Modeling Hydrologic Change: Statistical Methods - Chapter 2 ppsx

Introduction to Time Series Modeling2.1 INTRODUCTION Time series modeling is the analysis of a temporally distributed sequence of data orthe synthesis of a model for prediction in which

Introduction to Time Series Modeling

2.1 INTRODUCTION

Time series modeling is the analysis of a temporally distributed sequence of data orthe synthesis of a model for prediction in which time is an independent variable Inmany cases, time is not actually used to predict the magnitude of a random variablesuch as peak discharge, but the data are ordered by time Time series are analyzedfor a number of reasons One might be to detect a trend due to another randomvariable For example, an annual maximum flood series may be analyzed to detect

an increasing trend due to urban development over all or part of the period of record.Second, time series may be analyzed to formulate and calibrate a model that woulddescribe the time-dependent characteristics of a hydrologic variable For example,time series of low-flow discharges might be analyzed in order to develop a model

of the annual variation of base flow from agricultural watersheds Third, time seriesmodels may be used to predict future values of a time-dependent variable A con-tinuous simulation model might be used to estimate total maximum daily loads fromwatersheds undergoing deforestation

Methods used to analyze time series can also be used to analyze spatial data ofhydrologic systems, such as the variation of soil moisture throughout a watershed

or the spatial transport of pollutants in a groundwater aquifer Instead of havingmeasurements spaced in time, data can be location dependent, possibly at someequal interval along a river or down a hill slope Just as time-dependent data may

be temporally correlated, spatial data may be spatially correlated The extent of thecorrelation or independence is an important factor in time- and space-series model-ing While the term time series modeling suggests that the methods apply to timeseries, most such modeling techniques can also be applied to space series

Time and space are not causal variables; they are convenient parameters bywhich we bring true cause and effect into proper relationships As an example,evapotranspiration is normally highest in June This maximum is not caused by themonth, but because insolation is highest in June The seasonal time of June can beused as a model parameter only because it connects evapotranspiration and insolation

In its most basic form, time series analysis is a bivariate analysis in which time

is used as the independent or predictor variable For example, the annual variation

of air temperature can be modeled by a sinusoidal function in which time determinesthe point on the sinusoid However, many methods used in time series analysis differfrom the bivariate form of regression in that regression assumes independence amongthe individual measurements In bivariate regression, the order of the x-y data pairs

is not important Conversely, time series analysis recognizes a time dependence and

2

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attempts to use this dependence to improve either the understanding of the underlyingphysical processes or the accuracy of prediction More specifically, time series areanalyzed to separate the systematic variation from the nonsystematic variation inorder to explain the time-dependence characteristics of the data where some of thevariation is time dependent Regression analysis is usually applied to unordered data,while the order in a time series is an important characteristic that must be considered.Actually, it may not be fair to compare regression with time series analysis becauseregression is a method of calibrating the coefficients of an explicit function, whiletime series analysis is much broader and refers to an array of data analysis techniquesthat handle data in which the independent variable is time (or space) The principle

of least squares is often used in time series analysis to calibrate the coefficients ofexplicit time-dependent models

A time series consists of two general types of variation, systematic and tematic For example, an upward-sloping trend due to urbanization or the annualvariation of air temperature could be modeled as systematic variation Both types

nonsys-of variation must be analyzed and characterized in order to formulate a model thatcan be used to predict or synthesize expected values and future events The objective

of the analysis phase of time series modeling is to decompose the data so that thetypes of variation that make up the time series can be characterized The objective ofthe synthesis phase is to formulate a model that reflects the characteristics of thesystematic and nonsystematic variations

Time series modeling that relies on the analysis of data involves four generalphases: detection, analysis, synthesis, and verification For the detection phase, effort

is made to identify systematic components, such as secular trends or periodic effects

In this phase, it is also necessary to decide whether the systematic effects aresignificant, physically and possibly statistically In the analysis phase, the systematiccomponents are analyzed to identify their characteristics, including magnitudes,form, and duration over which the effect exists In the synthesis phase, the informa-tion from the analysis phase is used to assemble a model of the time series andevaluate its goodness of fit In the final phase, verification, the model is evaluatedusing independent data, assessed for rationality, and subjected to a complete sensi-tivity analysis Poor judgment in any of the four phases will result in a less-than-optimum model

2.2 COMPONENTS OF A TIME SERIES

In the decomposition of a time series, five general components may be present, all

of which may or may not be present in any single time series Three componentscan be characterized as systematic: secular, periodic, and cyclical trends Episodicevents and random variation are components that reflect sources of nonsystematicvariation The process of time series analysis must be viewed as a process ofidentifying and separating the total variation in measured data into these five com-ponents When a time series has been analyzed and the components accuratelycharacterized, each component present can then be modeled

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2.2.1 S ECULAR T RENDS

A secular trend is a tendency to increase or decrease continuously for an extendedperiod of time in a systematic manner The trend can be linear or nonlinear Ifurbanization of a watershed occurs over an extended period, the progressiveincrease in peak discharge characteristics may be viewed as a secular trend Thetrend can begin slowly and accelerate upward as urban land development increaseswith time The secular trend can occur throughout or only over part of the period

of record

If the secular trend occurs over a short period relative to the length of the timeseries, it is considered an abrupt change It may appear almost like an episodic event,with the distinction that a physical cause is associated with the change and the cause

is used in the modeling of the change If the secular trend occurs over a majorportion or all of the duration of the time series, it is generally referred to as a gradualchange Secular trends are usually detected by graphical analyses Filtering tech-niques can be used to help smooth out random functions External information, such

as news reports or building construction records, can assist in identifying potentialperiods of secular trends

Gradual secular trends can be modeled using typical linear and nonlinear tional forms, such as the following:

func-linear: y = a + bt (2.1a)polynomial: y = a + bt + ct2 (2.1b)

reciprocal: (2.1d)exponential: y = ae−bt (2.1e)logistic: (2.1f)

in which y is the time series variable; a, b, and c are empirical constants; and t istime scaled to some zero point In addition to the forms of Equations 2.1, composite

or multifunction forms can be used (McCuen, 1993)

Example 2.1

Figure 2.1 shows the annual peak discharges for the northwest branch of the costia River at Hyattsville, Maryland (USGS gaging station 01651000) for wateryears 1939 to 1988 While some development occurred during the early years of therecord, the effect of that development is not evident from the plot The systematicvariation associated with the development is masked by the larger random variationthat is inherent to flood peaks that occur under different storm events and when theantecedent soil moisture of the watershed is highly variable During the early 1960s,

Ana-y

a bt

=+1

e bt

=+ −

1

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1980, another “no-effect” flat line may be appropriate The specific dates of thestarts and ends of these three sections of the secular trend should be based on recordsindicating when the levels of significant development started and ended The logisticmodel of Equation 2.1f may be a reasonable model to represent the middle portion

of the secular trend

2.2.2 P ERIODIC AND C YCLICAL V ARIATIONS

Periodic trends are common in hydrologic time series Rainfall, runoff, and ration rates often show periodic trends over an annual period Air temperature showsdistinct periodic behavior Seasonal trends may also be apparent in hydrologic dataand may be detected using graphical analyses Filtering methods may be helpful toreduce the visual effects of random variations Appropriate statistical tests can beused to test the significance of the periodicity The association of an apparent periodic

evapo-or cyclical trend with a physical cause is generally mevapo-ore impevapo-ortant than the results

of a statistical test Once a periodic trend has been shown, a functional form can beused to represent the trend Quite frequently, one or more sine functions are used

to represent the trend:

in which is s the mean magnitude of the variable, A is the amplitude of the trend,

f 0 the frequency, θ the phase angle, and t the time measured from some zero point.The phase angle will vary with the time selected as the zero point The frequency

is the reciprocal of the period of the trend, with the units depending on the dimensions

of the time-varying variable The phase angle is necessary to adjust the trend so thatthe sine function crosses the mean of the trend at the appropriate time The values

of A, f0, and θ can be optimized using a numerical optimization method In somecases, f0may be set by the nature of the variable, such as the reciprocal of 1 year,

12 months, or 365 days for an annual cycle

Unlike periodic trends, cyclical trends occur irregularly Business cycles areclassic examples Cyclical trends are less common in hydrology, but cyclical behav-ior of some climatic factors has been proposed Sunspot activity is cyclical

Example 2.2

Figure 2.2 shows elevation of the Great Salt Lake surface for the water years 1989

to 1994 The plot reveals a secular trend, probably due to decreased precipitation inthe region, periodic or cyclical variation in each year, and a small degree of randomvariation While the secular decline is fairly constant for the first 4 years, the slope

of the trend appears to decline during the last 2 years Therefore, a decreasing nonlinearfunction, such as an exponential, may be appropriate as a model representation ofthe secular trend

The cyclical component of the time series is not an exact periodic function Thepeaks occur in different months, likely linked to the timing of the spring snowmelt.The peak occurs as early as April (1989 and 1992) and as late as July (1991) Peaksalso occur in May (1994) and June (1990, 1993) While the timing of the maximumamplitude of the cyclical waves is likely related to the temperature cycle, it may beappropriate to model the cyclical variation evident in Figure 2.2 using a periodicfunction (Equation 2.2) This would introduce some error, but since the actual month

FIGURE 2.2 Variation of water surface elevation of Great Salt Lake (October 1988 to August 1994).

Y

Record High 4,211.85 feet June 3–8, 1986, and April 1–15, 1987

Record Low 4,191.35 feet October-November 1963

if the time to urbanize is very small relative to the period of record The failure of

an upstream dam may produce an unusually large peak discharge that may need to

be modeled as an episodic event If knowledge of the cause cannot help predict themagnitude, then it is necessary to treat it as random variation

The identification of an episodic event often is made with graphical analysesand usually requires supplementary information Although extreme changes mayappear in a time series, one should be cautious about labeling a variation as anepisodic event without supporting data It must be remembered that extreme eventscan be observed in any set of measurements on a random variable If the supportingdata do not provide the basis for evaluating the characteristics of the episodic event,one must characterize the remaining components of the time series and use theresidual to define the characteristics of the episodic event It is also necessary todistinguish between an episodic event and a large random variation

Example 2.3

Figure 2.3 shows the time series of the annual maximum discharges for the SaddleRiver at Lodi, New Jersey (USGS gaging station 01391500), from 1924 to 1988.The watershed was channelized in 1968, which is evident from the episodic change.The characteristics of the entire series and the parts of the series before and afterthe channelization are summarized below

The discharge series after completion of the project has a higher average dischargethan prior to the channelization The project reduced the roughness of the channeland increased the slope, both of which contributed to the higher average flow rate

Discharge (cfs) Logarithms Series n Mean

Standard Deviation Mean

Standard Deviation Skew

The reduction in variance of the logarithms is due to the removal of pockets ofnatural storage that would affect small flow rates more than the larger flow rates.The skew is essentially unchanged after channelization

The flood frequency characteristics (i.e., moments) before channelization aremuch different than those after channelization, and different modeling would be nec-essary For a flood frequency analysis, separate analyses would need to be made forthe two periods of record The log-Pearson type-III models for pre- and post-channelization are:

x= 3.037 + 0.1928K

and

x= 3.402 + 0.1212K,respectively, in which K is the log-Pearson deviate for the skew and exceedanceprobability For developing a simulation model of the annual maximum discharges,defining the stochastic properties for each section of the record would be necessary

2.2.4 R ANDOM V ARIATION

Random fluctuations within a time series are often a significant source of variation.This source of variation results from physical occurrences that are not measurable;these are sometimes called environmental factors since they are considered to beuncontrolled or unmeasured characteristics of the physical processes that drive thesystem Examples of such physical processes are antecedent moisture levels, smallamounts of snowmelt runoff that contribute to the overall flow, and the amount ofvegetal cover in the watershed at the times of the events

FIGURE 2.3 Annual maximum peak discharges for Saddle River at Lodi, New Jersey.

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The objective of the analysis phase is to characterize the random variation

Generally, the characteristics of random variation require the modeling of the secular,

periodic, cyclical, and episodic variations, subtracting these effects from the

mea-sured time series, and then fitting a known probability function and the values of

its parameters to the residuals The normal distribution is often used to represent

the random fluctuations, with a zero mean and a scale parameter equal to the standard

error of the residuals

The distribution selected for modeling random variation can be identified using

a frequency analysis Statistical hypothesis tests can be used to verify the assumed

population For example, the chi-square goodness-of-fit test is useful for large

samples, while the Kolmogorov–Smirnov one-sample test can be used for small

samples These methods are discussed in Chapter 9

2.3 MOVING-AVERAGE FILTERING

Moving-average filtering is a computational technique for reducing the effects of

nonsystematic variations It is based on the premise that the systematic components

of a time series exhibit autocorrelation (i.e., correlation between adjacent and nearby

measurements) while the random fluctuations are not autocorrelated Therefore, the

averaging of adjacent measurements will eliminate the random fluctuations, with

the remaining variation converging to a description of the systematic trend

The moving-average computation uses a weighted average of adjacent

observa-tions The averaging of adjacent measurements eliminates some of the total variation

in the measured data Hopefully, the variation smoothed out or lost is random rather

than a portion of the systematic variation Moving-average filtering produces a new

time series that should reflect the systematic trend Given a time series Y, the filtered

series is derived by:

(2.3)

in which m is the number of observations used to compute the filtered value (i.e.,

the smoothing interval), and w j is the weight applied to value j of the series Y The

smoothing interval is generally an odd integer, with 0.5 (m− 1) values of Y before

observation i and 0.5 (m− 1) values of Y after observation i used to estimate the

smoothed value A total of (m− 1) observations is lost; that is, while the length of

the measured time series equals n, the smoothed series, , only has n−m+ 1 values

The simplest weighting scheme would be the arithmetic mean (i.e., w j= 1/m):

(2.4)

Other weighting schemes often give the greatest weight to the central point in the

interval, with successively smaller weights given to points farther removed from

the central point For example, if weights of 0.25 were applied to the two adjacent

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time periods and a weight of 0.5 to the value at the time of interest, then the

moving-average filter would have the form:

(2.5)Moving-average filtering has several disadvantages First, m − 1 observations are

lost, which may be a serious limitation for short record lengths Second, a

moving-average filter is not itself a mathematical representation, and thus forecasting with

the filter is not possible; a functional form must still be calibrated to forecast any

systematic trend identified by the filtering Third, the choice of the smoothing interval

is not always obvious, and it is often necessary to try several intervals to identify

the best separation of systematic and nonsystematic variation Fourth, if the

smooth-ing interval is not properly selected, it is possible to eliminate both systematic and

nonsystematic variation

Filter characteristics are important in properly identifying systematic variation

As the length of the filter is increased, an increasingly larger portion of the systematic

variation will be eliminated along with the nonsystematic variation For example, if

a moving-average filter is applied to a sine curve that does not include any random

variation, the smoothed series will also be a sine curve with an amplitude that is

smaller than that of the time series When the smoothing interval equals the period

of the sine curve, the entire systematic variation will be eliminated, with the

smoothed series equal to the mean of the series (i.e., of Equation 2.2) Generally,

the moving-average filter is applied to a time series using progressively longer

intervals Each smoothed series is interpreted, and decisions are made based on the

knowledge gained from all analyses

A moving-average filter can be used to identify a trend or a cycle A smoothed

series may make it easier to identify the form of the trend or the period of the cycle

to be fitted A model can then be developed to represent the systematic component and

the model coefficients evaluated with an analytical or numerical optimization method

The mean square variation of a time series is a measure of the information

content of the data The mean square variation is usually standardized to a variance

by dividing by the number of degrees of freedom, which equals n − 1, where n is

the number of observations in the time series As a series is smoothed, the variance

will decrease Generally speaking, if the nonsystematic variation is small relative to

the signal, smoothing will only reduce the variation by a small amount When the

length of the smoothing interval is increased and smoothing begins to remove

variation associated with the signal, then the variance of the smoothed series begins

to decrease at a faster rate relative to the variance of the raw data Thus, a precipitous

drop in the variance with increasing smoothing intervals may be an indication that

the smoothing process is eliminating some of the systematic variation When

com-puting the raw-data variance to compare with the variance of the smoothed series,

it is common to only use the observations of the measured data that correspond to

points on the smoothed series, rather than the variance of the entire time series This

series is called the truncated series The ratio of the variances of the smoothed series

to the truncated series is a useful indicator of the amount of variance reduction

associated with smoothing

Example 2.4

Consider the following time series with a record length of 8:

Y = {13, 13, 22, 22, 22, 31, 31, 34} (2.6)While a general upward trend is apparent, the data appear to resemble a series of

step functions rather than a predominantly linear trend Applying a moving-average

filter with equal weights of one-third for a smoothing interval of three yields the

following smoothed series:

= {16, 19, 22, 25, 28, 32} (2.7)While two observations are lost, one at each end, the smoothed series still shows a

distinctly linear trend Of course, if the physical processes would suggest a step

function, then the smoothed series would not be rational However, if a linear trend

were plausible, then the smoothed series suggests a rational model structure for the

data of Equation 2.6 The model should be calibrated from the data of Equation 2.6,

not the smoothed series of Equation 2.7 The nonsystematic variation can be assessed

by computing the differences between the smoothed and measured series, that is, e i=

−Y i:

e3= {3, −3, 0, 3, −3, 1}

The differences suggest a pattern; however, it is not strong enough, given the small

record length, to conclude that the data of Equation 2.6 includes a second systematic

component

The variance of the series of Equation 2.6 is 64.29, and the variance of the

smoothed series of Equation 2.7 is 34.67 The truncated portion of Equation 2.6 has

a variance of 45.90 Therefore, the ratio of the smoothed series to the truncated

series is 0.76 The residuals have a variance of 7.37, which is 16% of the variance

of the truncated series Therefore, the variation in the residuals relative to the

variation of the smoothed series is small, and thus the filtering probably eliminated

random variation

A moving-average filtering with a smoothing interval of five produces the

fol-lowing series and residual series:

= {18.4, 22.0, 25.6, 28.0} (2.8a)and

e5= {−3.6, 0.0, 3.6, −3.0} (2.8b)

The variance of the truncated series is 20.25, while the variances of and e5 are

17.64 and 10.89, respectively Thus, the variance ratios are 0.87 and 0.53 While the

variance ratio for the smoothed series (Equation 2.8a) is actually larger than that ofthe smoothed series of Equation 2.7, the variance ratio of the residuals has increased

greatly from that of the e3 series Therefore, these results suggest that the smoothingbased on an interval of five eliminates too much variation from the series, eventhough the smoothed series of Equations 2.7 and 2.8a are nearly identical The five-point smoothing reduces the sample size too much to allow confidence in theaccuracy of the smoothed series

Example 2.5

Consider the following record:

X = {35, 36, 51, 41, 21, 19, 23, 27, 45, 47, 50, 58, 42, 47, 37, 36, 51, 59, 77, 70}

(2.9)The sample of 20 shows a slightly upper trend for the latter part of the record, anup-and-down variation that is suggestive of a periodic or cyclical component, andconsiderable random scatter A three-point moving-average analysis with equalweights yields the following smoothed series:

= {41, 43, 38, 27, 21, 23, 32, 40, 47, 52, 50, 49 42, 40, 41, 49, 62, 69}

(2.10)The smoothed values for the first half of the series are relatively low, while the dataseem to increase thereafter The second point in the smoothed series is a local highand is eight time steps before the local high of 52, which is eight time steps beforethe apparent local peak of 69 The low point of 21 is nine time steps before the locallow point of 40 These highs and lows at reasonably regular intervals suggest aperiodic or cyclical component A five-point moving-average yields the followingsmoothed series:

= {36.8, 33.6, 31.0, 26.2, 27.0, 32.2, 38.4, 45.4, 48.4, 48.8, 46.8, 44.0, 42.6, 46.0, 52.0, 58.6} (2.11)This smoothed series exhibits an up-and-down shape with an initial decline followed

by an increase over six time intervals, a short dip over three time intervals, and then

a final steep increase While the length of the smoothed series (16 time intervals) isshort, the irregularity of the local peaks and troughs does not suggest a periodiccomponent, but possibly a cyclical component

If a smoothing function with weights of 0.25, 0.50, and 0.25 is applied to thesequence of Equation 2.9, the smoothed series is

The same general trends are present in Equation 2.12 and Equation 2.10.

To assess the magnitude of the nonsystematic variation, the smoothed series ofEquation 2.10 and the actual series of Equation 2.9 can be used to compute the

residuals, e:

e3 = {5, −8 −3, 6, 2, 0, 5, −5, 0, 2, −8, 7, −5, 3, 5, −2, 3, −8} (2.13)

These appear to be randomly distributed with a mean of −0.06 and a standarddeviation of 5.093 The residuals for the smoothed series based on a five-pointsmoothing interval follow:

e5 = {−14.2, −7.4, 10.0, 7.2, 4.0, 5.2, −6.6, −1.6, −1.6, −9.2, 4.8, −3.0, 5.6, 10.0,

The variances for the truncated series, the smoothed series, and the residual seriesare shown in Table 2.1, along with the ratios of the variances of both the smoothedand residual series to the truncated series The large decrease in the percentage ofvariance of the five-interval smoothing suggests that some of the systematic variation

is being removed along with the error variation

In summary, the smoothed time series of Equation 2.12 appears to consist of asecular trend for the last half of the series, a periodic component, and random variation.While the record length is short, these observations could fit a model to the originalseries

Example 2.6

Consider a time series based on the sine function:

Y(t) = 20 + 10 sin t (2.15)

in which t is an angle measured in degrees Assume that the time series is available

at an increment of 30 degrees (see column 2 of Table 2.2) For this case, the timeseries is entirely deterministic, with the values not corrupted with random variation.Columns 3 to 6 of Table 2.2 give the smoothed series for smoothing intervals of 3,

5, 7, and 9 Whereas the actual series Y(t) varies from 10 to 30, the smoothed series

vary from 10.89 to 29.11, from 12.54 to 27.46, from 14.67 to 25.33, and from 16.96

to 23.03, respectively As the length of the smoothing interval increases, the tude of the smoothed series decreases At a smoothing interval of 15, the smoothedseries would be a horizontal line, with all values equal to 20

Variance of Truncated

Series,

Smoothed Series,

Residual Series,

S S s t

2 2

S S e t

2 2

This example illustrates the dilemma of moving-average filtering The processreduces the total variation by smoothing both random and systematic variation.Smoothing of random variation is desirable, while eliminating part of the systematicvariation is not As the smoothing interval is increased, more of the systematicvariation is smoothed out However, if the smoothing interval is too short, aninsufficient amount of the nonsystematic variation will be smoothed to allow iden-tification of the signal.

To model time series in order to circumvent this dilemma, it is a good practice

to develop several smoothed series for different smoothing intervals and evaluateeach in an attempt to select the smoothed series that appears to provide the bestdefinition of the signal Developing general rules for determining the smoothinginterval is difficult to impossible

Example 2.7

A common problem in hydrologic modeling is evaluation of the effects of urbandevelopment on runoff characteristics, especially peak discharge It is difficult todetermine the hydrologic effects of urbanization over time because development occursgradually in a large watershed and other factors can cause variation in runoff charac-teristics, such as variation in storm event rainfall or antecedent moisture conditions

TABLE 2.2 Characteristics of Smoothed Time Series

This example demonstrates the use of moving-average smoothing for detecting

a secular trend in data The data consist of the annual flood series from 1945 through

1968 for the Pond Creek watershed, a 64-square-mile watershed in north-centralKentucky Between 1946 and 1966, the percentage of urbanization increased from2.3 to 13.3, while the degree of channelization increased from 18.6% to 56.7% withmost of the changes occurring after 1954 The annual flood series for the 24-yearperiod is shown in Table 2.3

The data were subjected to a moving-average smoothing with a smoothinginterval of 7 years Shorter intervals were attempted but did not show the seculartrend as well as the 7-year interval The smoothed series is shown in Figure 2.4.The smoothed series has a length of 18 years because three values are lost at eachend of the series for a smoothing interval of 7 years In Figure 2.4, it is evident thatthe smoothed series contains a trend Relatively little variation exists in the smoothedseries before the mid-1950s; this variation can be considered random After urban-ization became significant in the mid-1950s, the flood peaks appear to have increased,

TABLE 2.3 Annual Flood Series and Smoothed Series for Pond Creek Watershed, 1945–1968 Year

Annual Maximum (cfs)

Smoothed Series (cfs)

as is evident from the nearly linear upward trend in the smoothed series It appearsthat the most appropriate model would be a composite (McCuen, 1993) with zero-sloped lines from 1945 to 1954 and from 1964 to 1968 A variable-intercept powermodel might be used for the period from 1954 to 1963 Fitting this formulationyielded the following calibrated model:

where t = 1, 10, 20, and 24 for 1945, 1954, 1964, and 1968, respectively This modelcould be used to show the effect of urbanization in the annual maximum dischargesfrom 1945–1968 The model has a correlation coefficient of 0.85 with Pond Creekdata

It is important to emphasize two points First, the moving-average smoothingdoes not provide a forecast equation After the systematic trend has been identified,

it would be necessary to fit a representative equation to the data Second, the trendevident in the smoothed series may not actually be the result of urban development.Some chance that it is due either to randomness or to another causal factor, such as

FIGURE 2.4 Annual flood series and smoothed series for Pond Creek Watershed, 1945–1968.

Key:  , annual flood series; ∆, smoothed series.

Q

t

t i

for for 10 <

for