Modeling Hydrologic Change: Statistical Methods - Chapter 13 (end) docx

Frequency Analysis
Under Nonstationary
Land Use Conditions

Glenn E. Moglen

13.1 INTRODUCTION

Many hydrologic designs are based on estimates of flood magnitudes associated
with a specified return period. Flood-frequency analyses based on data collected by
stream gages can be used to determine these flood magnitudes. In the event of land
use change associated with urbanization, deforestation, or changes in agricultural
practices within the gaged watershed, the annual maximum time series recorded by
the gage includes a trend or nonstationary component that reflects the effect of the
land use change. Because urbanization typically increases the flood response of a
watershed, a flood frequency analysis performed on a nonstationary time series will
lead to underestimation of flood magnitudes and insufficient, underdesigned structures.
Accounting for this nonstationarity is, therefore, essential for appropriate design.

13.1.1 O

VERVIEW



OF

M

ETHOD

The method presented in this chapter may be used to adjust a peak-discharge time
series that is nonstationary because of changing land use within the gaged watershed
over the gaging period. The method has several parts. First, the method focuses on
deriving a spatially sensitive time series of land use. This step requires resourceful-
ness and creativity on the part of the hydrologist to obtain relevant data and to
organize these data into a format, most likely using making use of geographic
information systems (GIS) technology, that can be readily used to generate the values
necessary as input to the hydrologic model. The next step is to calibrate the hydro-
logic model over the gaging period being studied, while taking into account the
spatially and temporally varying land use. The final step is to use the calibrated
model to generate a synthetic time series of peak discharge, related to the observed
time series, but adjusted to reflect a single land use condition such as the current or
ultimate land use. This chapter examines the differences in derived flood-frequency
behavior between the observed (nonstationary) and adjusted (stationary) peak-
discharge time series.
13

L1600_Frame_C13.fm Page 367 Friday, September 20, 2002 10:25 AM
© 2003 by CRC Press LLC

13.1.2 I

LLUSTRATIVE

C

ASE

S

TUDY

: W

ATTS

B

RANCH

The process of accounting for nonstationarity in the flood record and ultimately
performing a flood-frequency analysis based on an adjusted flood series is illustrated
for a watershed in the Piedmont region of Maryland just north of Washington, D.C.
This watershed, Watts Branch, has a drainage area of 3.7 square miles at the location
of U.S. Geological Survey (USGS) gage station 01645200, which was active from
1958 to 1987. According to the Maryland Department of Planning’s assessment as
of 2000, it was composed of a mix of residential densities totaling 35% of the land
area. Commercial and industrial land uses cover 23%, and other urban uses (insti-
tutional and open urban land) cover an additional 15% of the land area. A significant
percentage (18%) of agricultural land remains within the watershed with the remain-
der (9%) made up of deciduous forest. By comparison, at the time of a 1951 aerial
photograph, the rough land use distribution was 15% urban, 64% agricultural, and
21% forest. Figure 13.1 shows a comparison of the spatial distribution of these land
uses in 1951 and 2000. As further evidence of the changes this watershed has
undergone, see the literature focused on channel enlargement and geomorphic
change (e.g., Leopold, Wolman, and Miller, 1964; Leopold, 1973; Leopold, 1994).

FIGURE 13.1

Spatial distribution of land use in Watts Branch watershed in 1951, 1987,
2000, and under ultimate development conditions.

L1600_Frame_C13.fm Page 368 Friday, September 20, 2002 10:25 AM
© 2003 by CRC Press LLC

The flood of record took place in 1975 with a discharge of 3400 ft

3

/sec. In 1972,
Hurricane Agnes, responsible for the flood of record in many neighboring water-
sheds, produced an annual maximum flow of 2900 ft

3

/sec. The Watts Branch water-
shed is used throughout subsequent sections to illustrate the various phases of the
modeling process.

13.2 DATA REQUIREMENTS

In a flood-frequency analysis for a stationary system, the only data required are the
observed annual maximum series,

Q

p,o

(

t

). Because of the land use change that
induced nonstationarity into

Q

p,o

(

t

),



other data are required: an observed causal
rainfall time series,

P

(

t

), and several GIS data sets such as digital elevation models
(DEMs), land use, and soils. These data sets are described in greater detail in the
following sections.

13.2.1 R

AINFALL

D

ATA

R

ECORDS

Rainfall data are collected through a nationwide network of rain gages and, more
recently, radar and satellite imagery. These data are archived and readily available
on the Internet at a number of websites, the most accessible being the National
Climatic Data Center (NCDC, 2001). This site provides free download access for
point rainfall data. Data are stored in a database that is accessed through the website
allowing the location and extraction of rainfall data that suits a range of selection
criteria such as latitude/longitude, state/county/city name, ZIP code, or station iden-
tification number.

13.2.2 S

TREAMFLOW

R

ECORDS

Streamflow data are collected and archived by the USGS and are similarly made
available for extraction via a web-based interface that allows for a range of potential
selection criteria (U.S. Geological Survey, 2001b). Data are organized and archived
in two forms: daily averaged flows and the annual maximum series. In the case of
the annual maximum series data, the discharge is accompanied by a field that also
identifies the date of occurrence of the annual maximum. This chapter will focus
on the annual maximum series and any trends that may be present in this series as
a result of land use change within the gaged watershed.

13.2.3 GIS D

ATA

Rainfall-runoff estimates of peak discharge could be developed with a range of
potential models. This chapter uses Natural Resources Conservation Service (NRCS;
U.S. Soil Conservation Service, 1985, 1986) methods to develop such estimates.
Although the details of using other models to perform a similar analysis would
certainly differ, the spirit of the approach presented here is consistent with any model.
As stated earlier, several different types of GIS data are required for the hydro-
logic modeling of the annual maximum discharges. First and foremost, topographic
data in the form of a DEM is probably the most fundamental data type required for

L1600_Frame_C13.fm Page 369 Friday, September 20, 2002 10:25 AM
© 2003 by CRC Press LLC

this analysis. The DEM serves first to determine flow paths and provide an automated
delineation of the study watershed that provides an estimate of the watershed area.
The DEM then allows for the estimation of slopes and times of concentration that
are central to the analysis as well. These data are made available by the USGS (U.S.
Geological Survey, 2001c) at several map scales. The data used in the case study

presented here are derived from the 1:24,000 map scale and have a resolution of 30
meters.
NRCS methods depend heavily on the estimation of a curve number, requiring
information about the area distribution of both land use and hydrologic soil type.
Thus, coverages of both land use and hydrologic soil type are required. Land use
coverages may be obtained from a number of sources. The USGS GIRAS (Mitchell
et al., 1977) is probably the oldest, widely available data set. It tends to reflect land
use of an approximately 1970s vintage. These data are now commonly distributed
as part of the core data set in the BASINS model (U.S. Environmental Protection
Agency, 2001a). Newer data may also exist on a more regionally varied level, such
as the MRLC data set (Vogelmann et al., 1998a; Vogelmann, Sohl, and Howard, 1998b;
U.S. Environmental Protection Agency, 2001b) that covers many states in the eastern
United States and dates to approximately the early 1990s. Other more high-quality
data sets will likely be available on an even more limited basis, perhaps varying by
state, county, or municipality. Generally, the higher-quality data will reflect condi-
tions from periods more recent than the GIRAS data mentioned earlier. Knowledge
of land use from before the 1970s will likely need to be gleaned from nondigital
sources such as historical aerial photography or paper maps.
Soils data are generally obtained from the NRCS. The NRCS publishes two
different sets of digital soils data: STATSGO (NRCS, 2001a) and SSURGO (NRCS,
2001b). The STATSGO data are the coarser of the two and are digitized from
1:250,000 scale maps (except in Alaska where the scale is 1:1,000,000) with a
minimum mapping unit of about 1544 acres. These data are available anywhere
within the United States. The SSURGO data are digitized from map scales ranging
from 1:12,000 to 1:63,360. SSURGO is the most detailed level of soil mapping done
by the NRCS. These data are in production at this time and availability varies on a
county-by-county basis. In the case study presented here, Watts Branch lies entirely
within Montgomery County, Maryland, one of the counties for which SSURGO data
are available.

13.3 DEVELOPING A LAND-USE TIME SERIES

The particular emphasis of this chapter is to consider the effect of changing land
use on peak discharge; land use is not static, but rather continually changing in both
time and space throughout the time series. A practical problem that generally arises
is that the GIS data to support the modeling of peak discharge on an annual basis
do not exist. In general, one has access to, at best, several different maps of land
use corresponding to different “snapshots” in time. This section provides and illus-
trates a method to develop a land-use time series on an annual time step.
The data required to develop a land-use time series are two different maps of
land use covering the extents of the watershed and spanning the same time period

L1600_Frame_C13.fm Page 370 Friday, September 20, 2002 10:25 AM
© 2003 by CRC Press LLC

as the available annual maximum discharge record. Additionally, data are required
that convey the history of land use development at times between the two land use
snapshots. Such data are typically available in the form of tax maps. In the example
provided here, the Maryland Department of Planning publishes such data (Maryland
Department of Planning, 2001) that indicate tax map information at the detailed
level of individual parcel locations. One of the attributes associated with these data
is the date of construction of any structure on the property.
The notation

LU

(

x,t

) is used here to indicate the land use across all locations in
the vector

x

within the watershed being studied, and

t

is any generic time in years.
The land-use time series is initialized to be the land use at time,

t

1

, indicated by the
earlier of the two land use maps. This land use is denoted by

LU

(

x

,

t

1

). If

t

1

is earlier
than approximately 1970, it is likely that the required land use data are not available
digitally, but rather in the form of a paper map or aerial photograph. Such data will
need to be georeferenced and then digitized into a hierarchical land-use classification
scheme such as the one created by Anderson et al. (1976). Figure 13.1 shows land
use over the study watershed at times

t

1



=

1951 and

t

2

=

2000. (For completeness,
this figure also shows the watershed at an intermediate time,

t



=

1987, and at some
future time corresponding to ultimate development conditions. These land use con-
ditions are discussed later in this chapter.) Using the

LU

(

x

,

t

1

)

coverage as a starting
point, the land use is then allowed to transition to

LU

(

x

,

t

2

) in the specific year,

t*

,
that the tax map information indicates is the year of construction for that individual
parcel. Applying this rule over all years

t

1

<

t*

<

t

2

and for all parcels within the
watershed allows the modeler to recreate land use change on an annual time step.
This process is illustrated in Figure 13.2, which shows a view of several rows
of parcels in a subdivision over the years 1969 and 1970. The date shown within
each parcel is the year in which the tax map data indicates that it was developed.

FIGURE 13.2

Parcel-level view of land-use change model. Parcels shown in white are devel-
oped and gray parcels are not developed. Note that all parcels shown become developed by
1970.

L1600_Frame_C13.fm Page 371 Friday, September 20, 2002 10:25 AM
© 2003 by CRC Press LLC

The GIS treats all parcels shown in gray (undeveloped parcels) to remain in their
original land use at time

t

=

t

1

, while those shown in white (developed parcels) have
transitioned to their final land use at time

t

=

t

2

. Figure 13.3 shows the aggregate
change in the land use distribution within the Watts Branch watershed between 1951
and 2000. Note that land use does not simply change linearly between these two
known times, but rather it changes in an irregular fashion following the actual
patterns of development as they were realized within the watershed.

13.4 MODELING ISSUES

A wide range of modeling issues confront the hydrologist performing this type of
study. First and foremost is the simple selection of the model to use. Other issues
include methods for calibrating the model and ultimately using the model to simulate
the discharge behavior of the study watershed in a predictive sense. This section
presents a discussion of these wide-ranging issues and argues for a particular series
of choices throughout the modeling process, recognizing that different choices might
be selected by others. The decisions presented here reflect the pragmatic needs of

the engineer wishing to make use of a valuable gage record, but also recognizing
the influence that urbanization has on that record.

13.4.1 S

ELECTING



A

M

ODEL

This study is concerned with observed and simulated peak discharges from a small
watershed subject to changes in land use over time. The model that is selected must
therefore predict peak discharges, be appropriate for a watershed of this size, and be
sensitive to land use change in its predictions of peak discharge. The NRCS TR-55
(U.S. Soil Conservation Service, 1986) is one such model. It is selected here over the
HEC-HMS model (U.S. Army Corps of Engineers, 2000) because the NRCS models

FIGURE 13.3

Aggregate land-use distribution in Watts Branch watershed over time. The bar
at the far right gives the ultimate development land-use distribution.

L1600_Frame_C13.fm Page 372 Friday, September 20, 2002 10:25 AM
© 2003 by CRC Press LLC

are the recognized analysis tools required by the Maryland Department of the
Environment for all flood discharge studies. TR-55 is chosen over the more general
TR-20 (U.S. Soil Conservation Service, 1984) because only peak discharge esti-
mates, not entire hydrographs are sought. Furthermore, this model is appropriate in
this case because of the small scale of the watershed being studied. A larger water-
shed where reach routing is a significant part of the overall travel time, or a watershed
with significant detention, would require the more sophisticated TR-20. In any case,
although the details specific to the TR-55 graphical method are presented here, the
general approach, which is essentially model independent, is emphasized.
The TR-55 model determines peak discharge using

Q

p

,

s

(

t

)

=



q

u

(

t

)

AQ

(

t

) (13.1)
where

Q

p,s

(

t

) is the simulated peak discharge in ft

3

/sec,

q

u

(

t

) is the unit peak discharge
in ft

3

/sec-in. of runoff,

A

is the area of the watershed in mi

2

, and

Q

(

t

) is the runoff
depth in inches. The functional dependence of runoff, unit peak discharge, and
simulated peak discharge on time is explicitly shown to emphasize the time-varying
nature of these quantities due to changes in land use.
The runoff depth is determined from
(13.2)
where the standard assumption is made that the initial abstraction,

I

a

(

t

), is equal to
20 percent of the storage,

S

(

t

)

.

Both storage and initial abstraction are in inches
units.

P

(

t

) is the

causal

precipitation depth associated with the observed annual
maximum discharge. This quantity is discussed in greater detail in Section 13.4.2.
Storage,

S

(

t

), is determined as a function of the curve number,

CN

(

t

) using
(13.3)
The curve number is determined using a standard “look-up table” approach given
the spatial overlap of land use and soils and using the NRCS-defined curve numbers
(U.S. Soil Conservation Service, 1985). The time-varying nature of

CN

(

t

) is due to
the time-varying land use within the watershed as discussed earlier in Section 13.3.
The unit peak discharge,

q

u

(

t

), is a function of two quantities, time of concen-
tration,

t

c

(

t), and the ratio of the initial abstraction to the precipitation, I
a
(t)/P(t).
Again, the dependence of these quantities on time is shown here explicitly. The unit
peak discharge is generally determined graphically; however, this procedure is auto-
mated with the GIS using the equation
(13.4)
Qt
Pt I t
Pt I t St
Pt St
Pt St
a
a
()
[() ()]
() () ()
[() . ()]
() . ()
=
−
−+
=
−
+
2

2
02
08
St
CN t
()
()
=−
1000
10
log[ ( )] [ ( )] [ ( )] log[ ( )] [ ( )] {log[ ( )]}qt Crt Crt tt Crt tt
ucc
=+⋅ +⋅
01 2
2
L1600_Frame_C13.fm Page 373 Friday, September 20, 2002 10:25 AM
© 2003 by CRC Press LLC
where r(t) is determined from
(13.5)
and C
0
, C
1
, and C
2
, are tabular functions of this ratio. Values of these constants for
the U.S. Soil Conservation Service Type II storm distribution, which is appropriate
for the study watershed are provided in Table 13.1.
For consistency, the time of concentration, t
c

(t), was determined in this study
using the SCS lag equation (U.S. Soil Conservation Service, 1973), rather than the
often-used velocity method. In this case, the dependency of the velocity method on
spatially and time-varied surface roughness would be too arbitrary to characterize
consistently. The lag equation’s dependency on curve number, which is characterized
very carefully throughout this study, was instead chosen as the basis for developing
t
c
estimates.
The SCS lag time, t
l
(t) in minutes, is determined using
(13.6)
where L is the longest flow path in the watershed in feet, and Y is the basin averaged
slope in percentages.
The lag time is converted to a time of concentration by multiplying by a factor
of 1.67 and also accounting for speedup in runoff rates due to imperviousness
introduced in the urbanization process. Time of concentration is thus determined
using the following equation (McCuen, 1982):
(13.7)
TABLE 13.1
Constants C
0
, C
1
, and C
2
Used in
Equation 13.4 for the SCS Type II Storm
Distribution

I
a
/PC
0
C
1
C
2
0.10 2.55323 −0.61512 −0.16403
0.30 2.46532 −0.62257 −0.11657
0.35 2.41896 −0.61594 −0.08820
0.40 2.36409 −0.59857 −0.05621
0.45 2.29238 −0.57005 −0.02281
0.50 2.20282 −0.51599 −0.01259
Source: U.S. Soil Conservation Service, Urban Hydrol-
ogy for Small Watersheds, Technical Release 55, U.S.
Department of Agriculture, Washington, DC, 1986.
rt
It
Pt
a
()
()
()
=
tt
L
CN t
Y
l

()
()
.
.
.
=
−






100
1000
9
1900
08
07
05
t t t t I t D D CN t D CN t D CN t
c
l
() . (){ ()[( () () ()]}=⋅−+⋅+⋅+⋅167 1
01 2
2
3
3
L1600_Frame_C13.fm Page 374 Friday, September 20, 2002 10:25 AM
© 2003 by CRC Press LLC

where I(t) is the time-varying imperviousness of the watershed in percentages, and
D
0
, D
1
, D
2
, and D
3
are −6.789 × 10
−3
, 3.35 × 10
−4
, −4.298 × 10
−7
, and −2.185 × 10
−8
,
respectively. Imperviousness was determined as a simple lookup function of land
use based on values determined by the Maryland Department of Planning for their
generalized land use data. These values are provided in Table 13.2.
13.4.2 CALIBRATION STRATEGIES
Rainfall data obtained from the NCDC Web site (National Climatic Data Center, 2001)
mentioned in Section 13.2.1 were obtained for the Rockville, Maryland rain gage
(Coop ID# 187705). Annual precipitation values were determined around a three-
dimensional window centered on the date of the annual maximum flood. Because of
potential time/date mismatches between the occurrence of precipitation and peak
discharge, the observed precipitation associated with the peak discharge was taken
to be the maximum of the sum over a 2-day window either beginning on ending on
the day associated with the annual maximum discharge.

Even allowing for potential time/date mismatches, the observed precipitation is
quite small on four occasions (1958, 1964, 1969, and 1981). In fact, no precipitation
is observed to explain the annual maximum in 1969. A plausible explanation for
this is the relatively small scale of the study watershed. Convectively generated
rainfall is highly varied in space compared to frontal-system generated rainfall, and
the rain gage, although close, is not actually within the study watershed. Because
of the size of the watershed, the annual maximum discharge is likely to be associated
with convective summer events rather than the frontal precipitation more common
TABLE 13.2
Percent Imperviousness Associated
with Various Land Uses Present in
Watts Branch, Maryland
Land Use
Percent
Imperviousness
Low-density residential 25
Medium-density residential 30
High-density residential 65
Commercial 82
Industrial 70
Institutional 50
Open urban land 11
Cropland 0
Pasture 0
Deciduous forest 0
Mixed forest 0
Brush 0
Source: Weller, personal communication.
L1600_Frame_C13.fm Page 375 Friday, September 20, 2002 10:25 AM
© 2003 by CRC Press LLC

in the cooler months. The data support this hypothesis with 19 of 27 annual maxima
observed in the months of June through September.
The hydrologic engineer has a number of parameters available to calibrate the
above model to the observed annual maximum series. Both curve numbers and t
c
values are frequent candidates for calibration in a typical analysis. Given the time
series implications of this analysis, it did not make sense to adjust either of these
quantities since an adjustment of a quantity might be made “up” in one year and
“down” in the next. The physical basis for such an adjustment pattern is unclear.
Instead, given the presumed inaccuracies in the observed precipitation record, pre-
cipitation was used as the only calibration parameter. The causal precipitation, P(t),
in Equation 13.2, was calibrated by setting the model outlined in Equations 13.1
through 13.7 into Equation 13.8, such that the observed and simulated annual
maximum discharges were the same within a small (0.1%) tolerance.
(13.8)
where Q
p,s
[t, P(t)] is the simulated peak discharge in year, t, and assuming a causal
precipitation depth, P(t). Q
p,o
(t) is the observed peak discharge in year, t. The
observed and causal precipitation were determined to have a correlation coefficient,
R, of 0.67. Table 13.3 and Figure 13.4 provide a summary and comparison of the
causal (simulated) and observed precipitation depths.
13.4.3 SIMULATING A STATIONARY ANNUAL
M
AXIMUM-DISCHARGE SERIES
With the causal precipitation time series determined, it is a straightforward process
to determine the annual maximum discharge that would have been observed had
land use remained constant over the period of record of the stream gage. In fact, the

only question facing the hydrologic engineer is what land use to employ in the
simulation. As a side-product of the calibration process, representations of land use
on an annual time step corresponding to each year from 1951 to 2000 are available.
Using any one of these years, t*, and the causal precipitation time series developed
in the calibration step, the annual maximum series that would be observed as if the
land use were fixed for that particular year, t* can be generated. In more mathematical
terms,
(13.9)
For illustrative purposes, the annual maxima corresponding to the following four
different land use conditions have been simulated: Q
p,1951
(t) (earliest land use),
Q
p,1987
(t) (last year of the gage record), Q
p,2000
(t) (“present” conditions), and Q
p,ult.
(t)
(projected ultimate condition of watershed given zoning data). These simulated peak
discharges are provided in Table 13.5 and shown in Figure 13.5. (For completeness,
| [ , ( )] ( ) |
()
.
,,
,
QtPt Qt
Qt
ps po
po

−
< 0 001
QtQtPtLUxt
pt
ps
,*
,
() [, (), ( ,*)]=
L1600_Frame_C13.fm Page 376 Friday, September 20, 2002 10:25 AM
© 2003 by CRC Press LLC
the observed [Q
o
(t)] and simulated runoff volumes, [Q
1951
(t), Q
1987
(t), Q
2000
(t), and
Q
ult.
(t)], are provided first in Table 13.4.)
With these new time series determined, it is important to clearly understand
what they represent. Each of the simulated time series represents the discharges that
would have been observed at the stream gage, had the land use been fixed at t*
conditions over the entirety of the gaging period. Thus, the 1951 time series, Q
p,1951
(t),
represents the peak discharges observed from 1958 to 1987 adjusted to the land use
TABLE 13.3

Observed and Causal Precipitation Associated with Annual Maximum
Discharge in Watts Branch, Maryland
Hydrologic
Year
Date of Peak
Discharge
P
obs
1 Day
before
Peak (in.)
P
obs
on
Day of
Peak (in.)
P
obs
1 Day
after Peak
(in.)
Maximum
Observed
2-Day P
(in.)
Causal
P (in.)
1958 07/08/1958 0.00 0.18
a
0.11 0.29 2.41

1959 09/02/1959 0.00 1.98 0.26 2.24 1.59
1960 07/13/1960 0.00 0.00 1.97 1.97 2.58
1961 04/13/1961 0.08 1.95 0.00 2.03 1.67
1962 03/12/1962 0.00 1.20 0.00 1.20 1.47
1963 06/05/1963 ————2.40
1964 01/09/1964 0.00 0.36 0.00 0.36 1.62
1965 08/26/1965 0.00 1.39 0.64 2.03 2.87
1966 09/14/1966 ————2.57
1967 08/04/1967 0.00 0.69 1.21 1.90 2.91
1968 09/10/1968 0.00 0.68 1.95 2.63 2.54
1969 06/03/1969 0.00 0.00 0.00 0.00 2.60
1970 08/14/1970 0.00 0.00 2.30 2.930 3.84
1971 08/01/1971 0.15 0.30 1.84 2.14 1.81
1972 06/21/1972 1.36 7.90 1.36 9.26 5.34
1973 06/04/1973 0.00 0.00 1.98 1.98 3.30
1974 09/28/1974 0.00 1.60 0.00 1.60 2.05
1975 09/26/1975 1.54 4.46 0.00 6.00 5.94
1976 12/31/1975 0.19 0.57 2.42 2.99 1.73
1977 03/13/1977 0.00 1.18 0.46 1.64 1.68
1978 07/31/1978 0.00 0.20 4.50 4.70 2.82
1979 08/27/1979 0.30 0.62 1.90 2.52 4.98
1980 10/01/1979 0.08 1.92 0.37 2.29 2.49
1981 08/12/1981 0.18 0.08 0.07 0.26 2.02
1982 09/02/1982 0.00 0.00 1.40 1.40 2.33
1983 05/22/1983 ————2.10
1984 08/01/1984 0.00 2.87 0.24 3.11 2.52
1985 07/15/1985 0.00 1.34 0.00 1.34 2.24
1986 05/21/1986 0.72 1.80 0.02 2.52 2.67
1987 09/08/1987 0.00 4.07 0.00 4.07 2.37
a

Numbers in italics represent a day contributing to the maximum 2-day precipitation.
L1600_Frame_C13.fm Page 377 Friday, September 20, 2002 10:25 AM
© 2003 by CRC Press LLC
FIGURE 13.4 Comparison of observed and causal (simulated) precipitation depths associated
with the annual maximum discharges in Watts Branch. The line shown corresponds to perfect
agreement.
FIGURE 13.5 Time series plot showing observed and adjusted annual maximum discharges.
1955 1960 1965 1970 1975 1980 1985 1990
10
2
10
3
10
4
observed
1951
1987
2000
ultimate
Discharge, Q (ft /S)
3
Year
L1600_Frame_C13.fm Page 378 Friday, September 20, 2002 10:25 AM
© 2003 by CRC Press LLC
conditions present in 1951. Note that the observed discharges lie between those
associated with the 1951 and 1987 land use conditions. This is to be expected since
the gaging period is from 1958 to 1987. It is also of interest to note that the 1987
discharge in the Q
p,1987
(t) time series is identical to that in the observed record. In

other words, under the method described here, adjustment is not made to this
discharge when determining the peak discharge under 1987 land use conditions. If
gaging were to have taken place in 1951 or to have continued in 2000, we would
expect the discharges in both of these years under these land use conditions to also
be identical to the observed record for the particular year in question. Each of these
time series is valuable, because each obeys a major assumption of flood-frequency
analysis that is violated by the actual, observed annual maximum series. These fixed
TABLE 13.4
Observed and Simulated Runoff Depths Associated with Annual
Maximum Discharge Events in Watts Branch, Maryland
Hydrologic Year Q
o
(in.) Q
1951
(in.) Q
1987
(in.) Q
2000
(in.) Q
ult.
(in.)
1958 0.76 0.70 0.81 0.82 1.02
1959 0.29 0.26 0.32 0.32 0.45
1960 0.88 0.81 0.93 0.93 1.15
1961 0.34 0.29 0.36 0.37 0.50
1962 0.25 0.21 0.27 0.27 0.38
1963 0.78 0.70 0.81 0.81 1.01
1964 0.32 0.27 0.34 0.34 0.47
1965 1.11 1.00 1.14 1.14 1.38
1966 0.90 0.80 0.93 0.93 1.14

1967 1.13 1.02 1.16 1.17 1.41
1968 0.88 0.78 0.90 0.91 1.12
1969 0.92 0.82 0.95 0.95 1.17
1970 1.85 1.72 1.89 1.90 2.19
1971 0.42 0.36 0.44 0.45 0.59
1972 3.11 2.94 3.17 3.17 3.53
1973 1.42 1.31 1.46 1.47 1.73
1974 0.56 0.49 0.58 0.58 0.76
1975 3.64 3.46 3.70 3.71 4.09
1976 0.37 0.32 0.40 0.40 0.54
1977 0.35 0.30 0.37 0.37 0.51
1978 1.07 0.97 1.10 1.11 1.34
1979 2.79 2.64 2.85 2.85 3.21
1980 0.86 0.75 0.87 0.87 1.08
1981 0.56 0.47 0.56 0.57 0.73
1982 0.75 0.65 0.76 0.76 0.96
1983 0.60 0.52 0.61 0.61 0.79
1984 0.88 0.77 0.89 0.89 1.10
1985 0.69 0.60 0.70 0.71 0.89
1986 0.98 0.86 0.99 0.99 1.22
1987 0.79 0.68 0.79 0.79 0.99
L1600_Frame_C13.fm Page 379 Friday, September 20, 2002 10:25 AM
© 2003 by CRC Press LLC
land-use time series are stationary; the trend of increasing discharges due to urban-
ization has been removed.
A few words are merited in the description of the ultimate land use condition.
Zoning data were obtained from the Maryland Department of Planning spanning
the area associated with the study watershed. As of 2000, the watershed is already
highly urbanized; however, zoning for this watershed indicates that the remaining
18% of agricultural land and a significant proportion of the forest cover would be

converted to commercial and/or residential land uses. Forest cover within 100 feet
of existing streams is assumed to remain due to efforts to retain a buffer zone adjacent
TABLE 13.5
Observed and Simulated Annual Maximum Discharges in
Watts Branch, Maryland
Hydrologic Year
Q
p,o

(ft
3
/sec)
Q
p,1951

(ft
3
/sec)
Q
p, 1987

(ft
3
/sec)
Q
p, 2000

(ft
3
/sec)

Q
p,ult.

(ft
3
/sec)
1958 592 486 710 735 1149
1959 194 147 248 257 467
1960 710 570 822 851 1309
1961 242 175 289 299 526
1962 164 112 194 202 388
1963 630 484 706 731 1143
1964 227 156 260 270 485
1965 932 726 1031 1067 1602
1966 740 568 820 849 1306
1967 954 743 1053 1090 1633
1968 730 551 798 826 1274
1969 770 583 840 869 1334
1970 1660 1306 1791 1852 2638
1971 325 229 361 374 635
1972 2900 2334 3093 3195 4259
1973 1250 971 1355 1401 2048
1974 443 326 490 508 829
1975 3400 2776 3619 3738 4934
1976 283 197 319 330 570
1977 259 179 294 305 533
1978 916 699 995 1029 1551
1979 2600 2075 2779 2872 3863
1980 739 525 762 789 1223
1981 458 314 473 490 804

1982 641 450 660 684 1077
1983 502 346 518 536 870
1984 763 541 783 811 1254
1985 589 409 605 626 997
1986 867 615 883 914 1395
1987 686 468 686 709 1114
L1600_Frame_C13.fm Page 380 Friday, September 20, 2002 10:25 AM
© 2003 by CRC Press LLC
to streams. Land use conditions under ultimate development are reflected in the
spatial distribution shown in Figure 13.1 and in the time series shown in Figure 13.3.
13.5 COMPARISON OF FLOOD-FREQUENCY ANALYSES
Hydrologic design is often based on peak discharges associated with various return
periods. In stream restoration, discharges that approximate bank-full flow conditions
have a return frequency of 1.5 to 2 years (Leopold, 1994). The design of bridges,
culverts, detention ponds, and outlet structures may depend on frequencies ranging
from 2 to 100 years depending on the structure and design specifications regulated
by the state. From an engineering perspective, the consequences of the analyses
presented in this chapter are on the differences in flood frequency between the various
time series we have determined.
13.5.1 IMPLICATIONS FOR HYDROLOGIC DESIGN
The USGS PeakFQ program (U.S. Geological Survey, 2001a) automates the deter-
mination of the flood frequency distribution for any annual maximum time series,
producing estimates consistent with Bulletin 17B (Interagency Advisory Committee
on Water Data, 1982) guidelines. The four time series presented in Table 13.5 were
analyzed with the results presented in Table 13.6 and Figure 13.6. The implications
for hydrologic design are addressed in this section.
The main, and most obvious, consequence of performing the discharge adjust-
ment procedure outlined here is the proliferation of peak discharge estimates as
shown in Tables 13.5 and 13.6 and Figures 13.5 and 13.6. Through comparison with
TABLE 13.6

Flood Frequency Distributions for Observed and Simulated Annual
Maximum Discharges in Watts Branch, Maryland
Return Period
(years)
Observed
Time Series
(ft
3
/sec)
1951 Land
Use (ft
3
/sec)
1987 Land
Use (ft
3
/sec)
2000 Land
Use (ft
3
/sec)
Ultimate
Land Use
(ft
3
/sec)
2 628 461 (−166)
a
678 (50) 702 (74) 1104 (476)
5 1215 929 (−286) 1292 (77) 1336 (121) 1952 (737)

10 1753 1373 (−380) 1864 (111) 1927 (174) 2708 (955)
25 2635 2123 (−512) 2818 (183) 2912 (277) 3929 (1294)
50 3459 2844 (−615) 3728 (269) 3852 (393) 5063 (1604)
100 4447 3727 (−720) 4836 (389) 4996 (549) 6416 (1969)
200 5625 4802 (−823) 6181 (556) 6385 (760) 8028 (2403)
500 7528 6580 (−948) 8400 (872) 8676 (1148) 10,640 (3112)
a
Numbers in parentheses represent the difference between simulated discharges and observed dis-
charges for the same return period.
L1600_Frame_C13.fm Page 381 Friday, September 20, 2002 10:25 AM
© 2003 by CRC Press LLC
the observed frequency distribution, the consequences of the discharge adjustment
method become clear. Consider, for example, developing a hydrologic design for a
bridge located at the stream gage based on a 50-year return frequency. In the absence
of an adjustment method, the design discharge would be 3459 ft
3
/sec. Given that
the gaging ended relatively recently in 1987, one might be tempted to simply use
the discharges adjusted to 1987 land use conditions. In this case, the design discharge
would increase by 269 ft
3
/sec to 3728 ft
3
/sec. If adjustments are made for urbaniza-
tion that has taken place since the end of the gage record, the design discharge would
increase to 3852 ft
3
/sec. In Maryland, where ultimate development determines the
design discharge for new bridges, the 50-year peak discharge based on adjustment
of the gage record to ultimate development land-use conditions is 5063 ft

3
/sec. This
is 1604 ft
3
/sec greater than the unadjusted discharge from the gage record. The
consequences on engineering design are clear: failure to adjust the observed dis-
charge record in the face of known urbanization can lead to systematic underdesign
of the structure. Depending on the structure, this underdesign might manifest itself
in increased flooding, a discharge structure failing to pass the required flow and
being overtopped, or in scour of a magnitude much greater than anticipated. In any
case, underdesign will lead to a greater likelihood of failure of the engineered
structure.
It is important to note that the adjustment procedure outlined here is of value,
even if land use is relatively unchanged over the period of record of the stream gage.
This is because there may still have been significant development since the gage
record was collected or ultimate development conditions may represent a large
degree of urbanization that was not present at the time of gaging. In both of these
cases, adjusted discharges would be greater, and potentially much greater, than those
associated with the observed flood-frequency distribution.
FIGURE 13.6 Observed and adjusted flood-frequency distributions.
2
5102550
100
200 500
Recurrence Interval (years)
1987
2000
Observed
1951
ultimate

10
3
10
4
Discharge, Q (ft /S)
3
L1600_Frame_C13.fm Page 382 Friday, September 20, 2002 10:25 AM
© 2003 by CRC Press LLC
13.5.2 ASSUMPTIONS AND LIMITATIONS
Although the method outlined here is a valuable approach for adjusting nonstationary
annual maximum discharges, it is not without its assumptions and limitations, which
should be enumerated explicitly. The method, as presented, lumps all uncertainty in
the modeling process into a time series of precipitation estimates that reproduce the
observed peak discharge using the TR-55 method. This precipitation time series has
been referred to throughout this chapter as the “causal precipitation time series.”
Actual differences in the hydrologic response may be due to any number of other
assumptions in the TR-55 model as employed here. For instance, the curve number
table that was used may not be appropriate, the antecedent conditions (assumed
normal) may be dry or wet, the impervious surfaces (assumed to be directly con-
nected to the drainage system) may instead be disconnected, subareas within the
watershed may need to be considered separately, and detention storage has been
ignored altogether. Design criteria instituted since the end of the gage record may
mean that the ultimate land use conditions include a significant storage component;
thus, peak discharges associated with ultimate development as portrayed here are
much greater than would be expected in the face of stormwater management prac-
tices. If a larger watershed were considered, TR-55 would need to be replaced with
the more sophisticated TR-20 or other rainfall-runoff model that both generates
runoff and routes this runoff through a channel network.
Putting the hydrologic model aside for a moment, there are also assumptions
and limitations associated with the land-use change model presented here. As

depicted in Figures 13.1 through 13.3, the land use is only allowed to transition from
one initial land use at time t = t
1
to another final land use at time t = t
2
. A more
robust model that would be relatively easy to incorporate would allow the landscape
to transition from/to multiple land use conditions at discrete points in time. This
model would then require n different snapshots in time and n – 1 spatially referenced
data sets that convey when the conversion of each bit of landscape from condition
at time t = t
i
, to condition at time t = t
i+1
, occurs. In the land use model presented
earlier, there were only n = 2 snapshots in time and the subscript, i, took on only
the value i = 1. The most likely limitation that the hydrologic modeler will encounter
in this context is a distinct lack of land use “snapshots” at fixed points in time, and
of spatially referenced data sets to convey the land use change history.
13.6 SUMMARY
This chapter has illustrated a GIS-based method to model land use change over time
on an annual time step and at the parcel level of spatial detail. These land-use change
data were used to drive a simple hydrologic model that is sensitive to land use
change. The hydrologic model was calibrated to reproduce the observed discharges
of the period of record of an actual gaged case-study watershed in Maryland. Once
calibrated, the hydrologic model was then used to determine the annual maximum
discharges that would have resulted in the case study watershed for fixed (rather
than time-varying) land use corresponding to past, present, and future land use
L1600_Frame_C13.fm Page 383 Friday, September 20, 2002 10:25 AM
© 2003 by CRC Press LLC

conditions. Using flood-frequency analysis, the fixed land-use, annual-maximum
flood-frequency distributions were compared to the flood-frequency distribution for
the actual observed distribution. The implications of not adjusting an annual maxi-
mum time series derived from a gaged watershed under urbanizing (nonstationary)
land use conditions are that design discharges associated with any return frequency
are likely to be underestimated and lead to the underdesign of a structure and a
greater likelihood for that structure to ultimately fail.
13.7 PROBLEMS
13-1 You are asked to develop the hydrologic analyses necessary to build a
bridge crossing at the location of the Watts Branch stream gage. The
bridge piers are to be sized based on the 25-year peak flow event. Express
the 1951, 1987, 2000, and ultimate development peaks for this return
period as a percentage of the observed 25-year peak. Comment on the
magnitudes of these percentages.
13-2 Assuming a fixed rectangular cross-sectional channel geometry that is 20
feet wide, with a slope of 0.005 ft./ft., and a Manning’s roughness of 0.03,
(a) Determine the normal depth associated with the observed, 1951, 1987,
2000, and ultimate development 100-year discharges.
(b) If the bridge was constructed in 1987 with the deck placed to just pass
the 100-year discharge at that time, by how much will it be overtopped
by the 100-year event in 2000? At ultimate development conditions?
13-3 Download the PeakFQ program from the Web: />ware/peakfq.html (U.S. Geological Survey, 2001a). Use the PeakFQ pro-
gram and the values in Table 13.5 to verify the Q
p,o
and Q
p,2000
flood-
frequency analysis values provided in Table 13.6. You will need to indicate
“STA” for the skew computation option to indicate that you are using the
station skew option rather than the default weighted skew option.

13-4 Let the runoff ratio time series, R
R
(t), be defined as:
where Q(t) is the runoff depth (in inches) defined in Equation 13.2 and
provided in Table 13.4 and P(t) is the causal precipitation depth (in inches)
also described as part of Equation 13.2 and provided in Table 13.3. Plot
R
R
(t) for the Q
p,o
(t) and Q
p,2000
(t) time series. Perform a regression on these
ratios versus time. Report the slopes that you determine from these regres-
sions in both cases. Are these slopes consistent with what you would
expect from the information contained in this chapter? Discuss.
13-5 You are asked to generalize the flood-frequency adjustment method pre-
sented in this chapter to apply to a 40-mi
2
watershed that includes several
detention basins and 15 miles of channels. Describe how you would go
Rt
Qt
Pt
R
()
()
()
=
L1600_Frame_C13.fm Page 384 Friday, September 20, 2002 10:25 AM

© 2003 by CRC Press LLC
about doing this. What additional information beyond that used in the
case study presented here would be necessary?
13-6 You are given access to additional land use coverages corresponding to
1964 and 1975 conditions. Describe how the land-use change model
presented in this chapter would be modified to accommodate such data.
Are there any inconsistencies that might arise? Would additional data be
necessary?
13-7 You are given access to an additional land use coverage corresponding to
1937 conditions. Describe how the land-use change model presented in
this chapter would be modified to accommodate such data. Are there any
inconsistencies that might arise? Would additional data be necessary?
13-8 This chapter makes the claim that the relatively small scale of the study
watershed makes it more likely to have convective summer thunderstorms
drive the annual-maximum peak flow series. Outline a small study that
would use only the USGS annual-maximum flow database to determine
the scale (in mi
2
) at which the annual maximum shifts from summer
thunderstorms to cooler-season frontal events.
L1600_Frame_C13.fm Page 385 Friday, September 20, 2002 10:25 AM
© 2003 by CRC Press LLC

Appendix A
Statistical Tables

TABLE A.1
Standard Normal Distribution

z 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

−

3.4 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0002

−

3.3 0.0005 0.0005 0.0005 0.0004 0.0004 0.0004 0.0004 0.0004 0.0004 0.0003

−

3.2 0.0007 0.0007 0.0006 0.0006 0.0006 0.0006 0.0006 0.0005 0.0005 0.0005

−

3.1 0.0010 0.0009 0.0009 0.0009 0.0008 0.0008 0.0008 0.0008 0.0007 0.0007

−

3.0 0.0013 0.0013 0.0013 0.0012 0.0012 0.0011 0.0011 0.0011 0.0010 0.0010

−

2.9 0.0019 0.0018 0.0018 0.0017 0.0016 0.0016 0.0015 0.0015 0.0014 0.0014

−

2.8 0.0026 0.0025 0.0024 0.0023 0.0023 0.0022 0.0021 0.0021 0.0020 0.0019

−

2.7 0.0035 0.0034 0.0033 0.0032 0.0031 0.0030 0.0029 0.0028 0.0027 0.0026

−

2.6 0.0047 0.0045 0.0044 0.0043 0.0041 0.0040 0.0039 0.0038 0.0037 0.0036

−

2.5 0.0062 0.0060 0.0059 0.0057 0.0055 0.0054 0.0052 0.0051 0.0049 0.0048

−

2.4 0.0082 0.0080 0.0078 0.0075 0.0073 0.0071 0.0069 0.0068 0.0066 0.0064

−

2.3 0.0107 0.0104 0.0102 0.0099 0.0096 0.0094 :0091 0.0089 0.0087 0.0084

−

2.2 0.0139 0.0136 0.0132 0.0129 0.0125 0.0122 0.0119 0.0116 0.0113 0.0110

−

2.1 0.0179 0.0174 0.0170 0.0166 0.0162 0.0158 0.0154 0.0150 0.0146 0.0143

−

2.0 0.0228 0.0222 0.0217 0.0212 0.0207 0.0202 0.0197 0.0192 0.0188 0.0183

−

1.9 0.0287 0.0281 0.0274 0.0268 0.0262 0.0256 0.0250 0.0244 0.0239 0.0233

−

1.8 0.0359 0.0351 0.0344 0.0336 0.0329 0.0322 0.0314 0.0307 0.0301 0.0294

−

1.7 0.0446 0.0436 0.0427 0.0418 0.0409 0.0401 0.0392 0.0384 0.0375 0.0367

−

1.6 0.0548 0.0537 0.0526 0.0516 0.0505 0.0495 0.0485 0.0475 0.0465 0.0455

−

1.5 0.0668 0.0655 0.0643 0.0630 0.0618 0.0606 0.0594 0.0582 0.0511 0.0559

−

1.4 0.0800.8 0.0793 0.0778 0.0764 0.0749 0.0735 0.0721 0.0708 0.0694 0.0681

−

1.3 0.0968 0.0951 0.0934 0.0918 0.0901 0.0885 0.0869 0.0853 0.0838 0.0823

−

1.2 0.1151 0.1131 0.1112 0.1093 0.1075 0.1056 0.1038 0.1020 0.1003 0.0985

−

1.1 0.1357 0.1335 0.1314 0.1292 0.1271 0.1251 0.1230 0.1210 0.1190 0.1170

−

1.0 0.1587 0.1562 0.1539 0.1515 0.1492 0.1469 0.1446 0.1423 0.1401 0.1379

−

0.9 0.1841 0.1814 0.1788 0.1762 0.1736 0.1711 0.1685 0.1660 0.1635 0.1611

−

0.8 0.2119 0.2090 0.2061 0.2033 0.2005 0.1977 0.1949 0.1922 0.1894 0.1867

−

0.7 0.2420 0.2389 0.2358 0.2327 0.2296 0.2266 0.2236 0.2206 0.2177 0.2148

−

0.6 0.2743 0.2709 0.2676 0.2643 0.2611 0.2578 0.2546 0.2514 0.2483 0.2451

−

0.5 0.3085 0.3050 0.3015 0.2981 0.2946 0.2912 0.2877 0.2843 0.2810 0.2776

(

Continued

)

L1600_Frame_App-A.fm2nd Page 387 Friday, September 20, 2002 10:29 AM
© 2003 by CRC Press LLC

TABLE A.1
Standard Normal Distribution (

Continued

)

z 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

−

0.4 0.3446 0.3409 0.3312 0.3336 0.3300 0.3264 0.3228 0.3192 0.3156 0.3121

−

0.3 0.3821 0.3783 0.3745 0.3707 0.3669 0.3632 0.3594 0.3557 0.3520 0.3483

−

0.2 0.4201 0.4168 0.4129 0.4090 0.4052 0.4013 0.3974 0.3936 0.3897 0.3859

−

0.1 0.4602 0.4562 0.4522 0.4483 0.4443 0.4404 0.4364 0.4325 0.4286 0.4247
0.0 0.5000 0.4960 0.4920 0.4880 0.4840 0.4801 0.4761 0.4721 0.4681 0.4641
0.0 0.5000 0.5040 0.5080 0.5120 0.5160 0.5199 0.5239 0.5279 0.5319 0.5359
0.1 0.5398 0.5438 0.5478 0.5517 0.5557 0.5596 0.5636 0.5675 0.5714 0.5753
0.2 0.5193 0.5832 0.5871 0.5910 0.5948 0.5981 0.6026 0.6064 0.6103 0.6141
0.3 0.6179 0.6217 0.6255 0.6293 0.6331 0.6368 0.6406 0.6443 0.6480 0.6517
0.4 0.6554 0.6591 0.6628 0.6664 0.6700 0.6736 0.6772 0.6808 0.6844 0.6879
0.5 0.6915 0.6950 0.6985 0.7019 0.7054 0.7088 0.7123 0.7157 0.7190 0.7224
0.6 0.7257 0.7291 0.7324 0.7357 0.1389 0.7422 0.7454 0.7486 0.7517 0.7549
0.7 0.7580 0.7611 0.7642 0.7673 0.7704 0.7734 0.7764 0.7794 0.780.23 0.7852
0.8 0.7881 0.7910 0.7939 0.7967 0.7995 0.8023 0.8051 0.8078 0.8106 0.8133
0.9 0.8159 0.8186 0.8212 0.8238 0.8264 0.8289 0.8315 0.8340 0.8365 0.8389
1.0 0.8413 0.8438 0.8461 0.8485 0.8508 0.8531 0.8554 0.8577 0.8599 0.8621
1.1 0.8643 0.8665 0.8686 0.8708 0.8729 0.8749 0.8770 0.8790 0.8810 0.8830
1.2 0.8849 0.8869 0.8888 0.8907 0.8925 0.8944 0.8962 0.8980 0.8997 0.9015
1.3 0.9032 0.9049 0.9066 0.9082 0.9099 0.9115 0.9131 0.9147 0.9162 0.9177
1.4 0.9192 0.9207 0.9222 0.9236 0.9251 0.9265 0.9279 0.9292 0.9306 0.9319
1.5 0.9332 0.9345 0.9357 0.9370 0.9382 0.9394 0.9406 0.9418 0.9429 0.9441
1.6 0.9452 0.9463 0.9474 0.9484 0.9495 0.9505 0.9515 0.9525 0.9535 0.9545
1.7 0.9554 0.9564 0.9573 0.9582 0.9591 0.9599 0.9608 0.9616 0.9625 0.9633
1.8 0.9641 0.9649 0.9656 0.9664 0.9671 0.9678 0.9686 0.9693 0.9699 0.9706
1.9 0.9713 0.9719 0.9726 0.9732 0.9738 0.9744 0.9750 0.9756 0.9761 0.9767
2.0 0.9772 0.9778 0.9783 0.9788 0.9793 0.9198 0.9803 0.9808 0.9812 0.9817
2.1 0.9821 0.9826 0.9830 0.9834 0.9838 0.9842 0.9846 0.9850 0.9854 0.9857
2.2 0.9861 0.9864 0.9868 0.9871 0.9875 0.9878 0.9881 0.9884 0.9887 0.9890
2.3 0.9893 0.9896 0.9898 0.9901 0.9904 0.9906 0.9909 0.9911 0.9913 0.9916
2.4 0.9918 0.9920 0.9922 0.9925 0.9927 0.9929 0.9931 0.9932 0.9934 0.9936
2.5 0.9938 0.9940 0.9941 0.9943 0.9945 0.9946 0.9948 0.9949 0.9951 0.9952

2.6 0.9953 0.9955 0.9956 0.9957 0.9959 0.9960 0.9961 0.9962 0.9963 0.9964
2.7 0.9965 0.9966 0.9967 0.9968 0.9969 0.9970 0.9971 0.9972 0.9973 0.9974
2.8 0.9974 0.9975 0.9976 0.9977 0.9977 0.9978 0.9979 0.9979 0.9980 0.9981
2.9 0.9981 0.9982 0.9982 0.9983 0.9984 0.9984 0.9985 0.9985 0.9986 0.9986
3.0 0.9987 0.9987 0.9987 0.9988 0.9988 0.9989 0.9989 0.9989 0.9990 0.9990
3.1 0.9990 0.9991 0.9991 0.9991 0.9992 0.9992 0.9992 0.9992 0.9993 0.9993
3.2 0.9993 0.9993 0.9994 0.9994 0.9994 0.9994 0.9994 0.9995 0.9995 0.9995
3.3 0.9995 0.9995 0.9995 0.9996 0.9996 0.9996 0.9996 0.9996 0.9996 0.9997
3.4 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9998

L1600_Frame_App-A.fm2nd Page 388 Friday, September 20, 2002 10:29 AM
© 2003 by CRC Press LLC

TABLE A.2
Critical Values of the

t

Statistic

Level of Significance, One-Tailed
0.25 0.20 0.10 0.05 0.025 0.0125 0.005 0.0025 0.0005
Level of Significance, Two-Tailed
df 0.50 0.40 0.20 0.10 0.05 0.025 0.01 0.005 0.001

1 1.000 1.376 3.078 6.314 12.706 25.452 63.657 318.300 636.600
2 0.816 1.061 1.886 2.920 4.303 6.205 9.925 14.089 31.598
3 0.765 0.978 1.638 2.353 3.182 4.176 5.841 7.453 12.941
4 0.741 0.941 1.533 2.132 2.776 3.495 4.604 5.598 8.610
5 0.727 0.920 1.476 2.015 2.571 3.163 4.032 4.773 6.859

6 0.718 0.906 1.440 1.943 2.447 2.969 3.707 4.317 5.959
7 0.711 0.896 1.415 1.895 2.365 2.841 3.499 4.029 5.405
8 0.706 0.889 1.397 1.860 2.306 2.752 3.355 3.832 5.041
9 0.703 0.883 1.383 1.833 2.262 2.685 3.250 3.690 4.781
10 0.700 0.879 1.372 1.812 2.228 2.634 3.169 3.581 4.587
11 0.697 0.876 1.363 1.796 2.201 2.593 3.106 3.497 4.437
12 0.695 0.873 1.356 1.782 2.179 2.560 3.055 3.428 4.318
13 0.694 0.870 1.350 1.771 2.160 2.533 3.012 3.372 4.221
14 0.692 0.868 1.345 1.761 2.145 2.510 2.977 3.326 4.140
15 0.691 0.866 1.341 1.753 2.131 2.490 2.947 3.286 4.073
16 0.690 0.865 1.337 1.746 2.120 2.473 2.921 3.252 4.015
17 0.689 0.863 1.333 1.740 2.110 2.458 2.898 3.222 3.965
18 0.688 0.862 1.330 1.734 2.101 2.445 2.878 3.197 3.922
19 0.688 0.861 1.328 1.729 2.093 2.433 2.861 3.174 3.883
20 0.687 0.860 1.325 1.725 2.086 2.423 2.845 3.153 3.850
21 0.686 0.859 1.323 1.721 2.080 2.414 2.831 3.135 3.819
22 0.686 0.858 1.321 1.717 2.074 2.406 2.819 3.119 3.792
23 0.685 0.858 1.319 1.714 2.069 2.398 2.807 3.104 3.761
24 0.685 0.857 1.318 1.711 2.064 2.391 2.797 3.090 3.745
25 0.684 0.856 1.316 1.708 2.060 2.385 2.787 3.078 3.725
26 0.684 0.856 1.315 1.706 2.056 2.379 2.779 3.067 3.707
27 0.684 0.855 1.314 1.703 2.052 2.373 2.771 3.056 3.690
28 0.683 0.855 1.313 1.701 2.048 2.368 2.763 3.047 3.674
29 0.683 0.854 1.311 1.699 2.045 2.364 2.756 3.038 3.659
30 0.683 0.854 1.310 1.697 2.042 2.360 2.750 3.030 3.646
35 0.682 0.852 1.306 1.690 2.030 2.342 2.724 2.996 3.591
40 0.681 0.851 1.303 1.684 2.021 2.329 2.704 2.971 3.551
45 0.680 0.850 1.301 1.680 2.014 2.319 2.690 2.952 3.520
50 0.680 0.849 1.299 1.676 2.008 2.310 2.678 2.937 3.496
(

Continued

)

L1600_Frame_App-A.fm2nd Page 389 Friday, September 20, 2002 10:29 AM
© 2003 by CRC Press LLC

TABLE A.2
Critical Values of the

t

Statistic (

Continued

)

Level of Significance, One-Tailed
0.25 0.20 0.10 0.05 0.025 0.0125 0.005 0.0025 0.0005
Level of Significance, Two-Tailed
df 0.50 0.40 0.20 0.10 0.05 0.025 0.01 0.005 0.001

55 0.679 0.849 1.297 1.673 2.004 2.304 2.669 2.925 3.476
60 0.679 0.848 1.296 1.671 2.000 2.299 2.660 2.915 3.460
70 0.678 0.847 1.294 1.667 1.994 2.290 2.648 2.899 3.435
80 0.678 0.847 1.293 1.665 1.989 2.284 2.638 2.887 3.416
90 0.678 0.846 1.291 1.662 1.986 2.279 2.631 2.878 3.402
100 0.677 0.846 1.290 1.661 1.982 2.276 2.625 2.871 3.390

120 0.671 0.845 1.289 1.659 1.981 2.273 2.621 2.865 3.381
inf 0.674 0.842 1.282 1.645 1.960 2.241 2.576 2.807 3.290

L1600_Frame_App-A.fm2nd Page 390 Friday, September 20, 2002 10:29 AM
© 2003 by CRC Press LLC

TABLE A.3
Cumulative Distribution of Chi Square

df

Probability of Greater Value
0.995 0.990 0.975 0.950 0.900 0.750 0.500 0.250 0.100 0.050 0.025 0.010 0.005

1 0.00 0.00 0.00 0.00 0.02 0.10 0.45 1.32 2.71 3.84 5.02 6.63 7.88
2 0.01 0.02 0.05 0.10 0.21 0.58 1.39 2.77 4.61 5.99 7.38 9.21 10.60
3 0.07 0.11 0.22 0.35 0.58 1.21 2.37 4.11 6.25 7.81 9.35 11.34 12.84
4 0.21 0.30 0.48 0.71 1.06 1.92 3.36 5.39 7.78 9.49 11.14 13.28 14.86
5 0.41 0.55 0.83 1.15 1.61 2.67 4.35 6.63 9.24 11.07 12.83 15.09 16.75
6 0.68 0.87 1.24 1.64 2.20 3.45 5.35 7.84 10.64 12.59 14.45 16.81 18.55
7 0.99 1.24 1.69 2.17 2.83 4.25 6.35 9.04 12.02 14.07 16.01 18.48 20.28
8 1.34 1.65 2.18 2.73 3.49 5.01 7.34 10.22 13.36 15.51 17.53 20.09 21.96
9 1.73 2.09 2.70 3.33 4.17 5.90 8.34 11.39 14.68 16.92 19.02 21.67 23.59
10 2.16 2.56 :3.25 3.94 4.87 6.74 9.34 12.55 15.99 18.31 20.48 23.21 25.19
11 2.60 3.05 3.82 4.57 5.58 7.58 10.34 13.70 17.28 19.68 21.92 24.72 26.76
12 3.07 3.57 4.40 5.23 6.30 8.44 11.34 14.85 18.55 21.03 23.34 26.22 28.30
13 3.57 4.11 5.01 5.89 7.04 9.30 12.34 15.98 19.81 22.36 24.74 27.69 29.82
14 4.07 4.66 5.63 6.57 7.79 10.17 13.34 17.12 21.06 23.68 26.12 29.14 31.32
15 4.60 5.25 6.27 7.26 8.55 11.04 14.34 18.25 22.31 25.00 27.49 30.58 32.80
16 5.14 5.81 6.91 7.96 9.31 11.91 15.34 19.37 23.54 26.30 28.85 32.00 34.27

17 5.70 6.41 7.56 8.67 10.09 12.79 16.34 20.49 24.7? 27.59 30.19 33.41 35.72
18 6.26 7.01 8.23 9.39 10.86 13.68 17.34 21.60 25.99 28.87 31.53 34.81 37.16
19 6.84 7.63 8.91 10.12 11.65 14.56 18.34 22.72 27.20 30.14 32.85 36.19 38.58
20 7.34 8.26 9.59 10.85 12.44 15.45 19.34 22.83 28.41 31.41 34.17 37.57 40.00
(

Continued

)

L1600_Frame_App-A.fm2nd Page 391 Friday, September 20, 2002 10:29 AM
© 2003 by CRC Press LLC

TABLE A.3
Cumulative Distribution of Chi Square

df

Probability of Greater Value
0.995 0.990 0.975 0.950 0.900 0.750 0.500 0.250 0.100 0.050 0.025 0.010 0.005

21 8.03 8.90 10.28 11.59 13.24 16.34 20.34 24.93 29.62 32.67 35.48 38.93 41.40
22 8.64 9.54 10.98 12.34 14.04 17.24 21.34 26.04 30.81 33.92 36.78 40.29 42.80
23 9.26 10.20 11.69 13.09 14.85 18.14 22.34 27.14 32.01 35.17 38.08 41.64 44.18
24 9.89 10.86 12.40 13.85 15.66 19.04 23.34 28.24 33.20 36.42 39.36 42.98 45.56
25 10.52 11.52 13.12 14.61 16.47 19.94 24.34 29.34 34.38 37.65 40.65 44.31 46.93
26 11.16 12.20 13.84 15.38 17.29 20.84 25.34 30.43 35.56 38.89 41.92 45.64 48.29
27 11.81 12.88 14.57 16.15 18.11 21.57 26.34 31.53 36.74 40.11 43.19 46.96 49.64
28 12.46 13.56 15.31 16.93 18.94 22.66 27.34 32.62 37.92 41.34 44.46 48.28 50.99
29 13.12 14.26 16.05 17.71 19.77 23.57 28.34 33.71 39.09 42.56 45.72 49.59 52.34

30 13.19 14.95 16.79 18.49 20.60 24.48 29.34 34.80 40.26 43.77 46.98 50.89 53.67
40 20.71 22.16 24.43 26.51 29.05 33.66 39.34 45.62 51.80 55.76 59.34 63.69 66.77
50 27.99 29.71 32.36 34.76 37.69 42.94 49.33 56.33 63.17 67.50 71.42 76.15 79.49
60 35.53 37.48 40.48 43.19 46.46 52.29 59.33 66.98 74.40 79.08 83.30 88.38 91.95
70 43.28 45.44 48.76 51.74 55.33 61.70 69.33 77.58 85.53 90.53 95.02 100.42 104.22
80 51.17 53.54 57.15 60.39 64.28 71.14 79.33 88.13 96.58 101.88 106.63 112.33 116.32
90 59.20 61.75 65.65 69.13 73.29 80.62 89.33 98.64 107.56 113.14 113.14 124.12 128.30
100 67.33 70.06 74.22 77.93 82.36 90.13 99.33 109.14 118.50 124.34 129.56 135.81 140.17

L1600_Frame_App-A.fm2nd Page 392 Friday, September 20, 2002 10:29 AM
© 2003 by CRC Press LLC