double
mass extension for the synthetic data generation portion of the
y
model.
This decision allowed the computer programming to be done
such that information from the data analysis section could be very
easily transferred to the data generation section.
Had the multiple
linear regression technique been used for data synthesis, a more
complicated program structure would have been required.
Chapter 4
MODEL STRUCTURE
The data analysis and generation model consists of two distinct
units with the output from the first unit (the data analysis sub­
routine) serving as partial input to the second unit (the data
correction and generation subroutine).
DATA ANALYSIS SUBROUTINE
Since the primary purpose of the "Data Analysis Model" is to
detect discontinuities in existing data, a method had to be developed
whereby the computer would accurately detect both the presence and
the location of these discontinuities.
Discontinuity Detection
The method used is presented below.
1.
(Refer to Figure 4)
First the coordinates of the points on the double mass
curve are calculated by a subroutine of the program.
These points would be similar to those plotted in ILgure 4.
2.
Using the following least squares^equation
Y = Y + M (X
where M = i=l(Xi Yi) - NXY
2 (Xi - X )2
i=l
the model next calculates the line of best fit through all
of the data points (one point at a time).
That is the
first line of best fit is passed through the first double
mass point only, the second line is passed through the
Slope
Slope
Slope = m
o,
CUMULATIVE
DEPEN DEN T^ STATION
DATA
-23-
CUMULATIVE
FIRGURE 4.
SURROUNDING
STATI ON
DATA
DETECTION OF DISCONTINUITIES IN DOUBLE MASS ANALYSIS
-24two points etc.
The most important parameter
obtained from these calculations is the slope
line of best fit.
of the
This parameter is used in determing if
a discontinuity exists in the data.
3.
Along with obtaining the slope "Mi", the model simulta­
neously calculates the slope of another straight line.
This line passes through the particular double mass point
and the origin.
For example, when the model fits a line
through the first four double mass points, it also calcu. Iates the equation of the line passing through point #4
and the origin.
The slope of this two point line is
labeled "M2".
4.
After each calculation of the slope values "Mi" & "M2",
the program checks for a discontinuity by calculating the
ratio.
R1
= M1 ~ M2
Mi
If the absolute value of "Ri" is greater than the
prescribed tolerance, then a discontinuity is assumed.
In Figure 4,
line I represents the line of best fit
through all points up to point "P•u
Line 2 represents the
line passing through point."?" and the origin.
The slopes
of lines I and 2 are "Mi" &-"M2" respectively.
Assuming
that the value of "Ri" exceeds the tolerance, the
-25presence of a discontinuity has now been established.
This also establishes that point "P" lies "above" the
discontinuity.
Discontinuity Location
5.
Once the presence of a discontinuity is established, the
program next attempts to accurately determine its location.
To do this the program first fits a line through the first
twenty points above point "P".
The slope of this line is
defined as "M3 ." Now another "two point" line is calculated
through point "P + 20" and the points lying below point "P"
one point at a time.
The slope of this line is labeled
"M4
This process continues until the ratio:
5.2="■
is exceeded.
that the ratios
It must be noted here
& Rg must be set by the programmer prior
to the running the program.
It was found that values of
Rl Sc R 2 in the range .I - .5 were the most satisfactory,
taking into account that
^ g should be chosen pro­
portionally with the "scatter" of the double mass points.
In Figure
,4' , line 3 represents the line of best fit.
through all points between point "P” and point "P + 2.0" the slope of which is"Mg." Point "Q" is the point at which
"R" is assumed to exceed the tolerance.
-2 6 -
6 . The point of discontinuity is assumed to lie half way
between point "P" and point 11Q.'"
Break point = P + Q
2
7.
Once a point of discontinuity is located, the program now
neglects all points below the "break point," thus treating
the break point as the new origin of the double mass curve
Starting from this new origin, the model now repeats steps
I - 6 to check for additional discontinuities.
Evaluation of the Data Analysis Subroutine
The method used in constructing the data analysis model was
chosen because of the following advantages:
1.
The method allows the model to detect more than one
discontinuity.
2.
The method lends itself easily to a computer solution.
3.
The logic behind this method is simple and straight
forward.
This method of data analysis also has certain disadvantages.
These include:
I.
The accuracy with which the program locates slope changes
is proportional to the "degree of scatter" of the double
mass points.
This scatter is measured by computing the
standard error of estimate of the points about the line
of best fit.
-272.
The tolerance which determines when a break point occurs
must be set by the programmer.
It is important to keep these disadvantages in mind, especially
when interpreting the results of the Data Analysis Model.
DATA GENERATION AND CORRECTION SUBROUTINE
Keeping in mind the methodology presented in Chapter 2, the
program structure is as follows:
Data Correction
1.
If discontinuities are detected in the data by the Data
Analysis Model, the first step of this program is to
correct the existing inconsistent data.
To accomplish this,
the programmer must first decide how the data is to be
corrected based on examination of existing records.
Once
this has been determined, the proper correction equations
are placed in a special subroutine and the data is corrected.
Data Generation
2.
Having corrected all faulty data, periods of missing records
are now synthesized by the data generation section of the
model.
The procedure used to accomplish this was simply
to program the methodology outlined in Chapter 2.
Once
this was done, the input needed for data generation were
(I)
surrounding station data for the periods of missing
base station record, and (2)
the equation of the line of
best fit for the correct period of data on the double mass
curve.
-2 8 -
As an example of how the data correction and generation model
works, consider Figure 5. .
Assuming the data on segment I to be in error, the programmer
must first decide how the data is to be corrected.
That is, it
must be decided to move the double mass points on segment I either
vertically, horizontally or in both directions.
As stated before,
this is a judgment decision based on examination of the records for
the stations involved.
Once this has been decided the points on
segment I are corrected in the proper manner to lie on segment'll.
Now assume that the known period of base station record ends _
at point A.
Assume also that the equation of line segments II & III
is the correct model to be used for data generation.
The known
abscissa values are now substituted into the correct equation to
yield the ordinate value for the missing period.
These ordinates
are then subtracted in the proper manner to produce the missing data,
values.
Evaluation of Data Correction and Generation Subroutine
The advantages which justify the use of this approach to
data generation and correction are:
1.
The exact method of data correction is in each case
left to the judgment of the programmer.
2.
The model provides for the correction and generation of
more than one period of data.
— 29-
25.0 - 22.5= 2.5
points on segment I
to be corrected to
on segment II.
X =22.0
CUMULATIVE
SURROUNDING
STATI ON
DATA
FIGURE 5. GENERATION OF SYNTHETIC DATA BY DOUBLE MASS EXTENSION
Chapter 5
PRESENTATION OF RESULTS
The effectiveness of the data analysis and generation model
was tested using data from various gaging stations in Montana.
The
test procedure was first to check the consistency of the data, and
then after making any necessary corrections, to synthesize a period
of data.
For the purpose of testing, periods of record were syn- -
thesized which in fact were already known.
Thus a comparison could
be made to determine the effectiveness of the model.
DESCRIPTION OF RESULTS
The results which follow consist of double mass curves (which
check the consistency of a particular set of data) and graphs of the
generated synthetic data plotted with the actual data.
Both stream-
flow and precipitation data are represented in the results.
The
streamflow data tested came primarily from the Gallatin, Madison,
and Yellowstone River drainages, while the precipitation data came
from both Gallatin County and Southeastern Montana.
SIGNIFICANT PARAMETERS
Significant parameters which are used in the discussion of the
results are the following:
I.
Correlation Coefficient
defined as:
- x)
(yj - y)
n Sx Sy
-31where
Xi = known base station data
x
= mean of x data
Sx = STD. deviation of x data
= known surrounding station
data
y
= mean of y data
Sy = standard deviation of y data
n
2.
= total number of data points.
Per Gent Error
defined, as :
Mg - Ma I x 100
where
Mg = mean of the generated
synthetic data
Ma - mean of the actual data for
the period of synthetic data
Per Cent Error is used to estimate the accuracy with which the
synthetic data for a given period compares with the actual data for
that same period.
3.
Monthly Per Cent Error
defined as :
'
- ms^ ) . ; MaW
Ma (■1-'
where
X 100 (1 . 1,...12)
Mg (i) = mean of the synthetic
data for a given month
of the year.
)
-32(i) = mean of the actual data
for a given month of the
year.
e (i)
= Average monthly deviations.
-33TABLE II
DATA FOR THE PRECIPITATION AND STREAMFLOW
STATIONS WHICH WERE STUDIED
STREAMFLOW STATIONS
1.
Hyalite Creek (U.S.G.S. # 6-0500)
LOCATION -- At Hyalite Ranger Station 7.3 miles south
of Bozeman, ,Montana..
DRAINAGE AREA ---48.2 sq. mi.
REMARKS -- Records fair. Flow regulated by Middle
Creek Reservoir since 1951.
2.
Gallatin River near Gallatin Gateway (U.S.G.S. #60435)
LOCATION.-- 7.3 miles south of Gallatin Gateway, Montana.
DRAINAGE AREA ---825 sq. mi.
'.
REMARKS -- Records good. Diversions for about 1400
acres above station.
3.
Madison River near West Yellowstone Montana (U.S.G.S. #6-0375)
LOCATION-- 1.6 miles east of West Yellowstone iMontana.
DRAINAGE AREA ---420 sq. mi.
REMARKS -- Records good.
above gage.
4.
No diversion or regulation
Yellowstone River at Yellowstone Lake outlet (U.S.G.S. #6-1865)
LOCATION -- .2 miles downstream from outlet of
Yellowstone Lake.
DRAINAGE AREA -- 1006 sq mi.
REMARKS -- Records good except for winter months which are bad.
-34PRECIPITATION STATIONS
1.
Wyola
LOCATION -- On Interstate 90 fourteen miles from
Montana-Wyoming border.
2.
Sidney
LOCATION -- On the Yellowstone River seven miles
from Montana-North Dakota border.
3.
Bozeman
LOCATION -- On campus of Montana State University
4.
Trident
LOCATION.-- Thirty one miles west of Bozeman,,Montana,
. 5.
Pryor
LOCATION -- Thirty seven miles south of
Billings, Montana.
6 . Billings
LOCATION. -- Downtown Billings, Montana
D e p endent
HYAl
ITF
Surrou ndi ng
S tction
CPK
G
(@ Hyalite Ranger Sta.)
a l l a t i n
r i
V
f r
(@ Gallatin Gateway)
F E E T ) X IO5
(ACRE
.08IX + 3320
DEPENDENT
ST A.
STREA M F L O W
DOUBLE MASS DIAGRAM
( ACRE
S U R R O U N DI N. G
FIGURE 6-A.
STA.
FEET) X 10
ST REAM F L O W
DOUBLE MASS DIAGRAM : STREAMFLOW
D e p e n d e n t
H Y A L IT E
C R K
C o r re lc itio n
( @ Gallatin Gateway)
C o e t t
D a t a
FEET)
x
IO
3
Ic
Q A U l AT|N— R IVER
-3 6 -
( A C R E
STREAMFLOW
S y n t n e t
(@ Ranger Sta.)
D a t a
A c t u a l
MONTHLY
RIntinnS
S t a t i o n
12
1959
3 4 S 6 7 S 9 10 11 12 I 2 3 4 5 6 7 S 9 10 11 12 I 2 3 4
i960 T I M E
(m o n t h s )
FIGURE 6 -B. COMPARISON OF DATA FOR HYALITE CREEK
I
1961
5 6 7 S 9 10 11 12
»/ .
ERROR
CORRELATION
COE FFIOfFNT
Observed data
Vo
E R R O R
Average observed
data for .I inter­
vals of r.
( DIMENSIONLESS)
CORRELATION
FIGURE 7.
COEFFICIENT
PER CENT ERROR vs. CORRELATION COEFFICIENT
S u r r o u n Li nr
L X - ^ c r n L e n t ___ £ . t c t i £ . . Q
P o W n F
YELLOWSTONE
(@ Sidney)
17 R l V F R
(@ Locate ).
RlVER
DEPENDENT
X 105
FEET)
(ACRE
5 TA.
S T R E A M F L OW
DOUBLE MASS DIAGRAM
020X - 4.05
s 'i9 6 o
DISCONTINUITY
C ACRE
SURROUNDING
FIGURE 8-A.
S TA.
FEET ) X 10
STREA M F L O W
DOUBLE MASS DIAGRAM:
STREAMFLOW
Dependent
LOCATE
(p O W D E R
RIVER)
S ID N E Y
D a t a
( YELLOW STONE
C o r r e l a t i o n
S y n t h e t i c
D a t a
(Discontinuity Ignored)
- - - - X '- ®
S y n t h e t i c
D a ta
(Discontinuity Corrected)
RIVER)
Coeff.
0/o Error
.6 1 7
1 1 .8 #
FEET)
x
I O 3
-------- e
Stnt inn S
w ( A C R E
-39-
STREAMFLOW
A c t u a l
MONTHLY
S u r r n i i nrii ng
S ta tio n
/Z r
12
3 4
1 9 5 0
FIGURE 8 -B.
S 6 7 8
9 10 11 12 I
1 9 5 1
'
2
J
3 4
1
5
6 7 S
(V) £
COMPARISON OF DATA FOR POWDER RIVER
(
9 10 11 12 I
M O N T H S
)
2 3 4
5 6
7 8 O 10 11 12
Surrou n rfi nn
Ga I
I
ATf N RIVER
M ADISO N
Stot i n n s
PiVFn
X IO6
FEET)
(ACRE
DEPENDENT
5TA.
ST R EA M F L O W
(@ Gallatin Gateway)
(@ West Yellowstone)
DOUBLE MASS DIAGRAM
(ACRE
S U R R O U N DI N- G
FIGURE 9-A,
FEET)
STA.
DOUBLE MASS DIAGRAM:
X
101
ST RE AM F L O W
STREAMFLOW
Dependent
Ga t e w a y
o
— •®
Siirrnu nrl i
( Ga l l a t i n r i v e r )
Actual
Synthetic
Data
C o r r e l a t i o n
Data
-c ® ^ AD j LISTED S yn th et i c
WEST YELLOWSTONE (M ADI SO N R I V E R )
C o e f T
°/o E r r o r
D at a
•Apr. -2.4
■May
3 .0
•W \ ■
Aug. -1.8
Nov. -1.9
Dec. -2.8
12
3 4
1950
FIGURE 9-B .
5
6 7 8
9 10 11 12 I
2 3 4
5 6 7 8 9 10 11 12 I
1951 J i M E
( months )
COMPARISON OF DATA FOR GALLATIN KlVEK
2 3 4
1952
5 6 7 5 9
10 11 12 ^
-If?-
acre
FEET) x IO5
Average Monthly
Deviations
(
MONTHLY
STREAMFLOW
—
Stntlfin
-42D c p cnct e r t
S tction
Surrounding
Stct ions
*
Y ELLOW STONF
I7IVFP
MAniSON
(@ Yellowstone Lake)
Ri \/ F n
(@ West Yellowstone)
x 1Q6
FEET)
(ACRE
DEPENDENT
STA.
STREA M F L O W
DOUBLE MASS DIAGRAM
5.0
( ACRE
S U R RO U N D I N.G
FIGURE IO-A.
7.5
F E E T ) X 10*
STA.
DOUBLE MASS DIAGRAM:
I
STREA M F L O W
STREAMFLOW
S y n t h e t i c
A dj ust ed
< (YELLOWSTONE
RIVER)
WEST YELLOWSTONE
C o r r e l o t i o n
D a t a
FEET)
( M A D I S O N RiVE R )
C o e ft.
D a t a
Synthetic
Dat a
Average monthly Deviations
Jan.
-2.15 x 10
x
1Q 5
e
( A C R E
MONTHLY
b T REA M F L O W
YELLOWSTONE R.
Actual
Ft ntinnS
S t a t i o n
12.1
x 10
Jul. 13.9xlOj
Aug. 6 .IxlOj
Sep.
.ExlOj
Oct. -.8x10,
- 1 .8 x 10
-E+f-
D e p e n d e n t
D ep en d en t
S rrtin n
S u rro u n d in g
W Y O L A
<,tat i n n s
S ID N E Y
DOUBLE MASS DIAGRAM
I.IOX + 2.20
( I N C H E S )
SUR RO U N Dl N-G
S TA.
PRECl Pl TATl ON
FIGURE Il-A. DOUBLE MASS DIAGRAM:
FRECITIIATIOU
S u r r n u nrti nn
S ta tio n
WVQ LA
0
A c t u a l
®
S y n t n e t i c
Si n + i n r , ^
S I D NEY
D a t a
Coett
C o r r e l a t i o n
D a ta
-U-JO
1 5 .5 #
°l» E r r o r
6
( I N C H E S )
MONTHLY
PRECIPITATION
Dependent
I 2
3 4
I 958
FIGURE 11-B.
5
6 7 8
9 10 11 12 I
1959
2
3 4
T I M E
5 6 7 8
9 10 11 12 I
( MONTHS )
2 3 4
296O
COMPARISON OF PRECIPITATION DATA FOR WYOLA, MONTANA
5
6
7 8 9 10 11 12
4»
D ep end ent
S trtin n
Surr
MONTANA STATF UNIVF RSl T Y
PU
ndi
nn
Stct
inn <;
TRI DENT
DOUBLE MASS DIAGRAM
726 x + .8l4
( I N C H E S )
S U R R O U N d i n . G STA.
FIGURE 12-A. DOUBLE MASS DIAGRAM:
PRECI PITATION
PRECIPITATION
D e p e n d e n t
S u r r n i i nrii ng
S t a t i o n
Actual
__
6
S y n t n e t i c
T R ID E
D a t a
NT
C o r r e l a t i o n
D a ta
%>
C o e f t
E r r o r
MONTHLY PER CENT ERRORS
- Z t 1-
( I N C H E S )
MONTHLY
PRECIPITATION
B O Z E M A N
9,1
12
3 4
1948
FIGURE 12-B .
S 6 7 S
D 10 11 12 I
1949
2
3 4
T I M E
5
6
7 S 9 10 11 12 I
(
m o n t h s
)
2 3 4
1950
COMPARISON OF PRECIPITATION DATA FOR BOZEMAN,MONTANA
5
6 7 8
9
10 11 12
-48D e p e n d e rt
S tation
S u rro u n d in n
^ lo t io n s
W
BH u Nn <
PMYnn
DOUBLE MASS DIAGRAM
Z
O
t
-
<
200
HCL
U
LU
1 5 0
CL
_
CL
to
LU
1.12X + 1.42
/
1938
( I N C H E S )
SURRO UNDI N- G
FIGURE 13-A.
S TA.
DOUBLE MASS DIAGRAM:
PRECIPITATION
PRECIPITATION
Snrrnii
S t a t i o n
PRYOR
A c t u a l
D a t a
©
S y n t h e t i c
SMnt i n n s
LL I N G S
C o r r e l a t i o n
D a t a
0Zo
C o e f f
E r r o r
~6+H
(IN C H E S )
m
e
BI
nr l i n g
^
MONTHLY
PRECIPITATION
D e p e n d e n t
12
3 4
19 5 0
FIGURE 13-B.
5 6 7 8 9
10 11 12 I
2
3 4
1951 T I M E
5 6 7 8 9
10 11 12 I
( MONTHS )
2 3 4
1952
COMPARISON OF PRECIPITATION DATA FOR PRYOR, MONTANA
5
6
7 8
9
10 11 12
Chapter 6
DISCUSSION OF RESULTS
The results obtained from an analysis of data from Hyalite
Creek and the West Gallatin River (see Figures 6-A and 6-B) are
typical of the type of results obtained for both streamflow and
precipitation data.
The double mass curve (Figure 6-A) shows that
no significant discontinuities exist in the data from 1935-1954, and
therefore correction of data for this period is not necessary.
On
the basis of this double mass curve, the period of monthly streamflow
record from 1955-1962 was synthesized and compared with the actual
data for that same time period (Figure 6-B).
The results for the
remaining test data are presented in a similar manner in Figures 8-13.
ACCURACY VS. CORRELATION
The most obvious conclusion from these results is that the
accuracy of the synthesized data is dependent directly upon the degree
of correlation of the real data.
That is, the correlation between
the real data used to construct the double mass curve, can be used
as a reliable indicator of the accuracy to be generated.
For
example, the synthetic data in Figure 6-B was generated using real
data which highly correlated (r=.955).
The accuracy of this generated
data is expressed by a per cent error value (4.1%) which was one of
the lowest obtained.
Examination of the rest of the results proves
conclusively that there is an inverse relationship between the
-51correlation coefficient (for the dependent station data and the
surrounding station data) and the per cent error values for the
generated synthetic data.
This is demonstrated in Figure 7 in
which per cent error is plotted against correlation coefficient.
To better illustrate this relation, the possible correlation co­
efficient values (0.0-1.0) were divided into ten equal intervas and
all observed per cent error values falling in each interval were
averaged.
This result is also shown in Figure 7, and serves to
reinforce the existence of the relationship between correlation
coefficient and per cent error.
From the preceding discussion it is apparent that it is de­
sirable, when generating synthetic data, to find the combination of
surrounding stations which have the best correlation with the de­
pendent station.
In this study it was found in every case that the
correlation coefficient was always highest between the dependent
station data and a single surrounding station.
In other words, when
dependent station data was correlated with every possible combination
of data from a group of surrounding stations, the result was that the
highest correlation always came from the dependent station and a
single surrounding station.
This observation is not meant to imply
that the above result will always hold true.
However, this informa­
tion may be useful for any hydrologic study in which it is desirable
to find the combination of sets of similar data which has the highest
correlation.
-52EFFECTIVENESS OF THE DOUBLE MASS ANALYSIS
The curve in Figures 8 -A and 8.-B demonstrates the effectiveness
of using the double mass analysis as a consistency check. This curve
shows that a discontinuity occurred in the data around 1950, thus
causing a significant slope change.
Examination of the U. S . Geological
Survey records revealed that the discontinuity was caused when the gage
at the dependent station (Locate, Montana) was changed in 1947 from a
"staff gage" to a "continuous recorder;"
To obtain .the average daily flow using a staff gage, the observer
'will read, the gage twice a day and average the readings.
However, if
-a large daily fluctuation in the streamflow is characteristic at the
particular location, the staff gage is not likely to provide a good
estimate of the average daily flow.
In fact, depending on the time of
day at which the readings are taken, it is possible to either con­
sistently overestimate or consistently underestimate the average
•daily streamflow.
The continuous recorder, because it does provide a continuous
.record, allows a more accurate estimate of the average daily streamflow to be made.
In this case it was assumed that since the continuous recorder
provides a more accurate record, it is the period of record taken
using the staff gage (1939-1947) which needs correction.
Figure 8 -B shows the results of predicting synthetic data with­
out first checking the consistency of the existing data. The syn-
-53thetic data was obtained by using the line of best fit through all
the double mass points in Figure 8 -A.
Synthesizing data using this
line, which disregard's the discontinuity in the double mass points,
caused the synthetic data values to be consistently high.
In this case the double mass line segment which should be used
for data generation is the line in Figure 8 -A which passes through
only the double mass points above the slope change. When this line
'..was used for data generation, the resulting synthetic data was
definitely more accurate, as can be seen by comparing the different
sets of data in Figure 8-B.
It is also evident in this case, that in early summer large
quantities of water are sometimes diverted above the Locate gage.
This is illustrated by the actual data plotted in Figure 8-B, which
:
■ shows that several years of record have very low flows recorded for
the high runoff months of May, June, and July.
Since diversion records
for Montana are practically non-existent, the accuracy with which
synthetic data can be generated for rivers subject to diversion is
adversely affected.
However, it should be noted that the procedure
for estimating synthetic data for these river's is no different than
- it is for rivers with no diversion.
That is, an attempt should still
be made to find the combination of surrounding stations whose data
show the best correlation with the data from the dependent station.
It might happen that a particular surrounding station (stations)
measures a flow which is subject to. the same diversion pattern as the
-54flow measured by the dependent station.
If so, it may be that a
rather high correlation exists and thus, synthetic data may be generated'
which is fairly accurate.
Another observation that can be made from the results is that in
some cases the deviation of synthetic data from actual data follows a
definite pattern.
(See Figure 9 -B).
Specifically, it appears that
rniany synthetic data values are consistently higher than the actual
.data during some months and consistently lower during other months.
In cases where this "cyclic variation" appeared to be well defined, it
,was found the accuracy of the synthetic data could be greatly improved
by the following procedure.
I.
First, a period of synthetic data is generated and compared
with the actual data.
'2.
Next, the deviations of.the synthetic data from the actual
data were computed for each month throughout the period of
synthetic data.
The deviation values for each month are
then averaged to produce an "average monthly deviation"
value for each month of the year.
3.
An additional period of synthetic data is then generated,
and the average monthly deviation values are added alge­
braically to each monthly synthetic data value.
In instances where cyclic variations were well defined, the
above procedure resulted in altering the synthetic data to more
-55accurately compare with the real data.
9 -B.
An example is shown in Figure
First, the average monthly deviations for this data were de­
termined by comparing actual and synthetic data for the ten years
previous to 1950.
Next the synthetic data shown in Figure 9 -B was
generated (dashed line).
Then, by applying the average monthly devia­
tions , the adjusted synthetic data (dotted line) was found. When
these data are compared with the actual data from 1950-52 (solid line),
it is obvious that the adjusted synthetic data is a significant im­
provement over the original synthetic data.
This fact can be expressed
numerically by comparing the per cent error value of 7.1% for the
synthetic data with 3.2% for the adjusted synthetic data. Another
example of increasing the accuracy of the synthetic data can be seen
in Figure 10-B.
In this case the per cent error value for the original
synthetic data was 13.2% and the value for the adjusted synthetic
data was 5.1%.
After examination of all the results, it was apparent that
cyclic variations are much more likely to occur in streamflow data
than in precipitation data.
Several attempts were made to increase
the accuracy of synthetic precipitation data, but because of the
erratic nature of the particular data, none were successful.
This
does not suggest however, that cyclic variations do not exist in
precipitation data.
Rather, the point to be made .is that unless
cyclic variations do exist in the data (either streamflow or
precipitation data), any attempt to use average monthly deviations
-56
to improve the accuracy of the synthetic will be unsuccessful.
If it is necessary to further quantify the accuracy of the syn­
thetic data, monthly per cent error values can be calculated by the
data generation model.
An example of a set of these values is listed
in Figure 12-B.- These monthly per cent error values provide an indica­
tion of how accurately the synthetic data values compare with the
actual data for each month of the year.
By analyzing both the per cent
error value and the monthly per cent error values for a set of syn­
thetic data, it is possible to better judge how well additional syn­
thetic data can be generated for missing periods of real data.
PROCEDURE FOR OPERATION OF THE DATA ANALYSIS AND GENERATION MODEL
The final objective of this study is to determine the optimum
operation procedure for the analysis of existing data and the estima­
tion of synthetic data.
The procedure is developed on the basis of
the results obtained in this study and if followed, should yield the
best possible results from the data analysis and generation model.
I.
First, determine which combination of data from surrounding
stations exhibits the highest correlation with the data for
the dependent station.
The results of this study indicated
that the accuracy of the synthetic data is proportional to
this correlation coefficient.
Also it was found that a
single surrounding station will probably have the highest
correlation.
-572.
Using the data analysis model, check the existing data for
consistency and make necessary corrections.
3.
Next, generate synthetic data for a period for which the real
data is known. Determine the accuracy of the synthetic data
by calculating the per cent error value and also the monthly
per cent error values.
It was found necessary to have about
ten years of actual data for proper data analysis and an
additional five years of actual data (concurrent with the
synthetic data) for the true accuracy to be determined.
Results which are obtained using shorter periods of actual
data should be used with caution.
■4.
Analyze the synthetic and actual data to determine if the
variations of the monthly values tend to be cyclic.
If
cyclic variations are suspected, then the average monthly
deviations should be calculated and applied to the synthetic
data.
If the application of these deviations increases the
accuracy of the synthetic data, only then is their use
justified.
5. ■ Estimate synthetic data for the periods of missing record.
The per cent error and monthly per cent error values determine
the confidence to be placed in this synthetic data.
If
cyclic variations exist, then add the average monthly devia­
tions to the synthetic data to obtain the best possible
accuracy.
Chapter 7
SUMMARY AND RECOMMENDATIONS FOR
FURTHER RESEARCH
Since 1968 the departments of Civil Engineering & Engineering
Mechanics and Industrial & Management Engineering at Montana State
University have worked to develop a state-wide water planning model.
When completed, this "state water model" will be capable of determine
the distribution of ground and surface water throughout the state for
any specified time interval.
At present, about 80% of the existing precipitation and streamflow data for Montana have periods of missing record and are, there­
fore, unsatisfactory for use in the state water model.
Also, all
inconsistent periods of record must be found and corrected before
they can be used in model studies.
To solve these problems with the existing data, a computerized
data analysis and generation model was developed.
This model utilizes
the double mass analysis to check the consistency of the existing
data, and then after making any necessary corrections, the moded.
uses the double mass equation to fill in periods of missing record.
The following represents the procedure used to test the effect­
iveness of the data analysis and generation model.
First a group of
20 precipitation and streamflow stations in South Central and South
Eastern Montana were selected.
These stations were chosen so that a
-59
variety of topographic and precipitation conditions were represented.
Next, a period of data from each station was analyzed to determine if
inconsistencies were present. At this point any necessary corrections
were made and then a period of synthetic data was generated. This
synthetic data was always generated for a time period for which the
actual data was known.
In this way a comparison could be made to
determine the accuracy of the synthetic data.
The results indicated that the correlation coefficient (between
the dependent station data and the surrounding station data) could be
used to predict the expected accuracy of the synthetic data..
In this
study the "accuracy"'of the synthetic data was defined as:
accuracy
= •— ---Hs.1 = E
M=
Where,
M s = The mean of the .
synthetic data.
Ma = The mean of the
actual data.
Also .present in the results was evidence which demonstrated the
necessity for testing the consistency of existing data before syn­
thetic data is generated.
In addition it was found that in some cases, especially with
streamflpw data, the synthetic data differed from the actual data
in a definite cyclic pattern.
When these "cyclic variations" were
well defined, it was possible to improve the accuracy of. the synthetic
-60-
data values by adding to them "average monthly deviations."
These
average monthly deviation values represent the average of the
differences between the actual and synthetic data for each month of
the year.
After reviewing the results of the cases' which involved cyclic
variations, it became evident that further research in this area is
indeed warranted.
The first step should be to determine and analyze
the natural phenomena which cause cyclic variations to occur - both in
streamflow as well as other types of hydrologic data.
Also, additional
analytic techniques should be developed to take advantage of the cyclic
variations in predicting synthetic data.
One such method might be to fit a cyclic function to the double
mass points by use of Fourier series.
If this could be done success­
fully, the synthetic data obtained from this cyclic function could be
much more accurate than if a straight line equation had been used.
APPENDIX A.
LITERATURE CONSULTED
1.
Beard. Regional Frequency Analysis, (Sacramento, California:
U. S. Army Corps of Engineers Hydrologic Engineering Center,
1967).
2.
Chow, V. T. Handbook of Applied Hydrology, (New York:
1961).
3.
Cochran, G. F. Optimization of Conjunctive use of Ground and
Surface Water for Urban Supply, (unpublished thesis; Reno,
Nevada; University of Nevada Library, ]968).
4.
Crawford, N . H., R. K. Linsley. Digital Simulation in Hydrology:
Stanford Watershed Model IV, (Report //39, Department of Civil
Engineering, Stanford University, 1965).
5.
Cross, W. D. Johnstone.
Koland Press, 1949).
Hill,
Elements of Applied Hydrology, (New York:
6 . Fiering, M. B., Barbara B. Jackson, Synthetic Streamflows,
(Geophysical Union Washington D. C. , Water Resources Monograph //I,
McGregor and Werner Inc., 1971).
7.
Foster.
Rainfall and Runoff, (New York:
Macmillian Press, 1948).
8 . Linsley, Kohler, Paulhus. Applied Hydrology, (New York:
McGraw-Hill, 1949).
9.
Linsley, Kohler, Paulhus.
McGraw, 1958).
Hydrology for Engineers, (New York:
10.
Meyer, A. F. The Elements of Hydrology, 2nd Edition, (New York:
John Wiley and Sons, 1946).
11.
Mount-,- J. R. Manual of Computing and Modeling Techniques and
their Application to Hydrologic Studies, (Texas Water Com­
mission report #100165, Austin, Texas, 1965).
12.
Sing, R. "Double Mass Analysis on the Computer."
Journal for Hydraulics Division, January 1967..
13.
Wisler, C. E., F.' Brader. Hydrology, 2nd Edition, (New York:
John Wiley and Sons, Inc., 1965).
A.S.C.E.
APPEIiDIX B.
. PROGRAM LISTING
ceVV£»N x(o:?»0), Y(i:3so), xsu'Konso),
y s u m (o
;3s o ),
I X > (130), XYSUM(3S3),XY(380),SjMnIF(380), SLOPEf380),
1YINT (33?), SUV3ir(?PO), S S - D E V (330), D E V (330), SLSy T (0:
)':ATl3(3aO), E S w O P E (333), B D E V (350),S L 2 C W K (380), Y TJ T B(3
lXX(ji",r),YY(33:,2),
?II(3P0), XXX(383),YYY(3«0),
I X 'Os?), YBAfOsn),YAR(P), STDEV(P) , SKEW (2),
I C5-E9, N, v, L> A\, IT, JJ,
I YSI\|( 12,31,5), YSJVSt 380, 5),
DEVMGf 12,50), AVMGDE (12 )
E
D V a "SlpN TySI'Tf330), I=NTl(3S0), I=NTP(330), IBREAKfO
I XiSV(330), RAD=V(380), XCALC(320), P(380,2),JXY(IO)
I, IYR (31 ), IDO)
C
C RrAC IN' N,r,L,NX N= NUMBER
C
STA. Ms* P= Y DATA CARDS,
REAr
. (105,1 ) \, m ,L
1 = p R “ AT(3llP)
R t rE (IOS,30=)
=03 Fp:'-T{IN = TMr MJMS=R r r BAS= STATIONS...M,NQ, GE Y
1.., L =XP- '1E PTS• PV m a s s CURVE',//)
V R T ^ r (103,191)
ID1 Epx 'AT (1OX, ":',13X, 'M *, I OX, fL l )
IlRITrt 10%, I 92) N , M ,L
193 F O R " A T (3110,//)
C
C READ IN ALL Y DATA IN ORDE^
C
PUTdUT « '
PJT=JT ' TD NJM =ERS 9r SUtTRGUvDING STAT IGNS '
PUTr JT
'
'
=P 10 J = I,N
iy 11 = 1 ,M
TC(J), IYRf I I),(YSIN(K,I I,J), < = I, 12)
2 FpPwiA T ( I4, I3, FS • I, 11F6 • I )
dp
REAC(105,2)
15 CBNt INUEPUTdUT ID(J)
10 CPNTI NUE
CP 361 J = I,N
II = I
K = O
—6 3 “
4
r = i,L
r K + I
VSI'C(IyJ) = VSTMfX, I I, J)
ir(- ■ .F-, 1 P) I I
'Ti'
•3 N-• I2 ) < = C
u-r L RN/ Is RF
161 CL RT I' RE
II
re 362
r
r RE AD I • X : ATA F"=R STAT ISN
IN d u e s Y iSN
c
re 2o U = IyV
REA : (I CS,B ) ICx, IYR(II) / (XSlNGfK,I I )
S FQpl A"I C TH 13,FE. I, MFfe. I )
Pr CQNT IN Lr
II = I
3
H O
= H -O
C N 3C I = O L
< Z I.+1
= YSI N 3 ( k , I I ) 4 X(T-I)
H D
IF I' •C . I") I I = I I*!
IF (• .E f . T E 5 < = O
CS-* I(■'u E
5iir\T
' '
K Trv i
I
n N U M B E R RlR BASF
FUTf- L T
IEX
FLTfjL-T
' '
C l Tr L T
-
STATIRNi
''
C CALCL LATf Y SRDILAtE Fee* SUPR&UT I\E QRDSEN,'
> F I7 E ( *CS i I B S )
CTF-N AT (//, 1 CX, lL T S T T x a Sr X(I) AND Y(I) WHI CH AR[ THE
If a Ih S CE? Te CQNSTRUCT t h e DBUBLE v a 5S CURVE I , / / )
*r ITE ( U ' r , I Cfe )
]c
C S R A I ( I - X/ 1 * ******* + * » * * * * * * * , ******* + * t * * * * * * * * * * * * * *
A R I T r (I r s , in=)
I
99
5x , 1X (
IYEAR = IYR(I) .i
RSR A lc / / , P
I)' ,
2-
X ,IY(I)I,/)
—64—
TH = I ^
1= 1 ,L
D?
, . r !:.=:(I r ? , ^OC)
ECT F 9 FV A T ( 1CX,
X ( I ) , Y ( T ) , I , I v 9 N T H , IYEAR
?E2C*5,3I7)
IE (I"t* TH .Er. 12) IV9VJM =Gj IYEAR = IYEAR + l
= T V9KTV + I
I TE CffYTlNi jE
I“ S '
1TH
C
C
C
CALCULATE.
THE
X(O)
=
V ( O)
= r.o
X
AVD
CpVT I
CALCULATE
E9 R
STAT
C.C
DO £ " ; 5 I = I , L
XX(1 ,1 )
= X (I)
X X(IiE)
= Y (I)
C
Y
-X ( T - I)
- Y ( I - I )
KVE
STATISTICAL
DATA
r R 5 M SUBROUTINE
STAT
C
Cr ir t’-lti. r
i » ?* tf t( * * Vit * <!Htt .
*■ii# Ut *■*■**■# Kit
Utl
a Hfftt Hite *i
AN = L
CALL BTAT
»RIT E (IC S , laH)
19=5 F3R Ca T(//,10X, 'OUTPUT OE STATISTICAL DATA F 9R X(I)
,-R I Tc ( l o g , I
AND
96 5
I °6 EQRviAT (I OX, 1**************************************** I,
a W I T £ ( Ic?-, S )
4
E 9 RV A T ( / / , 1 C X, ' Y E A K ' i l C X , »V A R I A N C E ' , 1 0 X , ’ STD
c 9 J = I ,E
DEV ,IO X ,'
DO
, - R
I Tc ( 1 0 < , 1 )
X V.R ( J ) i V A R ( J ) , S T D E V( J ) , SXEW( J )
a E 9 Rv A T (E S C ,I ,E I 5 , I,E E O •I,E I C •I )
CO XT I N JE
• ’? I T r ( I c 3 , 1.1 )
ii
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nE I T e i l c ? , I ? )
C9 RCQ
qn
coeee.
1)
IE EQRY a T (E45.5,//)
W R IT e (lcs,197)
I ° 7 (T9 Rv AT ( I D X , ' * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
C* <i& tf u ti t; z H tin H t;it it tH $ 11#Uttttait# a IHitiftMtt ttfttt tta twit itit^ttU H T*-MHtmatittti X unit
C
—65—
C CALCULATE* TL E* "1 BF ST^AJ g ^T LINE THRBU g H PTg.
C
XE.U (C) =C •D
YSU-(C) = : .o
XYSL-I(U) = 0.0
CALCULATE ApAR AvD YBA r
O O
H = C
KK = 0
INET = I
17 De -6 I =
KK = K U l
XSUv (I) =
YSUv (I) =
AK = KS
XE A R ' I ) =
YpA-M I ) =
calculate
IN P T i L
X S UIM I- I ) + X(I)
YSUIM 1-1) + Y(I)
/SUK(I)ZAf
Y S U f ( I )Z AK
SU *'1 ac" X(I)
AND Y(R)
XY(I) = X( D *Y( I )
XYSU y ( I ) = XYSUv (I-I ) f XY(I)
C CALCULATE SUM
X-XBAR SQUARED
S U r- Mr = C . r
XP = X ‘-Ad ( I )
PB SR , = T'■PT /I
/r [Fr = > ( M -X :
DI-Sl
r XDirr*XDTFF
SLMIr = SUOIE + D IrsQ
SC CE »TTN-.-E
SL1
--IF (I) s SUDIF
C
C CALCULATE SLBdE P f LlNr
SLSciE d ) = (XVSJM( I )-( (A<)*XEAR( I )*YBAR( I ) ))/(SUMDIF ( I )
XM(I) = X B A R ( D t S L B P E d )
—66—
Y I N T (I) = YBAR(I) . XM(I)
scY^r = 0,3
YB = Y --ARt I )
De 13 K = IXPT,!
YCIpr = Y(KT)-YB
YC IprSU = Y-'Ir + YDTF
S V Y D I = SUV)! + YlTFSS
1:3 CONTINUE
S L Y D IF" ( I ) = SUYDI
SSSD f V(I) = S J Y D I F (I)-(((XYSUYt I) -AK*XB a R ( I )*YBAR(I > >
I/ (GL-YU IC(I)))
L = (SSSnEY(I)ZAK)+*.5
ICD = S S S D r V ( I )
r
^
GET Th: ST". EST. cE VERTICAL DEVIATIONS
C rI Y v p M CARDS
UE-(T) = (vSCDE V ( I )/(A <))**,5
IFt 11D *L T » ')) DEV(T) = 0.000)39 T9 19
In S C - T ( I ) = ( Y ( I ) - Y T M ( I ) ) Z X ( I )
RAilDt I ) = CSLB3 Et I I-SLB d T (I ))ZSLQpE (I )
R - P AT 19 ( I )
IF ( I •L T • IN = TtO) 39 T° 56
IF ( I.EO.L) 99 T 3 70
IF tAp.S (P AT I9 ( I ) ) .L T , *3 ) 38 T9 56
IF CA B S (RAT IP(I)) .GT. .35 G& TO 40
>U VJ
C'-EC< C l OP p 03 LIVr BrvgNl BREAK POINT rgR DEVIATIONS
40
II * I
OUTp UT I I
41 CALL SLCHCK(CSLO d E, B D EV,X,Y<I I,Y IN TS j XXX,YYYi I I I )
-R IT e ( 1C3, 7? )
7C reP-iATtz/Z.lOX, 'BACK SLOPE M O X , 'DEVIATION', 10 X, IY INTE
I '.*.( I ) ', IOX, 'Y ( I ) ', 5X, 'POINT' )
Df' U
I = 1 1, I If PO
-RIt P (109,73) 3S L B P E (I ',BDEV(I),Y l N TB(I),XXX (I),Y Y Y (I),
73 FC-Pv AT £BRlS »2, I4 )
14 CeNTINUF
U U
DETERMINE POINT I-HFRr SECOND SLOPE TOLEtU N C E
IS M£T
-67-
C PKr 'Pe ;' T
FIRST TpLEr;AVCF FA5 MET
C
CAL'- LIf-EC-<
•VI Tr (I o s , - 7 ) Tj
F7 FCT ( / / / , I C X , ' XY WHE=E FIRST TOLERANCE WAS M E T ' , 14)
WRITE( : r S , 4 C )
JJ
6 :. Pf V a T(///, l e x , 'XY /MEr3E S^C'
T9LERANCE WAS MET', J4)
I VFr = (JJ + I I )/2
WR IT E ( IC ^ - 1 3 ) INFT
Pis PER AT(//,lOX,'NEW PFCCIN AFTER SLPFE CMANSE IS ASSUMED
X S v •" ( I 1' F T - I ) =
Y g L ' (I'P T - I ) =
X Y S 1. V t T k F T - I )
KK *
sc Tq
0,0
0,0
= 0 . 0
O
e
:
B a ccv rIN,r
SC rP /I
Y
C v[ ^
17 A 1 ' CALCULATE LINE EQ, FRpk- r-n T t P L
C
8 C M = L-+1
JBr-! AMt ) = IN- T
GC 'S 17
C<C<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<
70 TL ( Ik.PT. NE, I >
5
P TR
90
..RI fE ( I r F i -iI )
31 FOR Al ( //,rx,
,Y r i ,2 x , 'PRINT' ,5X, fSLRFE', 8 X, 'F o
I, 'YlNTERCFPT t, EX, •STD. CEV ',5x, 'SLRRE r3A ',2X, 'X D E V )
IYE Ar = IVR(I) -I
IMt'TH a IS
,k2 I = I , L
VCA 1
-C(I) = (T(I)-YJNT(L-P) )/ S L S F E d -3)
XCE' (I) = V(J) - X C A L C U )
n 11
DC
CALCULATE t h e STC' DEVIATION R f X DEVIATIONS
S = 0*0
DR <3 I I = 1,1
DSC = V C E V ( I J )+ X D E V (11 5
S = S+CSQ
—6 8 —
61 CC-v r If')-''
Air = T
SN. = s/4 I I I
cr.V ( i )
S'. * *■« s
s
IP q Jv-T ( I ) = T
-\ U T T 4 7 T | / j Z c ^ ; | " - * * ' I * 9 r T ( I ) , S L q P E ( n , S L 2 P T ( I ) ,
* (I )
I- ry a rA T (C Is ,: r I h .P, Sr t 4 .P ,F IO •2 )
ir ( / 0 ' rw «E » 1 2 ) I mOVJH = C; I VEAP
IYEA2+1
Iv r ' I H : I 10 \ I M + 4
4^ CpN T IfNu1E
ALL
SI' ''c ^ = 0 ' "
CO ( 6 I = • ,L
SC ' X =
I )»XDrV( I )
Si ' I = Su "SC 4. SCuAP
f*O ‘ •
'ICvC
SI • v = C J m ^ c z a l l
r i~ . C V = S " 5 * * %
” T
T
' '
/K J
£)'; T- 'j
s;><-v-vS T ^ ,r)E:VrArieN 0F ALL IU| X DAT A D E V I A T I O N
output
' '
•'U t" .1T
1 *T<6$ » * * $ $ $ $ $ * $ ■ $ : $ i
P! 1Et l' S,226) SLO c5E(L)^VIVT(L)
3 3 ^ AT U C X , . NO SIGNIFICANT SLGPE CHANGE...
.► / • Ji j X f" * j F~"7f ' I )
PU' : JT
1
* v|;rS$.r.$«F$5$s$$«r s$-6
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V M -t E ( I - T j II)
BEST EQ.
IS
''-ESrElFessissi1-"2"':A R & R
I V Th = 11
IBr1NAk (C ) = I
DO 03 V
= I,H
DO °1 I = IUSE a M H h - I ),(IBRrAK(HH)-I)
IPv-t H I )
I
< 0 E ' - t n ,/ x ( i i i J c l L c I n 8REAK<H,),’3,,/SL,,PE<IB* E A K , w o *'1
—
69
—
C
C
C
r I CUL A TE THE c TD, DEVIATION Ar X DEVIATIONS
S = CD
ND DS7 Ti = I, I
NS
= NDEV(Ti),/OEV(II)
q = o + c ;’
O f CO', TI' h :-r
T
a 11 :
SK = S / A l I T
DEV(I) = SN**.5
RAr
-Fv(I) =
XDEV(T)ZDCV(I)
f-H e (Ir * / 1 4 ) I'"RNTH. IvEARj I •
S L O P E (I)/ S L E P T (I )
I )/ r-A T ID-( I ), yDEv ( I ), DAr-EV(I)
34 ^ - V a H e TS, I4,E-10.?j2r V . ? , 3 r ' 5 . 2 }
I R d - O Th .[0. IE) Im o n t h = C ; IVEAR = IV E A P + 1
p. = - TH = I "OK T m + I
SI CO"" I N .E
R P E l - r a J TP )
T- t O
Al (/Zj : D y , f :**#*•**•<$*<«•4;$SFt.$r SLSPF C-jANlE
I - '/ZZ)
9' Sf N" [N -L
HH = H
.0 Sc I = ( TPPEA<(Hu)),L
IF ( I
TH .ED. 12) !MONTH =C #*IYEAR = IVrAR +1
I('Dr Th = IvfjNT^ + I
TPNTp(I) = I
XCAL C ( I ) = ( V ( I ) - Y p iT(L))ZSLDPir(L)
X C E V (I ) = y ( I )- x C A L C (I 5
CALfLLr TF 'HE STD. DEVIATION OF X DFVIATIDNS
S = C •D
CD E SM U = I j I
DSD = XD E V ( I I )*XDEV<1 1 )
S = S + ESO
C G N t IKLE
Ain = r
SC « SZA I I I
D C I (T) - SN**.5
R ADCv(I) =
XDEV ( D Z D C V d )
NRI7Et ICS/DR) IMONTH/ IvFARj IP N T E (I )/ S L O P E (I )/ S L E P T (T >
I I )Jt-ATI = ( I ) , XDEV ( I ), RArEVt I )
—70—
3
cp..;' \ T ( r ? ? , T t , 3 r i o , ? , 2 r i 4 . 2 , 3 F i 5 . 2 )
P- CO'.1 J\\ c
;iT- T ’* * * < ***■## * 4 # » * * # * * * # * » * # * # * * # * * * ♦* ****# *#*# ##*<
" IT- c I
o’a )
•I „ Cf
..'I '//, ::' t »t.-FS ?$*$::-**$*.?*s /MXARY O U T P U T
IPC = I
$$s »$$$$$$*«
-/"I?) !P E 3 / 13 - r *x (i )
I V 7 r(V<--AT (//, 1~X, 'CPR PO I .TS F P S M '/ 13/ ' TO' / 13)
•P ITr (< "-, " I") SLOPE ( I =1PEA X ( I ) - I )/ V I\'T ( IPPEAK ( I ) - D
114 F C :\ AT ( iCX/ 'Y = '/ C7»3» 'X +
'/-7.0)
MjT-'JT ............................................... '
CO
37
T iU
t 1J
s
J. / - 1 - 1
-P I’P ( :rS,32C) TPPCi<( -IK), I n R E A K ( H 1-Hl)
CQT 'AT ( In X, 'Pb ■? n P T siTS trnP -1' /13/' T O ' , I 3)
IH ■ “ I : 'E.A < (u H 1 ) -2
JkDEC
i;v
Cf-
at
-/I??) S L O nP ( H H K ) , Y INT (UHH)
(; :x, 'Y = ',ci.3 , ' x +',F7»O//)
JUT' IT •. . . . . . . . . . . . . . . . . . . . . . . . . - . . . . . . . . . . . . . '
— • <- 3I T T Jr"
'I T l I is, H O ) IlRCAK(A) ,L
13' CO v \ 1 («-x, 'FOi POINTS F R B X ' , 13,'T9',13)
/I-:: L ( I - 3/135 ) SLPnE ( U , V J N T t u
333 FBk 1A T ( IEX/ ' Y = ' ,C7.3> 'X
*',r7.0)
C
C
c c a l c u l a t e “Bn t h l y m E a n s a n d s t d d e v i a t i o n s
C t -;* m cl - -s>.; *$*■ ^ I $ ' t *$■-$*$*'?$$$*$ i£ s $ f s t i & » $ $ $ $ $ $ $ $ $ $ $ $ $ $ $
C
?9'T I I = I
<
=
9
DB '7 D I = I,L
< = <+l
DEA11BCK, II) = X1DEV(I)
ire
cr.
A ? T C B vl D C
1 2 ) < = o; 11 * 11*1
•
A S S J m E DE-Vm K I, I I ) = O C T e i E S
o o n
NIC- AH = I I
C
Calculate
monthly
ANYr AS = NjYEAS
-f a n s
-71-
DO ^<9 '< = 1,1?
T = O • .)
Dt5 4OO II = I / v J Y E a R
t
=
I + J r v Y J ( K ^ I I )
*1
)
4 0 D Jt v . '
,r
•'.V- :L (> ) = I / A VYE a P
4«0 Ce V IVvE
CALCULATE STD, CE V
ce ^oi K = 1,12
T = 0*0
CO iOH TI = I,XJYEAP
TSC = Jr v/ijo (Y , I I )* Cr YYO
T - T+Tgl
4O :■ C e 1-T IXv:.
3T / 4(<) = ( T / ( 4 N Y [ 4 R . 1 . 0 ) ) * * . 3
>qi CexrINvv
-.jT- r
K T iXjt
1 •
1
vTAN
SyvDEV
Oji**-' .-r f I
DO livI1* K = 1,12
-'CItE ( lc8,49?) <, AVMfiCE(K), STCMfi(K)
T 2 - 4T(i ox, I ? , l 5 X , F i o * 1 # l o X , r i j . l )
404 CQXf Ii
NL1E
Ct-HST***
+
C
C CALCULAtE TvE PARAMETERS NECESSARY Tfi PLfiT V VS Y
X = L
Cfi CvH I = I, N
=(I<1) = X(I)
=(IxH) = Y(T)
3S2 Cfi-r IXd;
fivT'^jT
' '
OUTn UT
' '
OLTn JT
' v EAXS CF Y AvC X '
S L1T-vU T XPAR(L)
OUTnJT
YfiAR(L)
CUT=JT
' '
fiLT-vT
1 '
tMO
%+ $$ + '
-72-
SU-'' )UTI\'E SL C H : < ( S s L e ^ r , D D E V , X , Y, I I,YINTS,XXX, YYY.» I
n <>
;I■
-I•■S In'• Y ? M (Ci3®^)> YSUvU O : 3S0)/ XYSl'M (OI 380 ), X 3
I Y
3 ( 1 ),XY(SyO),
XM(SSO)/ SU m D I F (380)
• : U , XXt(SSO), vYY (SSO ), IIIfSfj''), X( 0:380),Y
YIVt
I( ? " O ) .
YS-
M i ­ I )
n i - .1-)
SLYDTr ( 3 5 0 ' ,
SSODEv (3 8 0 )
= CM
/SU*
= O M
x Y3 . v i i - I ) = 3 . 0
SMCuLxTr XPAP AN;D Yr-AP
C
X = D
DB 56 I = 11/I I+SO
.XX(I) = X M )
Y Y Y ( T ) = Y(I)
IIJ(I) = I
..
3
<+ 1
Si/' ( I ) = \3;> (T-I)
YS1
- ( I ) = YSL-U T - D
+ X(I)
+ V(T)
X = <
XL-"' (I) = /S l v (I)X a x
h V- (I) = YSuM(I)ZAK
C ^ a l CJL A tF SuM ^F X(I)
AMD Y( 2 )
AY(I) = X(I) * Y (I )
XYSV--K M = XYSJM(I-I)
+ XY(I)
C
C CALCjLATF SLm Dr X-A3As SQUARED
r
GL-Orr = O M
X 8 = X ;A x ( T )
rU- £2
AHl-F
rK r So
SUDIF
J
=
=
=
= I I/I
X(J)-XS
X D T r FtXDIFF
SiJ-KF + OIr SO
82 CO'-tINJE
SUv-L IF(I)
= SLDIF
C
: -ALCULAOc SLSPE SF LINE
-73-
•SL'
H = (X Y S J M ( D - ( U K ) *XSA4( T ) YSAP(I)))
-M ! ! ) = X P a p ( D ^‘
S s l ? pE H >
l'I
'(I) = vlVAR ( J ) . XV(T)
-L VSr =
YP = Y S A R t D
''r
J Hj v>» !!,I
YO l t- = Y (K O - Y P
YCIcrSQ = YC IP* YC TF
SJYCi
n
= SUv D l
+ YDiFSS
cbntinue
SUYCTF- ( J ) = SUYD?
SS ' CYfT ) = S J Y n i n I )-(((XYSUMt I ) - A K*XBAC(I )#
!/(SUjUIF(T)))
'-Df ' ( D r r S-CRVt I ) / ( A D )** ,5
r. C N \ 1 TNiUf
UETv p n
o
fn
—74—
Stiif U T T V F L I N E C H K
^
a
IS SSP
AN#
Y .''(
12, 31, 5 ) /
I I/
D T l-' SIPN %4SL(3S0)
YB/Sr
r
XP a S a
- Y(H)
JJ1
Y S I NS ( SgQ,
5)
D E V M p (1?,50)>
AVMPDEC
Y(TI)
SL ' E - K ( I )
= C. Q
"f r" J = 3,|_
w
'NCw-) = ( Y3 a SE - v (J ) )/ (XP A S - ■ X (J ' )
“F < ^ s ^ ,
Ct ~ ’N -F
Ju
-
J
u[T ,S\
ENC
jT
'1!
’/ 3 ' L 9 P ' 11' l2£
-75-
Sample o u t p u t fro m d a t a a n a l y s i s p ro g r a m .
L = Number of data points.
1 2 .)
'Br
1'5
r F s V-RbUN OI NG STATI ON' S
GO 90
9005
NV11-1F-' - e
2 't:
LISTING
OASfT S tATISV
-T
<(I )
AND V ( I )
WHICH
ARE
TH[
G RDEFrD
PAIRS
*******<**#^***»******#+***********##******
V(I)
X(I)
Si *' 0000
70,00900
1?5. O 11OCO
:t s , "9000
166,99C C 0
493.90900
g 1-0.99000
I 0 5 A .39 9C 0
1555,00'00
!£^7,90000
2016.OOOCO
6 °lA.COOOr
135s2,90009
18473.0,COOC
2 1 2 4 6 . COOOO
25522,90009
31597.90303
38308.00000
5 4 7 5 9 . “0000
85569.90000
95977.90000
99250,COCOO
I
2
3
4
5
6
7
8
9
10
U
10
11
12
I
2
3
4
5
6
7
8
—76-
3
.St?:- :I
£
T
/) 36tiy,
T
u'RE CH.\5
XP" TH
I
?
3
4
5
6
7
S
9
IC
II
I9
nEST ES.
TS
$$ • *
vRAN
-P24.S
-331.3
-471.7
-553.9
-364,9
399.1
-93.7
733.7
477.0
2 3.5
-1.07,2
-271.3
STD.DrV
‘
1258.0
1313.3
1371,5
1409.0
1234,7
937.4
394.3
"O'). 4
1054.6
1:27.4
1124.7
1183.3
v
29.O33X +
-77
Data generation program.
CO' 'Oh
Ixv
Ip X10,P),
ImI3PT (lc#2), X(Oi^OC), Y (600 )> IMD(GOO) >I
M . / - F r?EA<,NPE=?>
I Y S t' (I'',40, = ), vsi\l(5"3,5), 10(10), IYPR ( 5 0 / XSI NG (I?, SC
I IX" NtM * O/*’), B E E N , M C T I ’',
c > r ' R ( ’OC),
VAC(P), STDEV(P), SKE (2),CORED,
AN
o r -- SII-'' xKM* " ( I " , 2 ) , K^eNf lo), KM^NTW(IO), SAN INT (50/2) #U
''E "'(h. /), y ' A -0(5''')/ XFINAL(^OC), A V ‘ DDE (I 2) > ST D ID (12 ),r
- X N * ( ' ? , r o , H F ?(l?),XFrN(l2,20),
X ' t o X S (400,2), CE 112), X P (430), X N I (403),E R R M O f 12)
I INTt f- , XK'5 N
^
•
AL-L Y ''AT A AND C A L C U L A T E O R D I N A T E
S n R L a O IN T-3P( I, J) , \ S R E a K, Iu IS3 T (1 1, J) ,NPER
■*-A - ,
x
C
-
q P Si S 3 E CHANGES,
r ,E 'UV-ER
'FRp
QP P E R I 8 7 S
-L ■ (I 'L ,I ) v SR e a k j VP- p .!CHECK,
'C- AT(?I1 ,rric.o,?!!'1, 14)
*
m
q F d E r IODS QF ' " I S S I NG f
X DATA
Y ! \ T , SL 9P E, IMS(I),IYP(I)It
M
IF 'I^1-LCv •vr • ) 5 3° T I 339
OC
I ’ = I. M-M E AK
^ RE/"(105,8) ICd ( I N I ) , Id P ( I I ^ p )
dC ' AT(OlD)
C O t IN-E
READ ( 1 3 5 , 1 0
N , M , L , IDX
'M--v AT (4 H O
NVTdvT ’
Cv TdJT n ,m ,w
QLTpUT
QLtdUT
' *
1 1
1
ID N J v d ERS FQF S U P E Q u MOING S T A T I Q N S
1
1 1
CS H J = I ,V
OC 15 T I = H v
Cl Td v-T
CUt d Jt
CL Tp U t
•
RE A O (I NS,3 ) IC (J ) , Iv R p ( r ! }#(Y S IN (K / II/J )/ K « 1,12)
3 F Q M AT ( T4, 13,FS.I, H F f •! )
Ir- C M t INJE
—7 8 -
cH ’ c-. T
U
Il(J )
Cf' " r jr
91 T-'
» >
I
9L Tri T
- H T 1JT
(' I 1T V
1
1
I
BASE E-TATIq V ID NUMBER
*
##** ******* **********
*
I
ELt' -T
ID <
-M r
'
, 'I
Df ' L U II = I
< = ;
DC 3 ^ 2
I
I, V
I = I *L
< = K + I
Y E V C,( I, j)
= V 5 IV (Ku I I/ J )
IF
.DC. I?) I I = I 1+ 1
Ir ( < "DC. I?) < =" O
?f - Cf ' T V -1E
Df " Cf" t I^
C
r .'-LCLLf ’"t Y EDDTVATE cr^qv C:J3RCJT?VE 0RDGr V
C Cf. -EC"
LLC r S S A E Y
R f d ISDS S ITH S U B R O U T I N E
C S R e CT
IF(IChLCK*' E.I ) CS TC 71
CALL CTPECT
r
T--A- V
I 'ISPT ( I I/J)
7 • Df C'-)O T I = I, VRFR
AL-H < *
C, V ' C )
V IS r T dT , D iI^ IS P T d
1* 2 )
d ": r- D: H T (CIlD)
CD: C D -"IVLE
v > t.
r
Ir YDAtA
.FEPC TC SE CSRRr CTED OS IT HElE
x/xxxA/xxxxxxxx'/xxxxxxxxxyx
v/xaxxxxxxxxx'/xxxxxyYxxxxx
r
D n 300 I I = 1 1 L3FF
Of* 24b I = IVIS = K I I, I), IMIS = T C11,?)
X(I)
= ( Y d ) - YINT)/SLOPE
C^c CeNTlNJE
3CD C O N t INJE
n n
—/ 3 —
I X ^ O M = TME if P r I ?. VALUE X N 9 MN XDATA CApDS
* /N'9W:N = BpUND^IES PF K N S 1
VK' x DAT A
IE p T . NUMBERS
IFC n KX .EC, r.) 39 T « 731
DC >?O 7 KK = I,NKX
RfAE (1CE> '41) IX99\D( KK, I ), IX99ND (KK, 2 )
Al Ff}RN AT C2110)
2C7 Ce V1 INUE
Df " D KK = I, h:<X
DC - 7 1 1 = Iv B P N D C K K , I ),IXE6ND(K<v2)
RFADc 1 3 5 ,5) T^x,TVp C I P , CX S IN D CK / I I ), <» 1, 6 )
RE A C ( 1 0 5 ,?) IDX,I Yp (I I),CXSIN3(K,I I ), Ks7, 12)
"5 rpD-AT (AX, 21 A, 6r8«0)
17 c o n t i n u e
I? CONTINUE
DO 2 IA KK = IiNKX
OEA' (105,215) X K N O V N C K K / I )/XKNOWNCKK, 2)
715 P j Rv AT(EIID)
?V4 CONTINUE
n
OLTp
n
r .. .T
I
'
^tt=VIr
• '
PEST9D ep X(T)
I
X
M8N™
$$$$$$$$$
« «
STA..
II ' I
it = C
CO ?08 KK = IiNKX
DO 2D9 I = XK \ 9 K N ( K K i I IiXKNOWN(KKiP)
K = K+ l
X(I) * XSINSCKi II) + X(J-I)
n n
"E". 1 2 ) I i = T i+1
IF (“ .ED. 12 ) < = 0
PUTpL1T » »
'•p ITE( I C 5/ 2 1 ) <( I ) ,k , IVr( I I ).i, I^x
21 FOR a t (1CX/F10.2/214,5 X , 16)
PCD C O N t IN v E
2 CE CONTINUE
OUTpUT 1 '
0 U T ■■U T f •
OUT p U t
’ 1
CLTpUT
'
LISTING Or UNPANOOMIZED x v a l u e s
OUT' uI
'
******* ** ************ * * *****
OU rpUT
»
pT .
RtiTFtiT
' '
1
1
X (I)
X(O)
* r *o
De 777 I = I,L
-80-
X 1X (I)
= X(T)-X(T-I)
1C8/776 ) IiXX(T)
77» =-eQNAT(iOX, U O i 10X,F10.l )
v.R IT E (
7 7 7 C O X T IN U F
C
C "• : O I
? I'ST y 9 N T h OF S P E A K S AND V IS S INS
d T R j = DS
7 31 I F d R R E A K .ES. O
GO TO 732
DB ??! J = liN3PEA<
• REACc1 C 5 >7?2)KXONTH(J)
7 2 2 P t?R ‘ AT ( 1 1 0 )
721 C O N T I N U E
OUTEiT KXONTH(I)
782 DC 7-3 j = I > N d E d
T E A d l Si 7 ^ 4 ) < % O N ( J )
724 rPR
AT(IlO)
78 3 Cf n T I N J E
OUTrLd KKBN(I)
C
C - E a D I' '0 T H L Y V A L U E S Br
E a NS AND STP.DEV,
755 J = ! • i 2
DB
E E A7 ( I C5 i 7 8 6 )
7?-
v
A 7 M 9 D F ( JJiSTDYB(J)
FgP’ AT (2 2 10 ♦ O J
725 CONT I N U E
JC 337 I I = I iY
- E A d I'd,A) IDXi IY R ( II )i (X'<NB/, (KiIIJiK=Ii 12)
6 TBRHATt IBi I3,F 3 . I,11p 6 . I )
537 CONTINUE
TI = I
.
K = O
DB 833 I = IiL
<
=
K+l
-FIN(KiII) = XdNALf I )
81
I r ) n =o ; 11 = I r*i
—
ir(< .[r.
P33 r-R\T t n u e
r.
HJT[ T -if.; At1OrBRAIC AND ABSOLUTE
O O
C
—
I = t
^6 E 38 TI = I/ M
Db
rl9 ur = 1,1?
X K I (I ) = X K N 8 W ( K , I I )
I = 1+ 1
^3° CbN-1 IN 1JFP3? C O N t INUC
CC = C *C
C = 0*u
■DO '1.1 I = 1^L
C = xF In A U M + C
CC = XKT(I) + CC
"’ll 05'.7 TNUE
AL = L
.-.VCi-'N = C /AL
AVLACT r CC/AL
ERROR = 1.0-(A V C S V N / A V E A C T )
O U T p ,1
' '
OUTp -T ' ACTUAL PEP CENT ERRbR'
SLTPjT
' '
Rl lTP' :T rPRHR
'0 A 2 3 K = I # 12
RU' = ' •^
Db yD4 11 = I'O
jr: = xPT‘!(*r, I I ) - X K N S W ( K i H )
CU"
= H F T + SUM
??U C P k-r TNLC
AM = M
D IE1-1O (K ) = SUM/(AM/2,)
%23 CPk- I N U E
%
CRPHD
— 82““
DC 540 K = I , 12
V 1UT DIFMg(K)
gU
24 0
CF V IlNiUF
C CALClJLATr "4PNTHLV PrR CE NT FFRgRS
r
XF I' (1,1) = 1.0
Cd = OiC
t = 0 .J
OD V >2 '<= 1,12
DO 5/+3 I I = I, M
B = XF In.(K, I I ) + D
PB = X K V R W (K,I I ) + RB
04? CONTINUE
AM B M
A = R/m‘
AA = B3/AM
rrr
'I (<)
=
1.0
- ( A/AA)
04 ? C O v-T I N j F
111"'": 'T
' '3FR CENT
OUTPUT
error
VALUES rOR F AC h M O N T H ’
' '
% Lr RRR'
O
OtJTr
- .IT
'
M qi--TH
VJ c 44
r ',I?
rrTr ( IRS,%4B)<,PRRvg(K)
V B rpN‘.-A T ( I 10. "I 0.4)
',44 CONTINUE
C
C
C
C
C q M p a PE KNOWN dERIOO W ITH SYNTHETIC PrRlSD t
T g L ti "'E STAT
D 6 "AO '< = 1,12
IEADt I V , S7R)
OF(V)
372 “ O 2 AT (r I.O «C')
36G CONTINUE
C
C
I = D
Dfc r51 I I = I, M
DO F52 K= I, IP
I= V l
CK(I) = X K N R W (<,I I 5
S50 CS N'T INUF
351 C Q N T [NJE
—6 3 —
C
( = 10
10 F63 I = I,L
<F( I ) = Xrp.'AK T )-ir (K)
]F(K,E3.12) K = 0
< = '<+ 1
CONTINUE
C
IP -34 1= 1 ,L
yC(Izl) = -'-(I)
xs(',%) = y<(i>
%5'+ LONT IN JE
C
AN *= L
CALL ST a T(XS)
C
OUTF5UT
f I
TjTt-LiT
' '
OJTF j T
'
vE A+J
VARIANCE
I
SXEN
“ J TD jT
I i
IP F 34 J=I,'1
i'' TT *.(Ir j , -■a 5 ) Xa AS (J ),v AR (J ), 5 TIEV (J ),S<r ,'(J)
16F r s ^ U A T (/,P23.1
-13.1, r ? C » 1 , F 1 0 . I )
;S'. CP‘»t IN-E
O U T 5JT
' '
HUT-JT
f C csRSEL a T IF*+' c o e f f i c i e n t '
D U T p UT C ntT D
rT c' END
—84—
S U P tVfirr INE 4? 0 T E N
Cn ' '
T * ^ ( 1 0 , P ) , I Y I S P T U C , 2 ) , X ( 0 : 6 0 0 ) , Y ( 6 0 0 ) , I MO( GOO)
4 .:/:•>I./ N3REA<, \PE?,
I YS!' (12 ,40 ,3 ), YSIXC-(5C0, 5), ID(IO), IYPR(SO), XSINGf 10,
1 IX" cNO (10,2), I3EE.N, VCTIu ,
2 XPA r (4c0),
VAP (g ), STDEV(Z), S K E N (2)#C9RC3,
/
DI M E NS IO N Y ^ (3^ 0 ), v o r i x (330)
r
C CAUC JLATe
t 3T A k PRICIP ESP ALL SURRSUNDINS STATIONS
on 5i
i = i,L
Y P( T)
= Q.C
J = 1, \
ns
YP(I)
= Y S r:r>( i,j) + Yri( I )
S-,. CON t TNNC
YPFlN(I) = VP(J)
53 CO NTINUE
C
CALCULATE Y(I)
UWICH
IS S R D INATE SN DSURLE v4ASS CURVE.
C
Y(O) = o»o
DG 7D I = I >L
V(D = YPC-TNf I ) + Y(I-I)
70 CONTINUE
PETvPN
END
-85—
:.l
' ‘‘ UT r 'jr
T
( 10 ,;»), IMISPTf 13 ,
2 ), X (0 : 6 0 0 ) , y (600) , IMS
v 9pr An', \-OE3,
(I:','+0,3), VSIV3( 503,5), 10(10), IY=R(SO), XSINC
,N
l'v,
o n o
I
II X ^ N D (I"',"'), IBEEN, n c T Im ,
d X:
jz 3 (4Q3 ),
VAR (2 ),
d
Jt £ OL a I lows or CORRECTION MERE
OP rO I = !I P ( 1 , 1 ) , I3 p (1,2)
<(I > = (Y ( t ) - 2 , 5 )/.5
°
C f ' T T N vT
3
‘
-vO *
T = TSR (2, I ), IBR f?, 2 )
f( ’ = (Y(T).2 , E)/,%
CO - I W r
-Et- RN
ENO
STOEV(Z), SKEW(Z),CORCf
SLlR^UT IME STAT
(XR)
,MISOTt»C»2),Xt3!400)<rt600l,JM9t6
IB , 1° 1' 'W R '5 0 1 ' X S I*46(
‘
V 4 R I2>'
STDEV(S)j S k e m (S) j CBRCO j
Sf Dss i
'0= J ^ r '21' S U M X IM C j 5),
XX(Ifl)
XX,400. 2,,XSI 430,2)
= XS(Ifl)
XX(TfZ) = XS(IfZ)
CBMTjNUE
C c-iM 0 U t e t h e
^e a m
D0 700 J «1,2
XXS
= c.^
70 IS I = 1,L
XXS v = XX ( If J) + YyTU-'1
IS
C ? MT I N i 1E
XoA- (J) =XXSi.v /i\J
Cevr-UTE VATIAMCE AMD STAMQ. 3[V
XS3SUM = 0.3
:e °o
I = i ,l
xsC = xx(Ifj) * Xx(J,j )
XSCRiJM = XRC + XSQRUv
3: CPMT JM-j e
XRA-RU = XTAR(J)
v AR t j
OXPAR(J)
) = (I . /a x )# (Y g c g
STQELv( J)
CdTrjTE
jv
= (V A R (J ))««(. F )
THE CQE-R PR S<[\
SUQL E 3 = 0.0
OQ vO I = IfL
O l VO = (XX(IfJ)-X 3 A R ( J ) )**3
SU1IOEa = DRVS+ SJMRr3
AO CQMTJNUE
(AM#X=3AC?SQ) )
—8 8 —
= (l./4\)*(SJX:[3)/(
CO ^ V T E
STDEV(J)
**3)
<U9T?SIS C c-rrp e PEAKEgNESS 9F g A TA
SJ
[4 S 0.D
DO 5 D I = I>!_
D^Vv = (XV(IzJ) - x a A R ( j ) )*#4
SV'Dr* = s JVDE4 ♦ D£V4
5" CO'n T IN jcC
AKJ-(J) = (I •/AN)»(SUs1DE4)/(STDEV(J)#«4)
DO 75 I = I,L
S U ' y (I,J) = (X<(r,J) - X O A R ( J ) )
7S CO >T IN jr
700 C r \TIK Jf
r
CALCjLATE Sj" O-
(X « A R 5■*(Y-YPAR
SL D- f s .0."
DO 7A 1=1,L
PROXy - SU-IX(1,1) * SUW X(I,2)
SU *'/ = 3ROXY + S U w XY
7 O C P L t INUC
O
CALCULATE TWE CORRELATION COEFF,
CPRfn =((I ./AN)* SUMX Y ) / ( S T D E V ( I )*5TDEV(2))
RETURN
EKD
-8 9 -
Sample o u t p u t f o r d a t a . g e n e r a t i o n p r o gr a m .
U 9 pr
(EXCC), (J?)
V= * SF S T A ,
c = o
=F y d a t a
C a ^d s l = # s f
pjs
.
N = D
N = TC
L
=
r
NU'-9 FPc Fpi. ?u:-P ?jv''Iki3 P T a t i s m F
$*•$*$*$«$
- *!*$+’* t T T ETfT
TD(J)
ID(J)
= 3D IC
= 32=5
AFE CTATISh ID vUMBE^
***» ******* **********
IDY = 326?
<^SN(I ) = IC
RAN"?"!.ZED. C""PLrTED SBSrPVATISN STATIS'
PT,
I
2
3
4
5
A
7
8
3
IC
MSNTH
ID
11
12
I
2
3
4
5
6
7
YEAR
49
49
49
50
50
50
50
50
50
50
RECSPD
RECORD
1*00
172.82
98.76
81.06
1 17,16
2 53.30
297,26
313,42
9 21 .80
770.00
—90—
A(*Tv'AL rER CFTMT ERR9R
-3-0?
= ? • 6 8 ' ' : 9 5 4 ? - o2
Ir
15 ™
t '"En
SYNTHETIC D a t a is . g t . ACTUAL
Xf=NT^LY DEVIATIONS 9F ACTUAL F?9X
D
IC
1
Ieco s I Hf. J5
5
D I c 1*
(V)
r
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IS O C D J
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CTrvc
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r.( < )
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Z
- Li. £ ' I C
- 'c ° . 7r
r rr •
(
DIP'"
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r
s
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-•'7o. 3 : 4
(F)
Z
D I F N L (V )
Z
-6H . r CCp
D I F M N ( !< )
S
3 * 6 •f IC
DIFNL(K)
S
IC C • I5^
DlFMN(K)
Z
1f T . f V ^
Prp CFNT EPR9R VA l LCS F?? EACH MgNTH
vONTH
I
"I .2 " 19
2
-I .0737
°
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A
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7
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n
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12
.0369
SYNTHETIC
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