A study was conducted of 78 U.S. Geological Survey gaged streams that have been subjected to varying degrees of urbanization over the last three decades. Flood-frequency analysis coupled with nonlinear regression techniques were used to generate a set of equations for converting peak discharge estimates determined from rural regression equations to a set of peak discharge estimates that represent known urbanization. Specifically, urban regression equations for the 2-, 5-, 10-, 25-, 50-, 100-, and 500-year return periods were calibrated as a function of the corresponding rural peak discharge and the percentage of impervious area in a watershed. The results of this study indicate that two sets of equations, one set based on imperviousness and one set based on population density, performed well. Both sets of equations are dependent on rural peak discharges, a measure of development (average percentage of imperviousness or average population density), and a measure of homogeneity of development within a watershed. Average imperviousness was readily determined by using geographic information system methods and commonly available land-cover data. Similarly, average population density was easily determined from census data. Thus, a key advantage to the equations developed in this study is that they do not require field measurements of watershed characteristics as did the U.S. Geological Survey urban equations developed in an earlier investigation.

During this study, the U.S. Geological Survey PeakFQ program was used as an integral tool in the calibration of all equations. The scarcity of historical land-use data, however, made exclusive use of flow records necessary for the 30-year period from 1970 to 2000. Such relatively short-duration streamflow time series required a nonstandard treatment of the historical data function of the PeakFQ program in comparison to published guidelines. Thus, the approach used during this investigation does not fully comply with the guidelines set forth in U.S. Geological Survey Bulletin 17B, and modifications may be needed before it can be applied in practice.

Introduction

The U.S. Geological Survey (USGS) maintains and publishes a set of rural regression equations for use in estimating peak discharges of varying return periods for each State in the United States (U.S.; Jennings and others, 1994; Ries and Crouse, 2002). Sauer and others (1983) developed a method for transforming these rural discharge estimates to estimates for urban watersheds based on several watershed characteristics, most notably two different indices of urbanization—the basin development factor (BDF) and the percentage of impervious area. A study was conducted to reassess the method of Sauer and others (1983) by taking advantage of 20 additional years of streamflow data, using geographic information system (GIS) data and techniques, and considering new approaches in the development of a set of urban regression equations. As in the earlier study, the goal of this study was to develop flood-frequency equations for urbanized sites by applying urban adjustments to the T-year flood values obtained from the regional equations for rural ungaged sites. The T-year flood is the flood that, on average, is equaled or exceeded once in any given number (T) of years.

The USGS typically conducts flood-frequency analyses consistent with Bulletin 17B (Interagency Advisory Committee on Water Data, 1982). For the study, the USGS flood-frequency analysis program, PeakFQ, was incorporated into an optimization program to calibrate several sets of urban regression equations. Because of limited-duration streamflow time series used in the analysis, however, the approach used in this study for the historical period input to the PeakFQ program does not comply with Bulletin 17B guidelines.

Previous Investigations

The effects of land-use changes on stream discharge are well documented (for example Carter, 1961; James, 1965; Viessman, 1966; Leopold, 1968; Andersen, 1970). The effects of urbanization, in particular, typically are related to characteristic changes in watersheds. Urbanization generally is associated with increased areas of impervious surfaces, such as pavement and rooftops. These surfaces usually have drainage features that quickly convey water away from structures or road surfaces and into nearby streams. In addition, urbanization generally reduces temporary stormwater storage, such as depression storage and tree-leaf interception. These urbanization processes create large volumes of stormwater that generally travel swiftly to the surface-drainage network and result in high flood peaks, which are the focus of this study.

Sauer and others (1983) presented a regression-equation approach of using rural flood-frequency estimates, such as those resulting from the equations summarized by Jennings and others (1994), and scaling these estimates upwards based on several measures of urbanization. Such rural flood-frequency estimates are currently (2006) calculated using the USGS National Flood-Frequency Program (version 3; Ries and Crouse, 2002).

Although Sauer and others (1983) calibrated several different sets of adjustment equations, their best model included seven parameters as presented in the following form:

(1)

where UQ_T		is the T-year urban peak discharge, in cubic feet per second;
A		is the drainage area, in square miles;
SL		is the main channel slope, in foot per mile;
RI2		is the 2-hour, 2-year rainfall, in inches;
ST		is the basin storage in percent;
BFD		is the basin development factor (an urbanization index that quantifies channel improvements, channel linings, storm drains/sewers, and curb-and-gutter streets);
IA		is the impervious area, in percent; and
RQ_T		is the T-year rural discharge as predicted from the appropriate USGS rural-regression equation;
(Note: The difficulty in determining the BDF index based solely on remotely sensed data was a motivating factor for conducting this study.)

In some States, the USGS has developed localized urban equations that supersede the use of the national urban equations developed by Sauer and others (1983; shaded areas in fig. 1). The localized urban equations vary in sophistication, but all generally require some measure of urbanization, such as the percentage of impervious or residential area. The general equations for the States of New Jersey (Stankowski, 1974), Pennsylvania (Stuckey and Reed, 2000), and Wisconsin (Conger, 1986) are presented here to illustrate the available range of localized urban equations.

Figure 1. Locations of urban and rural streamgages used in current (2006) study of rural regression peak discharges. [Shaded states have urban prediction regression equations that apply either statewide or in selected areas.]

New Jersey:

(2)

where UQ_T		is the T-year urban peak discharge, in cubic feet per second;
A		is the drainage area, in square miles;
SL		is the main channel slope, in foot per mile;
ST		is the basin storage in percent;
IA		is the impervious area, in percent; and

Pennsylvania (Region A):

(3)

where UQ_T		is the T-year urban peak discharge, in cubic feet per second;
A		is the drainage area, in square miles;
F		is forest cover, in percent;
U		is urban development, in percent;
C		is the watershed, in percent, underlain by carbonate rock; and
CA		is the watershed, in percent, controlled by lakes, swamps, or reservoirs.

Wisconsin (Milwaukee County urban areas):

(4)

where UQ_T		is the T-year urban peak discharge, in cubic feet per second;
A		is the drainage area, in square miles; and
IA		is the impervious area, in percent.

Overview of Peak Discharge Adjustment Method

The central premise of the study is that observed peak discharges at USGS streamgages are a function of watershed characteristics that exist at the time of the observed flood. Specifically, urbanization, measured as impervious area, varies over time and needs to be properly accounted for when performing flood-frequency analysis on an observed annual maximum time series.

Assumptions of Peak Discharge Adjustment Method

Assumption 1: The USGS rural regression equations for a State or region represent the best estimate of rural discharges for that particular location.

Assumption 2: In a watershed undergoing urbanization, the observed flood peak in any given year (t) depends on the amount of impervious area in the watershed. Because urbanization is an ongoing process, it may be necessary to obtain annual time series of impervious areas in rapidly changing watersheds.

Assumption 3: At streamgages influenced by urbanization, the observed annual maximum time series can be adjusted downward to be consistent with the predicted flood frequency obtained by applying the USGS rural regression equations for the gage location.

Assumption 4: The equations used to adjust the observed annual maximum time series can be inverted for use in scaling a USGS rural flood-frequency regression equation upward to account for urbanization effects. (Note: The objective function of a regression equation can be minimized only in the form in which it was calibrated, and the inversion process may cause a loss in optimality of the objective function.)

Assumption 5: Because calibrated coefficients and exponents vary with each return period, separate optimizations are required for each return period considered. This assumption is consistent with the approach in Sauer and others (1983), which produced different coefficients and exponents for each return period.

Assumption 6: Stormwater-management methods either are not present or do not have a measurable effect on flooding caused by urbanization. Although stormwater-management methods can mitigate increased flooding caused by urbanization, much of the urbanization quantified in this report predates the implementation of stormwater-management methods. Therefore, this report does not attempt to quantify any effects from implementing stormwater-management methods.

Enumeration of Steps in Peak Discharge Adjustment Method

The general steps in the adjustment method steps are described here. Because this is a calibration exercise, the process outlined here is iterative and is dependent on the minimization of an objective function that quantifies the difference between the observed rural flood-frequency values and the flood-frequency results obtained by using the PeakFQ program and adjusted annual maximum discharge data for urban streams.

Step 1. Assume (a) a functional form for an adjustment equation and (b) initial trial values for the coefficients and exponents of this adjustment equation.

Step 2. Adjust the annual maximum time series for the selected gage (i) as a function of the observed annual maximum time series and the impervious area time series. This produces an adjusted annual maximum time series.

Step 3. Write the adjusted annual maximum time series in the USGS National Water Data Storage and Retrieval System (WATSTORE) format.

Step 4. Use the PeakFQ program to determine the flood-frequency distribution at the selected gage (i) for the adjusted time series.

Step 5. Scan the output of the PeakFQ program for the T-year flood and incrementally adjust the objective function by the squared difference between the PeakFQ T-year flood and the rural regression value for the same flood-frequency. (Note that these differences will be in the logarithmic space to give similar weight to both small and large watersheds.)

Step 6. Modify the trial coefficients and exponents assumed in Step 1.

Steps 2 through 6 are repeated for all gages in the database and then performed iteratively until the objective function described in Step 5 is minimized. The entire process is conducted separately for the 2-, 5-, 10-, 25-, 50-, 100-, and 500-year flood-frequencies. A detailed flow chart of the general process described here is shown in figure 2. (Note: Steps 2-6 are automated by a nonlinear optimization program modified from McCuen (1993). The objective function that is minimized is the sum of the squared errors between the PeakFQ output for the T-year flood and the equivalent T-year flood estimated by the rural regression equation.)

After an equation is calibrated for each return period, the calibrated urban equation is determined in the following step (7):

: Step 7. Substitute the calibrated coefficients from the urban-to-rural adjustment function (f-inverse) into the original rural-to-urban function (f).

From the user’s standpoint, there likely is only an interest in the actual use of the function f from Step 7. The user’s experience will be limited to the following two steps:

Using the National Flood-Frequency (NFF) program (Ries and Crouse, 2002) or direct application of the USGS rural regression equation, estimate the rural flood-frequency for the ungaged location; and

Use the rural-to-urban function (f), transform rural T-year floods (from the rural regression equations previously determined) into corresponding T-year urban floods.

A critical challenge is the determination of the appropriate annual maximum discharge adjustment equation. The approach used during this study was to explore several functional forms and select the form that produced the best goodness-of-fit statistics across all frequencies.

Figure 2. Flow chart for the process for calibrating adjustment equations.

Analytical Presentation of Adjustment Method

Pursuant to Bulletin 17B (Interagency Advisory Committee on Water Data, 1982), FF(x, Q_o, HP, SS) is defined as the flood-frequency operator. This operator returns the flood-frequency vector, (Q_ff for the 2-, 5-, 10-, 25-, 50-, 100-, and 500-year floods) for a time series of annual maximum floods (x), for an identified high-outlier discharge threshold (Q_o), for a historical return period associated with the largest flood exceeding Q_o (HP), and assuming the station skew option (SS). An individual member of the flood-frequency vector, say the T-year flood, is identified as Q_ff(T) or Q_T.

The observed annual maximum time series is defined as Q_obs(t). The adjusted annual maximum time series is defined as Q_adj(t). Performing flood-frequency analyses on these two time series produced the flood-frequency vectors, Q_ff,obs and Q_ff,adj, respectively. Thus, the resulting equations take the following forms:

(5)

and

(6)

where HP_obs and HP_adj are the return periods associated with the largest floods in the respective annual maximum time series exceeding Q_o. For simplification, the dependency of the flood-frequency operator on a high-outlier threshold, a specified historical period, and a skew option was omitted from the notation. Thus, equations 5 and 6, respectively, reduce to simpler forms:

(7)

and

(8)

Two other flood-frequency vectors were relevant to this study—Q_ff,rural and Q_ff,urban, which are the rural and urban flood-frequency vectors, respectively. Because these vectors are determined from regression equations, they exist only as flood-frequency values and have no time series directly associated with them.

Adjustment of the Observed Annual Maximum Time Series for Nonstationary Urbanization

A central premise in the study by Sauer and others (1983) was that the observed flood-frequency record for a selected streamgage was stationary; in othet words, the record was unaffected by changes in urbanization over the period of record analyzed. To compensate for this premise, Sauer and others (1983) applied the rule that a streamgage record was usable if the amount of impervious area in the watershed had not increased more than 50 percent during the period of record.

In the current (2006) study, a different approach was taken. As discussed previously, an annual time series of impervious area, IA(t), was developed for each gage studied in order to relate a unique value of impervious area to each observed annual maximum flood value.

The observed and adjusted annual maximum time series are related by a set of nonlinear functions, f_T(.), such that

(9)

where Q_adj,T(t) is the adjusted time series of annual maximum flows that is keyed to the T-year flood. Each of the seven equations for the 2-, 5-, 10-, 25-, 50-, 100-, and 500-year floods (f₂, f₅, f₁₀, f₂₅, f₅₀, f₁₀₀, and f₅₀₀, respectively) must be calibrated.

When the flood-frequency operator is applied to each adjusted time series, it produces the rural flood-frequency value for the return period, (T):

(10)

Once all f_T(.) are determined and assumption 4 was applied, these functions were inverted such that the independent variables were the rural flood-frequency values and impervious area, and the dependent variable was the urban flood-frequency value:

(11)

where (.) is the inverted form of f_T(.).

The Adjustment Models

Several forms of f_T(.) were investigated—the null model, the simple imperviousness model, the simple density model, the imperviousness distribution model, the density distribution model, the scaled imperviousness model, and the scaled density model. These model forms are represented by the following equations (12 - 18, respectively).

Null model:

(12)

Simple imperviousness model:

(13)

Simple density model:

(14)

Imperviousness distribution model:

(15)

Density distribution model:

(16)

Scaled imperviousness model:

(17)

Scaled dentisy model:

(18)

The “distribution” models (eqs. 15 and 16) deserve a brief discussion. The terms ΔIA(t) and ΔPD(t) represent the difference between the 10^th and 90^th percentiles of the impervious area and population density, respectively. These terms quantify the homogeneity or uniformity of development within a watershed. Watersheds with relatively small values of either of these terms are considered to be uniform. Consideration of uniformity of development is helpful in predicting flood behavior.

The “scaled” models (eqs. 17 and 18) also deserve a brief explanation. The term “scaled imperviousness” varies nonlinearly from 1 to 100 as IA increases from 0 to 100 percent. For IA = I*, the scaled imperviousness equals 50.5. The scaled imperviousness function for typical parameter values is shown in figure 3. Scaled imperviousness is used because it could be argued that each incremental increase in imperviousness may not have an equivalent effect on flood-frequency. In figure 3, the effect of imperviousness increases most rapidly within the range of 5 to 15 percent imperviousness. This is consistent with studies of the effects of imperviousness on ecological health (for example, Schueler, 1987).

Figure 3. The relation between observed and scaled imperviousness for the scaled values I*=12.1 and c₄=0.2.

Censoring of Streamages with Non-elevated Flood-Frequency Values

Using PeakFQ, a preliminary flood-frequency analysis was performed for the observed annual maximum time series at each of the 78 streamgages. These analyses were conducted identically to those that are conducted in the optimization model with respect to identifying high outliers and determining the length of the historical record (discussed later). The flood-frequency results from these preliminary analyses were compared with the rural regression flood-frequency values described later in this report. If, at a specific streamgage, the flood-frequency from the observed annual maximum time series produced estimates for the T-year flood such that

(19)

this gage was eliminated from the database for the development of the T-year regression equation. The rationale for such data elimination is that if a streamgage does not have an increase in flood-frequency values relative to the rural regression equations for peak flows, then it does not make sense to use such a gage to calibrate an urban-scaling factor. The number of gages censored from the analysis will be presented later in the “Results” section.

Example—the Calibrated 5-year Simple Imperviousness Adjustment Model

As an example of the flow-adjustment process, consider the calibrated form for the 5-year, simple imperviousness adjustment model determined in this study:

(20)

Included in table 1 are the imperviousness time series, IA(t), the observed annual maximum time series, Q_obs(t), and the 5-year adjusted annual maximum time series, Q_adj,5(t) for USGS streamgage 01645000, Seneca Creek at Dawsonville, Maryland. The far right column of the table contains the ratios of the adjusted discharges to the observed discharges. Notice that the ratios are less than 1 for all by the three largest discharges in 1971, 1972, and 1975, indicating that the adjusted annual maximum time series generally is less than the observed annual maximum time series, which would be expected if the adjusted series approximates the comparatively small rural discharges that would have occurred in these years had urbanization not occurred. The adjustment ratios also are clearly smaller for smaller discharges and approach (and slightly exceed) 1 as the discharges become large. This indicates that relatively small discharges that correspond to more frequent 2- and 5-year flows are subject to adjustment from the effects of imperviousness. Such adjustments are consistent with the expectation that urbanization has the greatest effect on flood flows from storms of small magnitude. Further, in general, note that the adjustment ratios decrease with time, indicating the need to adjust the more recent discharges that correspond to greater urbanization in the watershed.

The behavior of the adjustment equation can be observed clearly by comparing years 1973 and 1999 (table 1). In 1973, the peak discharge was 3,020 ft³/s, and imperviousness was about 5.6 percent. The adjusted discharge for 1973 is 2,397 ft³/s. In comparison, the peak discharge in 1999 was 3,060 ft³/s, and imperviousness was 12.8 percent. The adjusted discharge for 1999 is 2,144 ft³/s. Although the observed discharge in 1999 is only slightly greater (40 ft³/s) than the observed discharge in 1973, the adjusted discharge for 1973 is greater by 253 ft³/s than the adjusted discharge for 1999. This clearly indicates the role of the annual imperviousness time series in this study. It is worth noting that imperviousness in the watershed between 1970 and 2000 increased from 4.8 percent to almost 13 percent, which represents more than a doubling of impervious area. For this reason, USGS streamgage 01645000 would have been eliminated from the Sauer and others (1983) study because their threshold of a 50-percent increase in imperviousness.

Finally, it is instructive to look at the 5-year flood-frequency values and compare these to the 5-year rural regression equation value of 5,876 ft³/s determined from Dillow (1996). While the observed 5-year peak discharge is 1.27 times the rural regression estimate, the adjusted 5-year peak is only 1.05 times this estimate. The similarity of the adjusted 5-year peak to the rural regression estimate is indicative of the success of the calibrated adjustment equation in quantifying the effect of imperviousness on the annual maximum discharge.

Table 1. Example of the adjustment of the observed annual maximum time series using the 5-year simple imperviousness adjustment model for the U.S. Geological Survey streamgage 01645000, Seneca Creek at Dawson, Maryland.

[%, percent; ft³/s, cubic feet per second]

Year	Imperviousness (%)	Observed Q (ft³/s)	Adjusted Q (ft³/s)	Q_adj,5(t) /Q_obs(t)


1970	4.84	2,200	1,702	0.774
1971	5.10	25,900	28,787	1.111
1972	5.37	26,100	28,831	1.105
1973	5.63	3,020	2,397	0.794
1974	5.90	3,160	2,508	0.794
1975	6.16	16,000	16,094	1.006
1976	6.42	4,900	4,101	0.837
1977	6.69	3,770	3,016	0.800
1978	6.95	7,850	6,969	0.888
1979	7.22	16,000	15,716	0.982
1980	7.48	10,800	9,946	0.921
1981	7.85	1,340	896	0.668
1982	8.23	3,160	2,385	0.755
1983	8.60	3,260	2,455	0.753
1984	8.98	3,010	2,225	0.739
1985	9.35	3,620	2,734	0.755
1986	9.73	1,070	669	0.625
1987	10.10	4,950	3,871	0.782
1988	10.48	7,410	6,120	0.826
1989	10.85	8,250	6,886	0.835
1990	11.23	2,270	1,553	0.684
1991	11.40	5,120	3,947	0.771
1992	11.58	1,750	1,146	0.655
1993	11.76	3,350	2,412	0.720
1994	11.93	9,160	7,651	0.835
1995	12.11	2,080	1,388	0.667
1996	12.28	11,000	9,399	0.854
1997	12.46	3,880	2,829	0.729
1998	12.64	5,280	4,023	0.762
1999	12.81	3,060	2,144	0.701
2000	12.99	1,910	1,244	0.651

Q₅		7,435	6,157

Methods

The database used for this investigation contains watershed characteristics for 78 USGS streamgages across the United States (Appendix 1). The two most critical elements in this database are the flood-frequency values obtained directly from the gage record and the flood-frequency values obtained from the current USGS rural discharge equations that apply at each gage site. In this section, the method for determining these two sets of values for the gages used in this investigation is discussed. Other ancillary data, such as outlier values, development of impervious-cover time series from census data, and determination of watershed characteristics also are discussed.

Development of Rural Peak Discharge Estimates

The premise of the current study is that the USGS rural flood-frequency regression equations for each State represent the best estimate of the nonurbanized flood-frequency at any gage analyzed in the study. Thus, it is necessary to apply these equations to each gage included in the study and use the flood-frequency results in the development of the urban adjustment methods.

Delineation of Watershed Boundaries

For each gage used in the study, the digital elevation model (DEM) covering the extent of the gaged watershed was obtained from the National Elevation Dataset (NED; U.S. Geological Survey, 2004a). Flow directions were constrained by “burning in” stream locations from the 1:100,000-scale National Hydrography Dataset (NHD; U.S. Geological Survey and U.S. Environmental Protection Agency, 2004). Standard GIS techniques were used to fill depressions and determine flow directions using a D8 algorithm (Jenson and Domingue, 1988). The watershed boundaries were delineated automatically by indicating the location of the streamgage to the GIS and initiating the delineation algorithm. An automated delineation is considered acceptable and consistent with USGS standards if the delineated drainage area is within 10 percent of the drainage area reported by the USGS (U.S. Geological Survey, 2004b).

Estimation of Watershed Characteristics

Using the watershed boundaries generated as described in the previous section other watershed characteristics were determined as needed from additional interpretation of the DEM (for example, calculating the channel slope). If additional land-use and land-cover descriptors were needed, such as percentage of forest cover, these data were determined by using the most current land-use and land-cover data available. In most cases this meant using the 1992 National Land Cover Dataset (NLCD; Vogelmann and others, 1998a, 1998b; U.S. Geological Survey, 2005), although the 2001 NLCD (Homer and others, 2004; Multi-Resolution Land Characterization Consortium, 2005) was available for some locations during this study.

Development of Flood-frequency Estimates from Gage Record

Flood-frequency estimates were developed from the observed annual maximum time series recorded as peak streamflow by the USGS (U.S. Geological Survey, 2004b). In this section, the methods will be discussed for selecting the gages used in the analysis, obtaining the observed annual maximum time, and using the PeakFQ program (Flynn and others, 2005) to analyze the data and produce the flood-frequency values.

Selection of Gages

The following list of criteria was used for selecting gages for this study:

The gage must have been active for 20 or more years during 1970-2000, the first full period for which census population data are available in digital format.

The gage must not be subject to flow regulation.

The gage must be equipped to report measured peak discharges, not just stage measurements.

The current peak-flow regression equations for the gage cannot be dependent on a measure of urbanization for flood prediction.

The drainage area for the gage, as determined by GIS, must be within plus or minus 10 percent of the reported USGS drainage area for the gage, thus providing a high degree of confidence in the application of GIS methods for determining the rural discharge estimates.

The gage record must not have a large proportion (approximately 20 percent) of streamflow values identified as outliers by the PeakFQ program.

The gage drainage area should contain more than 5 percent impervious area as quantified by the NLCD 2001 (Multi-Resolution Land Characteristics Consortium, 2005).

Applying these criteria resulted in the final selection of 78 gages in 12 States for the study analyses. The States in which the gages are located are California, Florida, Illinois, Maryland, Massachusetts, Michigan, Mississippi, Nevada, New York, Rhode Island, Virginia, and West Virginia (fig. 1).

Observed Annual Maximum Time Series

The observed annual maximum time series was obtained from the USGS (U.S. Geological Survey, 2004b). Observed peak discharge values for each study gage were obtained for all recorded floods occurring between 1970 and 2000. Data were collected in WATSTORE format, which can be read directly by the PeakFQ program.

Use of the PeakFQ Program

The USGS developed the PeakFQ computer program (Flynn and others, 2005) to conduct flood-frequency analyses based on the guidelines in Bulletin 17B (Interagency Advisory Committee on Water Data, 1982). The PeakFQ program uses the sample-moment method to fit the Pearson Type III frequency distribution to the logarithms of annual flood peaks. The skew that is used may be (a) user-developed generalized skew for a region, (b) the skew map from Bulletin 17B, (c) computed from the data, or (d) weighted between the generalized skew and station skew computed from the data. Adjustments can be made for high and low outliers and historic information. Qualification codes can be used to censor data from the analysis.

Skew

The PeakFQ program offers several options for defining a skew value to use in developing a flood-frequency estimate. In developing rural flood-frequency estimates for a given State, a regionalized skew typically is developed that is weighted with the local skew at the gaging station being analyzed. For urban estimates, this method is problematic because of fewer urban watersheds from which to develop a regionalized value and the varying extent of urbanization. In this study, therefore, each gaging station was considered individually and the “station skew” option was used throughout the analyses presented in this report.

PeakFQ Run-Time Parameters

Because of the relatively short periods of record (31 years or less) for the flood-frequency analysis, there is a greater possibility of time-sampling errors than may be expected when examining longer periods of record. To reduce the potential effects of these errors, a systematic method for applying the high-outlier option of the PeakFQ program was applied. To apply this option, however, two quantities must be defined—the discharge threshold that defines the minimum high-outlier discharge and the return frequency of the largest discharge in the systematic record analyzed. Because of the nature of the model calibration process, the procedures outlined here were performed repeatedly (for each iteration, for each gage, for each return period examined) in calibrating various trial-adjustment models.

Identifying the High-Outlier Threshold

The high-outlier threshold is the largest discharge that is considered within the normal range of discharge for a particular time series being analyzed. Discharges greater than this threshold were not attributed a probability using the Weibull plotting position formula in the analyses. The high-outlier discharge threshold was determined by the following method: given an annual maximum time series of n discharges at a particular gage, it was arbitrarily decided that these n values would represent the flood-frequency characteristics for that location for a period four times as long. Although this is an arbitrary assumption, it adequately produced the desired effect of filtering out large extreme floods that far exceeded the periods of record being analyzed while allowing the consideration of moderate-sized floods. Thus, the approach for identifying outlier discharges for this study is not consistent with the approach outlined in Bulletin 17B; however, this outlier-identification approach was a repeatable criterion that worked effectively and could be applied uniformly.

The discharges determined from the rural flood-frequency regression equations were used to determine the high-outlier threshold. The flood-frequency distribution was treated as a piecewise log-normal distribution, and the high-outlier threshold was determined by linear interpolation within the bounds of these values by applying the following equation:

(21)

where T_l < 4n ≤ T_u and z(T_x) is the standard normal deviate corresponding to a flood of return period, T_x. Thus, the high-outlier threshold (Q_o), is determined simply as

(22)

Identifying the Historic Period for the Largest Flood in the Time Series

In a similar manner, the largest flood in the systematic record was assigned a value for the historic period that it represents given the rural regression flood-frequency equation estimates that apply to a specific watershed. The largest flood was compared to the rural flood-frequency distribution, and a return period was determined by interpolating from this distribution, thus treating the flood-frequency distribution as a piecewise log-normal distribution. If the largest flood in the systematic record exceeded the 500-year rural flood, a return period of 500 years was arbitrarily assigned to the flood. The interpolation was performed as follows:

(23)

where Q_l < Q_o ≤ Q_u and z(Q_x) is the standard normal deviate corresponding to a flood of magnitude, Q_x. With z(Q_o) known, the cumulative probability (P(Q_o)) associated with this value was determined from the standard normal distribution. Finally, the return period (T(Q_o)) was determined by using

(24)

Example Application of Outlier and Historical Period Identification Procedures

To illustrate the application of the outlier and historical period identification procedures, the procedures were applied to the record of USGS streamgage 02037800, Falling Creek near Midlothian, Virginia. The period of record for this gage is 1951-2003, with several breaks in the record. Because of limited census data, only the period 1970-2000 was considered. Further, the gage was inactive in 1978 and again from 1994 to 2000; therefore, only the records for the periods 1970-1977 and 1979-1993 were available for this study. The largest flood during these two periods had a magnitude of 5,170 ft³/s and occurred in 1979. The next largest flood had a magnitude of 1,400 ft³/s and occurred in 1985 (fig. 4).

The rural peak discharges determined from Bisese (1995) for the Southern Piedmont region produced peak-discharge estimates for the streamgage at Falling Creek of 380 ft³/s, 647 ft³/s, 862 ft³/s, 1,217 ft³/s, 1,552 ft³/s, 1,923 ft³/s, and 3,054 ft³/s for the 2-, 5-, 10-, 25-, 50-, 100-, and 500-year floods, respectively. To identify the high-outlier threshold (Q_m) and determine the return period corresponding to the largest flood, the following criteria were applied. The length of the time series used in the analysis was n=23 years, representing the 1970-1977 and 1979-1993 periods of record. Using 4n=92 years, the log-normally interpolated 92-year flood from equations 21 and 22 is 1,876 ft³/s. No interpolation is necessary to determine the return period corresponding to the largest flood for either the observed or adjusted (discussed later) annual maximum time series because the observed (5,170 ft³/s) and adjusted (4,220 ft³/s) values exceed the rural 500-year flood of 3,054 ft³/s. A 500-year return period is arbitrarily attached to the largest flood during the periods of record analyzed. Thus, the PeakFQ program was executed for both the observed and adjusted annual maximum time series using station skew, high-outlier threshold (1,876 ft³/s), and historical period (500 years).

Figure 4. Annual maximum time series for U.S. Geological Survey streamgage 02037800, Falling Creek near Midlothian, Virginia.

Land-Use Time Series

This study draws heavily on the idea of developing a time series of changes in the spatial characteristics of the landscape. As a single descriptor of land-use change, imperviousness is the most simple characteristic that can be identified as having a strong influence on flood magnitude. Historical records of imperviousness are scarce, however, in both the spatial and temporal dimensions. For this reason, historical census-derived population-density estimates were used as a surrogate of imperviousness and as a predictor of imperviousness from which to develop the necessary space-time estimates of the changing landscape.

Population Density

Census data are readily available in 10-year increments and digital format for 1970 through 2000. Assuming linear changes in population density between incremental census data, it is possible to develop space-time estimates of population density for any location in the United States at the resolution of a census tract. Census tracts vary considerably in scale depending on population density. For this study, the focus was on urban areas for which the census data tend to delineate small-scale census tracts. For example, Montgomery County, Maryland, is a densely populated county directly north of Washington, D.C. In the 2000 census, this county was composed of 177 census tracts (GeoLytics, 2003), ranging in size from 0.13 mi² to 70 mi² and averaging 2.86 mi². Population density is not uniformly distributed among these tracts, however, and smaller tracts tend to have greater population densities, as shown in figure 5.

Landscape changes, such as increases in imperviousness, introduction of curb and gutter drainage, and channelization can vary with population density. Using a single, average population-density value and the landscape changes for which it is a surrogate to quantify human influences within a watershed may be a poor model when a watershed is large or if human influences vary considerably. To quantify such variability or nonhomogeneity within a watershed, the distribution of population densities was quantified as a fraction of the total watershed area as shown in the example for USGS streamgage 01645000, Seneca Creek at Dawsonville, Maryland (fig. 6). The population densities that correspond to the 10^th and 90^th percentiles of the watershed area were tagged, and the difference between these densities was recorded. This quantity (ΔPD) was defined as follows:

(25)

The more homogeneous the distribution of population within a watershed, the smaller ΔPD will be. ΔPD is a potentially useful predictor, as discussed in the earlier section on Adjustment Models.

Figure 5. Population density in Montgomery County, Maryland, as a function of census-tract area from the 2000 census.

Figure 6. Distribution of population density as a fraction of the total watershed area for U.S. Geological Survey streamgage 01645000, Seneca Creek at Dawsonville, Maryland. [ΔPD is the difference in population density between the 10^th and 90^th percentiles of the distribution of population density in the watershed.]

Imperviousness

The 2001 NLCD includes imperviousness mapping product at 30-meter resolution. Each pixel reports a value of imperviousness as a percentage of the total area of the pixel. The determination of the percentage of imperviousness across an entire watershed is straightforward and requires simply calculating the average imperviousness from the set of pixels within the watershed.

Estimating Imperviousness from Census Data

A time series of historical land-use (or imperviousness) data covering several decades and uniformly available at the national scale does not exist. Because of the need to develop estimates of imperviousness for each year for each watershed being studied, it was necessary to develop a procedure that could be uniformly applied to each watershed.

The imperviousness layer from the 2001 NLCD was obtained for the central region of Maryland because it was available and because this region includes two highly urbanized areas—Baltimore, Maryland, and Washington, D.C. Similarly, tract data from the 2000 census were obtained. For each census tract, the population density (in thousands of people per square mile) was determined based on the reported total population and the area of the tract as determined by using GIS. Similarly, the tract outlines were used to sample the 2001 NLCD imperviousness layer and to determine the average imperviousness for each tract. Several different models were investigated for describing the relation between population density and imperviousness. A simple power model was selected to provide the best compromise between statistical power and model rationality. The calibrated equation took the following form:

(26)

where IA		is imperviousness, in percent; and
PD		is population density in thoursands of people per square mile.

This equation was determined for 0.0002 < PD < 176.4 and 0.08 < IA < 96.66. The median PD and IA values were 3.87 and 21.8, respectively. The regression was performed using data from 998 census tracts. The ratio of the standard error (S_e) of equation 26 to the standard deviation (S_d) of observed imperviousness was S_e/S_d = 0.6163, with an explained variance of 62.1 percent.

The regression equation developed by Stankowski (1972) to estimate imperviousness from population density in New Jersey had a different form from that of equation 26:

(27)

where IA		is imperviousness, in percent; and
PD		is population density, in persons (not thousands of persons) per square mile.

The results of both equations, however, are quite comparable, as shown in figure 7.

In focusing on predicted imperviousness between 0 and 50 percent, which is the typical range observed in most watersheds, the maximum departure between the two methods is about 1.2 percent (fig. 7). The estimated imperviousness obtained by using equation 27 is 31.0 percent and 29.8 percent by using equation 26 for a population density of approximately 5,660 persons per square mile. While the difference is small, it is even less significant considering that most of the watersheds in this study had lower population densities and, thus, smaller estimated imperviousness. The typical departure in estimates of imperviousness between the two methods for the watersheds included in this study was less than 1.0 percent.

Figure 7. Relations of imperviousness to population density as measured by Stankowski (1972) and as measured in this study.

Implementation of Historical Census Data

Historical census tract data were obtained for 1970, 1980, 1990, and 2000. Population density and estimated imperviousness were determined, using equation 26, for each census tract in each study watershed for each census period. The watershed boundary was then used to select all intersecting census tracts, and an area-weighted average of imperviousness for each census period was determined for each watershed. Imperviousness was assumed to vary linearly between each census measure.

In the event of decreasing population density over time, the estimate of imperviousness naturally would decline (eq. 26). Although population could decrease, imperviousness was assumed to be a strictly nondecreasing function with time and, therefore, was assumed to remain at the previous value. Of the 78 watersheds studied, 10 had decreasing population estimates and nonvarying estimates of imperviousness during the entire gaged period.

To illustrate the use of historical census data in estimating imperviousness, census tract data were obtained for the watershed of USGS streamgage 01645000, Seneca Creek at Dawsonville, Maryland (fig. 8). Equation 26 was applied to population densities for 1970, 1980, 1990, and 2000. The resulting imperviousness estimations for these years were 4.84 percent, 7.48 percent, 11.23 percent, and 12.99 percent, respectively.

Figure 8. Estimated imperviousness from census-tract data for U.S. Geological Survey streamgage 01645000, Seneca Creek at Dawsonville, Maryland. [Imperviousness in this figure ranges from 0 to 40 percent. Darker shades correspond to high degrees of imperviousness.]

Results

Previously, in the “Methods” section the need to censor gages with non-elevated flood-frequency was discussed. The actual numbers of streamgages that were used in the calibration of the adjustment equations as a function of the flood-frequencies being considered are given in table 2.

Table 2. Number of gages used in the regression analysis as a function of return period.

Return period (years)	2	5	10	25	50	100	500


Number of streamgages used	52	52	50	48	43	41	28

In total, information was obtained from 61 of the 78 streamgages (identified by italics in appendix 2) for at least one of the seven frequencies for which urban regression equations were calibrated in this study. In other words, of the 78 streamgages originally selected for use in this study, observed flood-frequency estimates for 17 of the streamgages were no greater than the rural regression estimates for any of the seven frequencies examined. The explanation for this is attributed collectively to error inherent in the rural regression equations, possible error in the discharge measurement or corresponding rating curve, error in the flood-frequency analysis, and other inherent characteristics (such as manmade storage or missing basin attributes) that may have served to reduce the flood magnitudes as predicted by the rural regression equations.

The goodness-of-fit characteristics for the model calibrations are presented in table 3, and the calibrated coefficients are shown in table 4. Table 3 presents standard error (S_e) calculated as

(28)

where n		is the number of observations,
m		is the number of coefficients or exponents being calibrated,
y		is the observed discharge (from the PeakFQ output), and
		is the predicted output calibrated by the nonlinear regression tool.

Standard deviation (S_y) is calculated as

(29)

where

is the mean of the discharges for the return period (T).

Explained variance (R²) is calculated as

(30)

where

(31)

in which is the mean of the predicted discharges for the return period (T).

Table 3. Goodness-of-fit characteristics for the models evaluated.

[All statistics presented are for log-transformed discharges: log₁₀(Q). Bold values are best values for the return period.]

Model	Explained variance (R₂)							Standard error ratio (S_e/S_y)


	Return period (years)							Return period (years)
	2	5	10	25	50	100	500	2	5	10	25	50	100	500


Null	0.821	0.859	0.864	0.902	0.899	0.889	0.814	0.390	0.378	0.341	0.265	0.261	0.277	0.446
Simple imperviousness	0.818	0.866	0.879	0.909	0.902	0.884	0.801	0.392	0.389	0.348	0.281	0.277	0.310	0.480
Simple density	0.818	0.866	0.877	0.907	0.897	0.881	0.804	0.397	0.363	0.328	0.257	0.298	0.311	0.478
Imperviousness distribution	0.832	0.867	0.874	0.909	0.901	0.897	0.855	0.385	0.374	0.338	0.292	0.268	0.271	0.356
Density distribution	0.833	0.870	0.874	0.902	0.893	0.889	0.779	0.383	0.368	0.335	0.278	0.289	0.297	0.526
Scaled imperviousness	0.822	0.873	0.869	0.904	0.900	0.884	0.810	0.409	0.396	0.365	0.288	0.280	0.310	0.528
Scaled density	0.823	0.861	0.873	0.904	0.900	0.887	0.821	0.409	0.414	0.361	0.289	0.287	0.307	0.464

Table 4. Calibrated values of the adjustment model coefficients and exponents.

[Variables for all models are defined in equations 12-18.]

Model	Variable	Return period (years)

Model	Variable	2	5	10	25	50	100	500

Null	c₁	0.275	0.489	0.305	0.330	0.305	0.330	0.330
Null	c₂	1.10	1.04	1.10	1.10	1.10	1.10	1.10
Simple imperviousness	c₁	0.289	0.331	0.33	0.33	0.228	0.225	0.263
	c₂	1.16	1.15	1.16	1.16	1.15	1.15	1.12
	c₃	0.177	0.173	0.176	0.176	0.0383	0.0446	0.035
Simple density	c₁	0.27	0.315	0.314	0.368	0.35	0.35	0.308
	c₂	1.10	1.10	1.10	1.10	1.10	1.10	1.10
	c₃	0.05	0.0987	0.0977	0.0642	0.172	0.172	0.0474
Imperviousness distribution	c₁	0.33	0.42	0.405	0.435	0.363	0.293	0.251
	c₂	1.10	1.10	1.10	1.10	1.10	1.10	1.10
	c₃	0.134	0.159	0.135	0.101	0.0819	0.0148	0.0229
	c₄	0.0687	0.025	0.0166	0.0425	0.0388	0.0666	0.108
Density distribution	c₁	0.27	0.313	0.312	0.315	0.304	0.345	0.332
	c₂	1.10	1.10	1.10	1.10	1.10	1.10	1.10
	c₃	0.171	0.228	0.111	0.155	0.140	0.136	0.198
	c₄	0.11	0.062	0.0264	0.0739	0.0963	0.126	0.113
Scaled imperviousness	c₁	0.288	0.315	0.3	0.3	0.3	0.315	0.285
	c₂	1.15	1.15	1.15	1.15	1.15	1.15	1.15
	c₃	0.11	0.148	0.0997	0.1	0.1	0.109	0.0898
	c₄	0.131	0.29	0.2	0.167	0.105	0.149	0.204
	I*	13.9	20.1	9	12.5	7.54	10.2	13.1
Scaled density	c₁	0.287	0.3	0.3	0.3	0.3	0.296	0.284
	c₂	1.15	1.15	1.15	1.15	1.15	1.15	1.15
	c₃	0.11	0.1	0.125	0.0944	0.108	0.108	0.109
	c₄	0.271	0.505	0.765	0.213	0.000488	0.00469	0.00907
	PD*	1.05	1.2	3.04	0.960	0.284	0.369	0.519

As shown in table 3, the null model performed almost as well as the more sophisticated models, especially in regard to the standard error ratio measure. It must be noted, however, that only urban streamgages were used in this study. In particular, the calibrated equations were developed only for urban gages where the observed flood-frequency was associated with greater discharges than those predicted by the rural regression equations. Given such a dataset from which to calibrate an equation, the null model was certain to perform well simply because the data were already screened for the tendency to produce elevated flood magnitudes. The null model could be expected to have a poorer performance if both rural and urban streamgages were in the dataset. Further, the major shortcoming of the null model is that it will not predict larger urban discharges in relation to larger amounts of urbanization (imperviousness or density). The value of the null model is to provide some information about the additional predictive power of the models that include imperviousness in the flow-adjustment scheme.

Provided that a model that accounts quantitatively for urbanization is needed, the question now turns to which model of those examined performs “best.” An absolute best model is not indicated in table 3 because only seven models were examined, and it is impossible to prove logically that the very best mathematical form was investigated. Nevertheless, the models examined encompass several different approaches to adjusting flows, and their performance probably is not far removed from the hypothetical best adjustment model. Further, the relative performance of the seven models varied, depending on the measure (explained variance or relative standard error) and the flood-frequency being considered (table 3). The goodness-of-fit values in table 3 were ranked within each measure and return period and the best performing model received a ranking of “1” with subsequent rankings increasing as model performance decreased (table 5). The rankings then were summed across both the explanatory measures and flood-frequencies, and the model producing the overall smallest sum was considered to be the best overall model among those considered. These summed rankings are presented in table 6.

Table 5. Rankings of model performance based on the goodness-of-fit values show in table 3.

Model	Explained variance (R₂)							Standard error ratio (S_e/S_y)


	Return period (years)							Return period (years)
	2	5	10	25	50	100	500	2	5	10	25	50	100	500


Null	5	6	7	6	5	3	3	3	4	4	4	1	2	2
Simple imperviousness	6	4	1	1	1	5	6	4	5	5	5	3	5	5
Simple density	6	4	2	3	6	7	5	5	1	1	1	7	7	4
Imperviousness distribution	2	3	3	1	2	1	1	2	3	3	3	2	1	1
Density distribution	1	2	3	6	7	2	7	1	2	2	2	6	3	6
Scaled imperviousness	4	1	6	4	3	5	4	6	6	7	7	4	5	7
Scaled density	3	7	5	4	3	4	2	6	7	6	6	5	4	3

Table 6. Summed rankings of model-calibration statistics based on goodness-of-fit values given in table 3.

[Bold values are best values for each column.]

Model	Explained variance, R₂ sum	Standard error ratio, S_e/S_y sum	Overall sum (Overall rank)



Null	35	20	55 (3)
Simple imperviousness	24	32	56 (4)
Simple density	33	26	59 (5)
Imperviousness distribution	13	15	28 (1)
Density distribution	28	22	50 (2)
Scaled imperviousness	27	42	69 (7)
Scaled density	28	37	65 (6)

This ranking scheme indicates that the best overall performing model was the imperviousness distribution model (table 6). This assertion can be confirmed by a closer examination of table 3 in which the imperviousness distribution model produced the largest explained variance in three of the seven flood-frequencies examined and was among the top three performing models in this measure. With respect to relative standard error, the imperviousness distribution model is the best model in two of the seven flood-frequencies and was always among the top three performing models in this measure. In ranking overall performance, the imperviousness distribution model performed the best (slightly better than the density distribution model) with regard to explained variance and relative standard error (table 5).

In regard to the remaining models, one could ask two questions. First, does population density serve as a stronger predictor than imperviousness? Second, does the “simple” approach, such as simple imperviousness or simple density, outperform the “distribution” or “scaled” approach?

With regard to the first question, the tabulated results (table 5) indicate a modest advantage for imperviousness over density as a predictor. The imperviousness distribution model ranked first overall, whereas the density distribution model ranked second. The simple imperviousness model ranked fourth overall, whereas the simple density model ranked fifth. The opposite trend occurred for the scaled models—the imperviousness-based model ranked seventh (last) and the density-based model ranked sixth. Thus, the results slightly favor imperviousness over population density as a predictor of flood-frequency magnitude.

With regard to the second question, the “distribution” approach (eqs. 15, 16) ranked first, followed by the “simple” approach (eqs. 13, 14); the “scaled” models (eqs. 17, 18) performed the worst. This result is interesting because the simple models require the calibration of three parameters; the distribution models require four parameters, and the scaled models require five parameters. Generally, a more complex model would be expected to perform best because it has more flexibility to conform to the observed data. In this case, the moderately complex model performs best. The ΔPD or ΔIA term of the distribution model structure apparently is an effective way of quantifying urban heterogeneity within a watershed and, therefore, is useful in predicting flood behavior.

The calibration results for all models are presented in table 4. (Appendix 3 contains plots for comparing predicted and observed discharges for the calibrated simple imperviousness model, which is fairly representative of other model calibrations as well.) While the values shown in table 4 represent optimum calibration results, a brief examination of this table reveals somewhat erratic trends in many calibration parameters as return period varies. It would be expected that these parameters vary smoothly from one return period to the next. Towards this end, a secondary smoothing was performed for each calibration parameter using a linear regression model:

(32)

where T		is the return period and
α and ß		are determined by regressing the calibrated values of c_x,T on log₁₀(T).

These smoothed parameter values were used in all subsequent analyses, because it was felt these values were more representative of the true relations between rural and urban flood magnitudes (table 7). Appendix 4 contains the calibrated and smoothed coefficients plotted against return periods for each adjustment model.

Table 7. Smoothed values of the adjustment model coefficients and exponents.

[Variables for all models are defined in equations 12-18.]

Model	Variable	Return period (years)

Model	Variable	2	5	10	25	50	100	500

Null	c₁	0.289	0.297	0.303	0.311	0.317	0.324	0.338
Null	c₂	1.10	1.10	1.10	1.10	1.10	1.10	1.10
Simple imperviousness	c₁	0.327	0.312	0.301	0.287	0.276	0.264	0.239
	c₂	1.16	1.16	1.16	1.15	1.15	1.14	1.13
	c₃	0.200	0.170	0.147	0.117	0.095	0.072	0.019
Simple density	c₁	0.305	0.312	0.318	0.325	0.330	0.336	0.349
	c₂	1.10	1.10	1.10	1.10	1.10	1.10	1.10
	c₃	0.0856	0.0910	0.0950	0.100	0.104	0.108	0.118
Imperviousness distribution	c₁	0.414	0.393	0.378	0.357	0.341	0.325	0.289
	c₂	1.10	1.10	1.10	1.10	1.10	1.10	1.10
	c₃	0.161	0.136	0.118	0.0927	0.0740	0.0552	0.0115
	c₄	0.0269	0.0361	0.0431	0.0523	0.0592	0.0662	0.0823
Density distribution	c₁	0.289	0.297	0.304	0.313	0.320	0.326	0.342
	c₂	1.10	1.10	1.10	1.10	1.10	1.10	1.10
	c₃	0.166	0.165	0.164	0.163	0.162	0.161	0.159
	c₄	0.066	0.073	0.079	0.087	0.093	0.098	0.112
Scaled imperviousness	c₁	0.303	0.302	0.301	0.300	0.300	0.299	0.298
	c₂	1.15	1.15	1.15	1.15	1.15	1.15	1.15
	c₃	0.123	0.117	0.113	0.108	0.104	0.100	0.0911
	c₄	0.189	0.185	0.182	0.178	0.175	0.172	0.165
	I*	14.4	13.7	13.1	12.3	11.8	11.2	9.88
Scaled density	c₁	0.298	0.297	0.296	0.295	0.295	0.294	0.292
	c₂	1.15	1.15	1.15	1.15	1.15	1.15	1.15
	c₃	0.108	0.108	0.108	0.108	0.108	0.108	0.107
	c₄	0.512	0.418	0.347	0.253	0.182	0.111	-0.0539
	PD*	1.67	1.45	1.28	1.06	0.893	0.725	0.336

As stated previously, the functions, f_n(.) can be rearranged to express the urban peak discharge as a function of the rural peak discharge and imperviousness. For simplicity, the equation 11 can be revised as follows:

(33)

Thus, the null model is rearranged as

(34)

the simple imperviousness model is rearranged as

(35)

the simple density model is rearranged as

(36)

the imperviousness distribution model is rearranged as

(37)

the density distribution model is rearranged as

(38)

the scaled imperviousness model is rearranged as

(39)

and the scaled density model is rearranged as

(40)

where IA is the impervious area, in percent, for the watershed conditions at which the urban discharge estimate is desired.

The coefficients and (or) exponents, c_x,T, refer to the coefficients outlined in the models presented in equations 12-18 for the T-year prediction equation. Note that equations 12-18 refer to adjustment models that convert urbanized discharges to equivalent rural ones, whereas equations 33-40 convert rural discharges to urban ones.

Based on the smoothed coefficients and (or) exponents given in table 7, seven sets of urban equations are given based on the calibrated null, simple imperviousness, simple density, imperviousness distribution, density distribution, scaled imperviousness, and scaled density adjustment models. For the null model, the resulting urban equations are as follows:

(41)

(42)

(43)

(44)

(45)

(46)

and

(47)

The calibrated urban equations for the simple imperviousness model are as follows:

(48)

(49)

(50)

(51)

(52)

(53)

and

(54)

The calibrated urban equations for the simple density model are as follows:

(55)

(56)

(57)

(58)

(59)

(60)

and

(61)

The calibrated urban equations for the imperviousness distribution model are as follows:

(62)

(63)

(64)

(65)

(66)

(67)

and

(68)

The calibrated urban equations for the density distribution model are as follows:

(69)

(70)

(71)

(72)

(73)

(74)

and

(75)

For the scaled imperviousness model, the calibrated urban equations are as follows:

(76)

(77)

(78)

(79)

(80)

(81)

and

(82)

Finally, for the scaled density model, the calibrated urban equations are as follows:

(83)

(84)

(85)

(86)

(87)

(88)

and

(89)

A positive exponent on any predictor variable indicates a positive relation between that predictor and the urban discharge; a negative exponent indicates a negative relation. All of the model forms have positive exponents on all predictors except the ΔIA and ΔPD terms (eqs. 62-75), which have negative exponents. (One other exception, which will be discussed later, occurred for c₄ in equation 89.) It can be concluded quickly that all exponents are rational because the urban flood magnitude is expected to increase with increasing rural flood magnitude, increasing imperviousness, and increasing population density. Although the negative sign on the ΔIA and ΔPD terms is less obvious, it indicates that as the watershed becomes more varied spatially in urbanization, the predicted discharge decreases. In essence, the ΔIA and ΔPD terms serve to temper the IA or PD terms, which convey only the average imperviousness or population density in a watershed but not variability. Finally, the positive signs on the two parameters that control the scaled sigmoid function presented in figure 8 are worth noting.

Trends in the parameter values as return period increases also are important to examine. Generally, as the return period increases, one might expect to see diminishing effects of urbanization on flood magnitude. For example, the 2-year flood would be expected to have greater dependence on urbanization than the 100-year flood, and trends in the parameter values would be expected to support this. The following discussion focuses on the rationality of the trends for each model:

Null model: The evaluation of trends in this model is not applicable.

Simple imperviousness model: Trends in this model are entirely rational. The model has a negative trend in the IA exponent.

Simple density model: The positive trend in the PD exponent for this model is not rational. This is of concern.

Imperviousness distribution model: Trends in this model are entirely rational. The model has a positive trend in ΔIA exponent and a negative trend in the IA exponent.

Density distribution model: The trends in this model follow the same pattern as in the imperviousness distribution model, although the negative trend in the PD exponent is weak.

Scaled imperviousness: Trends in this model are entirely rational. The model has a negative trend in the scaled IA exponent. It is not clear whether a positive or negative trend in the two scaled imperviousness terms is expected but both terms have negative trends, which is consistent with the trends in the analogous terms in the scaled density model.

Scaled density: This model has a weak positive trend in the scaled PD exponent. This is not rational, but because the trend is so weak, the non-rationality similarly is weak. The negative trend in c₄ actually leads to a slightly negative value for the 500-year flood. The physical interpretation of this result is not clear, and the trends for this model are of concern. Again, the trends in both terms in the scaled density model are negative, consistent with the analogous terms in the scaled imperviousness model.

The results of the rationality analysis are summarized as follows:

All models had rational signs in the urbanization (imperviousness (IA) or population density (PD)) exponents.

All imperviousness models (simple, distribution, and scaled) had rational trends in the exponents.

Two population density models (simple and scaled) had one or more non-rational trends in the exponents. The results for these two models should be used with caution, if at all.

Each of the above equations (eqs. 41-89) was divided by RQ_T to produce the ratio of urban to rural discharge (UQ/RQ). Assuming a vector of rural flood-frequency values (Q_ff,rural), the UQ/RQ was determined as a function of varying imperviousness. To visualize results of this approach, each model was applied to values in two watersheds, one fairly large and one fairly small. The fairly large watershed (102 mi²) was represented by USGS streamgage 01645000, Seneca Creek at Dawsonville, Maryland; the fairly small watershed (2.2 mi²) was represented by USGS streamgage 01585200, West Branch Herring Run at Idlewylde, Maryland. The UQ/RQ ratios are shown in figures 9-12.

The three imperviousness models that were investigated have several common characteristics (figs. 9, 11). First, all models indicate a general decline in UQ/RQ, approaching 1, as the return period increases. Because the effects of urbanization are most profound for small magnitude and high-frequency flooding events, the greatest adjustment ratios are required for the small return periods. Also, all models had a greater UQ/RQ value as imperviousness increased. This was not surprising because, all things being equal, higher values of imperviousness would be expected to lead to greater urban discharges. In most cases, the null model results were roughly centered within the spread of imperviousness values and were considered representative of average imperviousness across the datasets evaluated (figs. 9, 11).

The three calibrated population density models were applied to discharge data from the two selected gaging stations to obtain the UQ/RQ values across all return periods (figs. 10, 12). The irrationality in the trend in the population density exponent in the simple density model is shown in figure 10A by the increasing spread of the constant population density traces in relation to return period increases. The tight clustering of the constant population density traces in figure 10C indicates that the calibrated parameters for the scaled population density term do not lead to greatly varying estimates of scaled population density. This is not irrational, but it diminishes the credibility of the scaled population density model.

By comparing the UQ/RQ ratios for imperviousness (figs. 9, 11) and population density (figs. 10, 12) in the two watersheds, the effect of watershed scale is apparent between the two datasets. Herring Run is roughly two orders of magnitude smaller than Seneca Creek, which results in a greater range in UQ/RQ ratios for this smaller watershed. In comparing figures 9A and 11A for the 2-year return period, a UQ/RQ range is observed of 0.8 to 1.6 for Seneca Creek and approximately 1.2 to 2.2 for Herring Run, respectively. The greater range in UQ/RQ ratios for the smaller watershed is because the exponent on RQ, which is positive but less than 1, has the effect of the UQ/RQ ratio varying more widely for small discharges associated with small watersheds.

Figure 9. Application of the (A) simple imperviousness model, (B) imperviousness distribution model, and (C) scaled imperviousness model to the ratio of urban to rural discharges as a function of return period and imperviousness at U.S. Geological Survey streamgage 01645000, Seneca Creek at Dawsonville, Maryland. [The heavy dashed line shows the ratio for the "null" model. The light lines indicate imperviousness increasing in 2-percent increments starting from 0 percent for the lowest trace.]

Figure 10. Application of the (A) simple density model, (B) density distribution model, and (C) scaled density model to the ratio of urban to rural discharges as a function of return period and imperviousness at U.S. Geological Survey streamgage 01645000, Seneca Creek at Dawsonville, Maryland. [The heavy dashed line shows the ratio for the “null” model. The light lines indicate population density increasing in increments of 1,000 persons per square mile starting from 0 persons per square mile for the lowest trace.]

Figure 11. Application of the (A) simple imperviousness model, (B) imperviousness distribution model, and (C) scaled imperviousness model to the ratio of urban to rural discharges as a function of return period and imperviousness at U.S. Geological Survey streamgage 01585200, West Branch Herring Run at Idlewylde, Maryland. [The heavy dashed line shows the ratio for the “null” model. The light lines indicate imperviousness increasing in 2-percent increments starting from 0 percent for the lowest trace.]

Figure 12. Application of the (A) simple density model, (B) density distribution model, and (C) scaled density model to the ratio of urban to rural discharges as a function of return period

Methods for Adjusting U.S. Geological Survey Rural Regression Peak Discharges in an Urban Setting

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