Flood frequency characteristics and estimated flood discharges for the 50-, 20-, 10-, 4-, 2-, 1-, 0.5-, and 0.2-percent annual exceedance probabilities were computed at 299 streamgaged locations in Wisconsin. The State was divided into four flood frequency regions using a cluster analysis to produce regions which are homogeneous with respect to physical basin characteristics. Regression equations relating flood discharges to basin characteristics within each region were developed and can be used to estimate flood discharges at ungaged locations in Wisconsin. Basin characteristics included in the final regression equations include drainage area, saturated hydraulic conductivity, percent forest, percent herbaceous upland, percent open water, and the maximum 24-hour precipitation with a 10-year recurrence interval. The standard error of prediction for regression equations ranges between 40 and 71 percent, and the pseudo coefficient of determination ranges between 0.8 and 0.95. Nonmonotonic trends in the annual peak flow time series in the southwest part of the State are producing bias in some flood discharge estimates at streamgages with shorter (less than 20 years) periods of record. This bias increases the uncertainty in regression equations in this flood frequency region.

Levin, S.B., 2022, PeakFQ inputs and outputs for 299 streamgages in Wisconsin through water year 2020: U.S. Geological Survey data release,

Levin, S.B., 2023, Model archive—Regional regression models for estimating flood frequency characteristics of unregulated streams in Wisconsin: U.S. Geological Survey data release,

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Multiply | By | To obtain |

Length | ||
---|---|---|

inch (in.) | 2.54 | centimeter (cm) |

foot (ft) | 0.3048 | meter (m) |

mile (mi) | 1.609 | kilometer (km) |

Area | ||

square mile (mi^{2}) |
2.590 | square kilometer (km^{2}) |

Flow rate | ||

cubic foot per second (ft^{3}/s) |
0.02832 | cubic meter per second (m^{3}/s) |

Multiply | By | To obtain |

Length | ||
---|---|---|

centimeter (cm) | 0.3937 | inch (in.) |

meter (m) | 3.281 | foot (ft) |

kilometer (km) | 0.6214 | mile (mi) |

Area | ||

square kilometer (km^{2}) |
0.3861 | square mile (mi^{2}) |

Flow rate | ||

cubic meter per second (m^{3}/s) |
35.31 | cubic foot per second (ft^{3}/s) |

Vertical coordinate information is referenced to the North American Vertical Datum of 1988 (NAVD 88).

Horizontal coordinate information is referenced to North American Datum of 1983 (NAD 83).

A water year is the 12-month period from October 1 through September 30 and is designated by the year in which it ends.

annual exceedance probability

average variance of prediction

drainage area ratio

expected moments algorithm

generalized least squares

Log-Pearson Type III distribution

ordinary least squares

U.S. Geological Survey peak-flow frequency analysis program

potentially influential low flow

predicted residual sum of squares

coefficient of determination

standardized distance

standard model error

standard error of prediction

U.S. Geological Survey

weighted-multiple-linear-regression program

Flood frequency analysis refers to the statistical analysis used to estimate the magnitude and frequency of floods at gaged or ungaged locations. Flood frequency information is used in a variety of infrastructure and public safety projects such as the design of dams, culverts, bridges, and highways and is used in flood insurance and flood-plain management. Annual peak streamflow collected at U.S. Geological Survey (USGS) streamgages (

Periodic updates of flood frequency estimates and regional regression equations are necessary to incorporate new data and methods. There is a lengthy history of flood frequency studies in Wisconsin starting with an initial report in 1961 (

Regression equations are used to estimate flood discharges corresponding to selected AEPs for ungaged locations on streams. Regression equations in this study were developed using the updated flood frequency estimates and basin characteristics at streamgages without substantial regulation or urbanization and do not include the main stems of the Wisconsin River, St. Croix River, or the Mississippi River. In large and hydrologically diverse states such as Wisconsin, regression equations can be improved by grouping the available streamgages into hydrologically similar regions before the development of the equations. For this study, four flood frequency regions were developed for Wisconsin using a clustering algorithm. The additional data and updated statistical methodologies used in this report increase the confidence in the resulting flood frequency estimates and supersede the frequency analyses and regression equations in previous reports.

The purpose of this report is to present methods for estimating the magnitude and frequency of floods for unregulated, rural streams in Wisconsin. This report (1) describes the statistical methods used to estimate the flood discharge magnitudes at gaged locations; (2) presents the estimated flood discharges for the 50-, 20-, 10-, 4-, 2-, 1-, 0.5-, and 0.2-percent AEPs at 299 streamgages in Wisconsin; (3) describes the methods used to develop regression equations for estimating the frequency and magnitude of floods at ungaged locations; and (4) presents the final regional flood frequency regression equations for Wisconsin with examples for their usage.

Flood frequency analysis and development of regression equations require two types of data: (1) annual peak streamflow and (2) physical basin characteristics. Annual peak streamflow, defined as the maximum instantaneous streamflow recorded during a water year, is needed to estimate flood discharge for selected AEPs at streamgaged locations. Regression equations relate flood discharges for selected AEPs to basin characteristics at each streamgage and are used to estimate probable flood discharges on ungaged streams.

Annual peak streamflow data were obtained for all active and inactive streamgages in basins in Wisconsin with at least 10 years of annual peak streamflow observations through water year 2020 (

In addition to regulation from dams, candidate streamgages were screened for urbanization. The percentage of developed land computed by the Wisconsin StreamStats web application (

Streamgages and flood frequency regions used to estimate peak flow frequencies and magnitudes in Wisconsin.

Basin characteristics were used in cluster analysis for identifying homogeneous regions for regression equation development and as explanatory variables in the regional regression equations. A suite of 24 basin characteristics consisting of geophysical, climatic, and land-use characteristics were determined for each streamgage in the study. Basin characteristics that were considered for this study included basin characteristics developed and used in previous flood frequency studies in Wisconsin (

Drainage area (

Channel slope (

Mean basin slope (

Mean annual precipitation (

Mean annual snowfall (

Saturated hydraulic conductivity (

24-hour precipitation indices (

Climate-factors (

Land-use categories (land use categories from the 2001 National Land Cover Dataset) were computed within the StreamStats web application as a percentage of total basin area. Land uses in this study included Forest (

One assumption in regression analysis is that the data are spatially independent. Redundancy happens when two streamgages of similar drainage areas are nested (meaning that one basin is contained within the other) and have similar basin characteristics and peak flow magnitudes. This can happen when two gages are on the same stream and there are no large confluences or incoming tributaries between them. In these cases, the basins will likely have the same response to a given storm and thus would represent only a single spatial observation.

A redundancy analysis outlined by _{ij}_{ij}_{ij}

is the distance, in miles, between centroids of basin

is the drainage area in square miles at site

is the drainage area in square miles at site

The DAR is computed as the ratio of drainage area of the larger basin to the drainage area of the smaller basin and was used to determine if two nested basins are similar in size. Previous studies have considered nested pairs of gages with a DAR less than or equal to five to be redundant (

The _{ij}_{ij}

Flood frequency analysis uses statistical techniques to estimate flood discharges associated with specific AEPs or recurrence intervals. An AEP is the probability that a flood of a specific magnitude will happen in a given year. Formerly, AEPs have been reported as recurrence intervals where the recurrence interval, in years, is the reciprocal of the AEP; for example, the flood corresponding to the 1-percent AEP is commonly referred to as the 100-year flood. The recurrence interval terminology is now discouraged because its interpretation can cause confusion; therefore, the AEP terminology is used in this report.

Flood frequency analyses were performed using the Expected Moments Algorithm (EMA) method with the Multiple Grubbs-Beck test for potentially influential low floods (PILFs), as recommended in Bulletin 17C (

A primary assumption of flood frequency analysis is that the mean and variability of annual peak flows at a streamgage are not changing with time. Previous studies have identified trends in annual peak flows in Wisconsin streams (

Annual peak flow data were assessed for the presence of monotonic trends using the Mann-Kendall test (

The results of Mann-Kendall tests of trends for all streamgages are in

Although there is evidence that trends in annual peak streamflow happen at a regional scale in Wisconsin, a detailed analysis of the causal factors of these trends is beyond the scope of this report. Consistent with previous flood frequency reports in Wisconsin (

Flood frequency analyses for 299 streamgages in this study were computed with USGS software PeakFQ version 7.3 (

The EMA method was used to estimate the LP3 distribution for all streamgages used in this study. This method is described in Bulleting 17C (

The EMA represents each peak flow as an interval with a lower and upper bound which enables the method to incorporate data with various forms of uncertainty, such as censored data or historical floods. A flow interval represents the uncertainty associated with a peak flow. For most annual peak flows, the upper and lower bound are set at the reported peak flow value. Censored data occur when an annual peak flow is only known to be above or below some threshold value. This can happen at crest-stage gages when the annual peak flow is known to be below a minimum recordable value. Previously used flood frequency methods described in Bulletin 17B omitted such values because of their lack of precision and high uncertainty; however, the EMA method can incorporate this information by representing the censored peak streamflow as an interval that is bounded by zero and the streamflow associated with the elevation of the gage’s minimum recording threshold. By using interval data, the EMA method can use more years of data to fit the distribution while also accounting for the greater uncertainty in these censored values.

PILFs are values in the peak flow data that have lower magnitudes than other peak flows at the same location and which exert a high influence on the fit of the LP3 distribution. The physical processes that result in PILFs are often different than those that result in large floods, and including PILFs in the data when fitting the LP3 distribution can bias estimation of the largest floods (the lowest AEPs). Because these large floods are typically of more interest, it is recommended by Bulletin 17C that PILFs be identified and removed from the data to improve the fit of the distribution to the larger floods that correspond to smaller AEPs.

Previous flood frequency analyses in Wisconsin used the standard Grubbs-Beck test recommended by Bulletin 17B to identify PILFs. This test is adept at identifying a single PILF within a dataset but is unreliable when there are two or more PILFs. Bulletin 17C recommends the use of the Multiple Grubbs-Beck test, which is a generalization of the former test that is more sensitive to the presence of several PILFs.

Differences between estimated flood discharges in this report and the previously published report (

[USGS, U.S. Geological Survey; ft^{3}/s, cubic foot per second]

USGS streamgage number | Station name | Period of record (water years) | Maximum recorded annual peak streamflow through water year 2010 | Maximum recorded annual peak streamflow from 2011 to 2020 | ||

Water year | Peak streamflow (ft^{3}/s) |
Water year | Peak streamflow (ft^{3}/s) |
|||

04024430 | NEMADJI RIVER NEAR SOUTH SUPERIOR, WI | 1974–2020 | 2001 | 15,800 | 2011 | 33,000 |

04027200 | TWENTYMILE CREEK AT GRAND VIEW, WI | 1960–2020 | 1992 | 1,920 | 2016 | 9,000 |

04074850 | LILY RIVER NEAR LILY, WI | 1970–2020 | 2005 | 190 | 2020 | 419 |

05331833 | NAMEKAGON RIVER AT LEONARDS, WI | 1996–2020 | 2001 | 952 | 2016 | 2,680 |

05379288 | BRUCE VALLEY CREEK NEAR PLEASANTVILLE, WI | 1996–2017 | 2010 | 560 | 2016 | 2,080 |

05382200 | FRENCH CREEK NEAR ETTRICK, WI | 1960–2020 | 2001 | 2,950 | 2017 | 7,100 |

Regional regression equations were developed to estimate the magnitude of flood discharges for selected exceedance probabilities at ungaged streams in Wisconsin. Before regression equation development, flood frequency regions were delineated by which had similar basin characteristics. Regression equations were developed using multiple linear regression, which relates streamflows corresponding to various AEPs to basin characteristics by region.

Climatic and physiographic characteristics that affect the flood responses of streams vary widely across Wisconsin. Dividing the State into hydrologically similar regions can help increase the predictive accuracy of regression equations. Hydrologic similarity refers to the tendency for streamflow in two or more basins to respond similarly to a given rainfall event. Streamgages in regions that exhibit hydrologic similarity typically have similar basin characteristics and produce regression equations with higher accuracy. Previous studies have divided Wisconsin into five regions (

A clustering analysis was used to divide the State into homogeneous regions with respect to basin characteristics using a process outlined in

Clustering was performed with the K-means clustering algorithm using the factoextra package in R (

After selecting the final cluster results, regional boundaries were manually adjusted to prevent overlapping regions or to maintain region boundaries consistent with 8-digit or 12-digit hydrologic unit boundaries. Although it may be unavoidable for a large drainage basin to cross into more than one region, it is desirable to have regions that generally follow drainage basin boundaries to avoid a situation where a stream crosses in and out of the same region multiple times, which could result in inconsistent flood frequency estimates at different points along the stream.

Four final flood frequency regions were defined for Wisconsin (

Distributions of selected basin characteristics for four flood frequency regions in Wisconsin.

Relation between drainage area and 1-percent AEP flood discharge for four flood frequency regions in Wisconsin.

Regional regression equations were developed for estimating flood discharges corresponding to selected AEPs at ungaged locations on streams in Wisconsin. The development of regional regression equations includes two steps: (1) exploratory analysis, in which variables were transformed and the pool of potential explanatory variables was reduced; and (2) final model selection, in which the final models are selected and fit for all AEPs. During exploratory analyses, ordinary least squares (OLS) regression was used because of the ease of use and the availability of variable selection techniques for this regression method. For selection and fitting of the final models, generalized least squares (GLS) regression was used. GLS regression accounts for unequal record lengths as well as spatial correlation of concurrent flows at different streamgages and provides better estimates of the predictive accuracy of the regression equations (

During exploratory analysis, the relation between explanatory variables and the flood discharge corresponding to the 1-percent AEP was examined for linearity and variables were examined for potential multicollinearity. Scatterplot matrices of the log-transformed (base 10) flood discharges, log-transformed (base 10) explanatory variables, and untransformed explanatory variables were generated to evaluate whether log-transformation of the explanatory variables was needed and to check for correlation of the explanatory variables with flood discharge. Multicollinearity happens when explanatory variables used in a regression model are highly correlated with each other. Regression models that include variables with multicollinearity are unreliable because the regression equation coefficients and standard errors may be biased. The potential set of explanatory variables was reduced before the variable selection process such that no two explanatory variables had a Pearson’s R correlation coefficient greater than 0.6.

The variable selection process identifies the best subset of explanatory characteristics to use in a regression model. To minimize predictive inconsistencies between flood frequency estimates among different AEPs, variable selection analyses were performed using the 1-percent AEP flood discharge. After a final set of explanatory variables was chosen for a region, equations for the other AEPs were fit with GLS regression using the same set of explanatory variables. The best subsets method (from the ‘leaps’ R package; ^{2}) values. Candidate regression models were then evaluated based on maximizing the ^{2}, while minimizing the predicted residual sum of squares (PRESS) and Mallow’s Cp. Additionally, explanatory variables for each candidate model were assessed for statistical significance and multicollinearity. Multicollinearity was evaluated by computing the variance inflation factors. For this study, candidate models were eliminated from consideration if variance inflation factors were greater than 2 or if coefficients for any explanatory variables had p-values greater than 0.05.

The best three equations, as suggested by the best subsets OLS analysis, were refit and examined using GLS. Final GLS regional regression equations were selected based on minimizing the standard model error (SME), standard error of prediction (Sp), average variance of prediction (AVP), and the PRESS statistic while maximizing pseudo coefficient of variation (pseudo ^{2}) as well as visual assessments of fit and residuals. The performance metrics pseudo ^{2} and SME indicate how well the equations perform on the streamgages used in the regression analyses. The Sp, AVP, and PRESS statistic are measures of the accuracy with which GLS regression models can predict streamflows corresponding to various AEPs at ungaged sites. Regression models contain sampling error and model error. Sampling error refers to uncertainty in the flood frequency data used to derive the regression equation. Model error refers to errors stemming from uncertainty in the coefficients of the model equation. SME measures the error of the model itself and does not include sampling error. The Sp represents the sum of the model error and the sampling error. The AVP is a measure of the average accuracy of prediction for all sites used in the development of the regression model and assumes that the explanatory variables for the streamgages included in the regression analysis are representative of all streamgages in the region. The pseudo ^{2} is a measure of the percentage of the variation in annual peak streamflow explained by the variables included in the model.

Streamgages that were flagged by the WREG program as having large influence or leverage were further examined for elimination. Leverage is a measure of how much the values of explanatory variables at a streamgage vary from values of those variables at all other streamgages. Influence is a measure of how strongly the values for a streamgage affect the estimated regression parameters. Residual scatterplots were compared to fitted values and explanatory variables were examined to determine if flagged streamgages with large influence and leverage were isolated hydrologic outliers and could be removed from the analysis.

Streamgages that had high leverage, influence, or substantial lack of fit within the selected regression and had fewer than 15 years of annual peak streamflow records were removed from the dataset because of the high level of uncertainty in the estimates of the selected AEPs. Although a period of record of at least 10 years of peak flow data is recommended for flood frequency analysis (

The final regression equations and performance metrics are shown in ^{2} ranged from 80.0 to 95.0 percent, and the SME ranged from 38.1 to 66.9 percent. The regression equations presented here are valid for estimating the magnitude and frequency of floods at ungaged locations on streams in Wisconsin for which (1) the streamflow is not substantially altered because of urbanization or regulation and (2) the basin characteristics at the ungaged location are within the range of those used to develop the equations (

[RR, regression region; AEP, annual exceedance probability; ^{2}, coefficient of determination; Sp, standard error of prediction; SME, standard model error; AVP, average variance of prediction; %, percent;

RR | AEP | Equation | Pseudo ^{2} |
Sp | SME | AVP | ||||||

1 | 50% | log |
0.936 | 0.840 | −0.668 | −0.374 | −0.514 | 0.423 | 0.92 | 53.58 | 50.83 | 0.05 |

1 | 20% | log |
0.437 | 0.834 | −0.654 | −0.440 | −0.583 | 0.610 | 0.91 | 55.98 | 53.05 | 0.05 |

1 | 10% | log |
0.205 | 0.830 | −0.638 | −0.473 | −0.621 | 0.698 | 0.90 | 58.47 | 55.35 | 0.06 |

1 | 4% | log |
−0.018 | 0.825 | −0.616 | −0.507 | −0.663 | 0.784 | 0.89 | 61.00 | 57.64 | 0.06 |

1 | 2% | log |
−0.148 | 0.822 | −0.599 | −0.529 | −0.691 | 0.835 | 0.89 | 62.72 | 59.19 | 0.06 |

1 | 1% | log |
−0.252 | 0.819 | −0.582 | −0.547 | −0.717 | 0.877 | 0.88 | 65.24 | 61.50 | 0.07 |

1 | 0.50% | log |
−0.339 | 0.816 | −0.566 | −0.564 | −0.741 | 0.913 | 0.87 | 67.77 | 63.80 | 0.07 |

1 | 0.20% | log |
−0.433 | 0.812 | −0.544 | −0.583 | −0.770 | 0.953 | 0.86 | 71.16 | 66.90 | 0.08 |

2 | 50% | log |
−1.980 | 0.945 | −0.720 | 1.002 | −0.007 | −0.007 | 0.95 | 40.04 | 38.08 | 0.03 |

2 | 20% | log |
−2.172 | 0.939 | −0.770 | 1.121 | −0.007 | −0.008 | 0.94 | 42.88 | 40.74 | 0.03 |

2 | 10% | log |
−2.283 | 0.936 | −0.794 | 1.183 | −0.007 | −0.009 | 0.94 | 44.83 | 42.54 | 0.03 |

2 | 4% | log |
−2.405 | 0.933 | −0.819 | 1.249 | −0.007 | −0.009 | 0.93 | 47.73 | 45.21 | 0.04 |

2 | 2% | log |
−2.490 | 0.931 | −0.834 | 1.293 | −0.007 | −0.010 | 0.92 | 50.55 | 47.83 | 0.04 |

2 | 1% | log |
−2.567 | 0.929 | −0.848 | 1.332 | −0.007 | −0.010 | 0.92 | 53.35 | 50.42 | 0.05 |

2 | 0.50% | log |
−2.641 | 0.927 | −0.860 | 1.368 | −0.007 | −0.010 | 0.91 | 56.15 | 53.00 | 0.05 |

2 | 0.20% | log |
−2.732 | 0.925 | −0.874 | 1.412 | −0.007 | −0.011 | 0.90 | 59.88 | 56.45 | 0.06 |

3 | 50% | log |
1.625 | 0.785 | −0.476 | −0.013 | - | - | 0.92 | 44.60 | 42.75 | 0.03 |

3 | 20% | log |
1.882 | 0.772 | −0.536 | −0.014 | - | - | 0.91 | 46.05 | 44.10 | 0.04 |

3 | 10% | log |
2.012 | 0.766 | −0.565 | −0.015 | - | - | 0.90 | 48.19 | 46.10 | 0.04 |

3 | 4% | log |
2.148 | 0.762 | −0.595 | −0.016 | - | - | 0.88 | 51.60 | 49.28 | 0.04 |

3 | 2% | log |
2.234 | 0.759 | −0.614 | −0.017 | - | - | 0.87 | 54.49 | 51.98 | 0.05 |

3 | 1% | log |
2.310 | 0.757 | −0.630 | −0.017 | - | - | 0.86 | 57.39 | 54.69 | 0.05 |

3 | 0.50% | log |
2.379 | 0.756 | −0.644 | −0.018 | - | - | 0.85 | 60.33 | 57.43 | 0.06 |

3 | 0.20% | log |
2.461 | 0.754 | −0.661 | −0.019 | - | - | 0.83 | 65.00 | 61.80 | 0.07 |

4 | 50% | log |
2.365 | 0.623 | −0.420 | - | - | - | 0.86 | 54.80 | 52.90 | 0.05 |

4 | 20% | log |
2.892 | 0.585 | −0.560 | - | - | - | 0.87 | 49.92 | 48.08 | 0.04 |

4 | 10% | log |
3.168 | 0.565 | −0.637 | - | - | - | 0.87 | 49.48 | 47.55 | 0.04 |

4 | 4% | log |
3.459 | 0.546 | −0.720 | - | - | - | 0.86 | 51.56 | 49.44 | 0.04 |

4 | 2% | log |
3.645 | 0.534 | −0.773 | - | - | - | 0.85 | 53.41 | 51.13 | 0.05 |

4 | 1% | log |
3.810 | 0.525 | −0.822 | - | - | - | 0.83 | 56.44 | 53.96 | 0.05 |

4 | 0.50% | log |
3.959 | 0.517 | −0.866 | - | - | - | 0.82 | 59.15 | 56.49 | 0.06 |

4 | 0.20% | log |
4.138 | 0.508 | −0.920 | - | - | - | 0.80 | 63.83 | 60.90 | 0.06 |

[

Basin characteristic | Region 1, |
Region 2, |
Region 3, |
Region 4, |
||||

Minimum | Maximum | Minimum | Maximum | Minimum | Maximum | Minimum | Maximum | |

Drainage area, in square miles | 0.58 | 2,085.76 | 0.83 | 2,243.41 | 1.01 | 5,994.64 | 0.30 | 1,034.01 |

Percent forest | - | - | 8.60 | 91.20 | - | - | - | - |

Maximum 24-hour precipitation with a 10-year recurrance interval, in inches | 3.61 | 4.17 | 3.40 | 4.00 | - | - | - | - |

Percent herbaceous upland | 0 | 25.16 | - | - | - | - | - | - |

Percent open water | 0 | 9.20 | 0 | 25.62 | 0 | 9.75 | - | - |

Hydraulic conductivity, in micrometers per second | 8.95 | 74.16 | - | - | - | - | 6.90 | 62.34 |

Percent wetland | - | - | 0.58 | 59.58 | 0 | 29.57 | - | - |

Flood discharges corresponding to the,

The accuracy and uncertainty of the regression equations is affected by imprecision, inaccuracies, or incomplete data in the basin characteristics and the estimates of selected AEPs at gaged sites on which the equations are based. Regression equations are a simplification of actual physical processes and may not adequately represent the physical flood dynamics in all cases; for example, the effect of lakes and wetlands on streamflow in regions 1–3 depends on their size and their location within the drainage network. A simple percentage of drainage area that is open water used in these regression equations may not always adequately account for the spatial and hydrologic connectivity of these basin characteristics.

Uncertainty in the regression equations also is affected by uncertainty in the LP3 flood discharge estimates at the streamgages used in the regression analysis. Bias in flood discharge estimates can result from short periods of record that do not adequately cover the full range of long-term climatic conditions at a stream or periods of record in which there is a monotonic trend. Uncertainty in estimated flood discharges resulting from the presence of a trend in peak streamflow propagates into the regression equations and adds additional uncertainty and, potentially, bias. Although trends in annual peak flows were detected at some locations, the causal attribution of those trends and development of an appropriate method for adjusting the estimated AEP discharge at locations with trends was beyond the scope of this study.

The equations in region 4 show some bias toward overestimating 1-percent AEP flood frequencies for smaller drainage basins with predicted flood discharge of 1,000 cubic feet per second (ft^{3}/s) or less and underestimating moderately sized basins with predicted flood discharge around 10,000 ft^{3}/s (^{3}/s; however, if there were only 20 years of data available at this stream, the resulting 1-percent LP3 estimated flood discharge could be as low as 8,700 ft^{3}/s (using data from 1975 through 1995) or as high as 24,400 ft^{3}/s (using data from 1914 through 1934). This issue is most prevalent in small- and medium-sized basins in region 4, which also have the shortest record lengths.

Flood frequency estimates for the 1-percent annual exceedance probability per unit drainage area for streamgages with different periods of record for four regions in Wisconsin.

Annual peak streamflow at U.S. Geological Survey streamgage 05436500, Sugar River near Brodhead, WI, from water year 1940 through 2020.

The goodness-of-fit metrics reported in

is the estimated flood discharge for a given AEP at an ungaged location predicted from a regression equation; and

is computed as:

is the critical value from a student’s t-distribution for an alpha level (

is the standard error of prediction for ungaged site

is the model error variance,

is a row vector of basin characteristics, starting with 1 as a placeholder for the intercept term, for ungaged site

is the covariance matrix for the regression coefficients, and

is the matrix transpose of _{i}

The flood frequency estimation methods presented in this study can be applied to three types of rural, unregulated sites. The first case is at a streamgage location; for this case, flood discharge estimates from the LP3 distribution and the regression equations can be combined to for a weighted flood discharge estimate. The second case is at an ungaged location near a streamgage; in this case, the estimated flood discharge at the location of interest is weighted with the estimated flood discharge at the streamgage using the ratio of the two drainage areas. The third case is at an ungaged location that is not near a streamgage; in this case, the regression equation is used to estimate the flood discharge. For each of these three cases, a description of the appropriate method and an example are presented.

Two estimates of flood discharge for a streamgage are available: one from the at-site log-Pearson Type III frequency curve and the other from the appropriate regional regression equation developed in this study. A theoretically improved estimate of flood discharge can be calculated if the individual estimates are assumed to be independent and the variances of the individual estimates are known. If the independent estimates of flood discharge are weighted inversely proportional to their respective variances, then the variance of the weighted-average estimate will be less than the variances associated with each individual estimate (

For a particular AEP, the variance of prediction from the log-Pearson Type III analysis at a streamgage (_{P}_{(}_{g}_{)}_{s}_{P}_{(}_{g}_{)}_{s}

The variances of prediction for flood discharges estimated using the regional regression equations (_{P}_{(}_{g}_{)}_{r}

Using the variances from the two independent estimates of flood discharge, the weighted-average estimate of flood discharge is computed using the following equation (_{P}_{(}_{g}_{)}_{w}

is the weighted flood discharge estimate for a ^{3}/s;

is the flood discharge estimate for a ^{3}/s;

is the flood discharge estimate for the ^{3}/s;

is the variance of prediction of a flood discharge estimate for the

is the variance of prediction of a flood discharge estimate for the

is the variance of prediction for a weighted flood discharge estimate for the

is the 90-percent confidence interval of a weighted flood discharge estimate for a ^{3}/s.

This example illustrates the calculation of a weighted estimate of flood discharge corresponding to the 1-percent AEP (_{1%}) for streamgage USGS 04086200, East Branch Milwaukee River at Kewaskum, WI (map number 80, _{P}_{(}_{g}_{)}_{s}^{3}/s (_{P}_{(}_{g}_{)}_{s}_{P}_{(}_{g}_{)}_{r}_{P}_{(}_{g}_{)}_{r}

For an ungaged location on a gaged stream with 10 or more years of annual peak flow record, the flood discharge estimate from the appropriate regional regression equation can be combined with the weighted-average flood discharge estimate, _{P}_{(}_{g}_{)}_{w}_{P}_{(}_{g}_{)}_{r}

is the regression-weighted estimate of flood discharge for the ^{3}/s;

is the weighted-average flood discharge estimate for the ^{3}/s;

is the flood discharge estimate for the ^{3}/s;

is the flood discharge estimate for the ^{3}/s;

is the drainage area associated with the streamgage, in square miles; and

is the drainage area associated with the ungaged location, in square miles.

If the drainage area associated with the ungaged location is between 50 and 150 percent of the drainage area associated with the streamgage,

This example illustrates the calculation of a regression-weighted estimate for the 1-percent AEP flood discharge for a stream in region 3 that is directly upstream of streamgage 04086200, East Branch Milwaukee River at Kewaskum, Wis. (map number 80, ^{2}, percentage of open water = 2.49 percent, and percentage of wetlands = 20.97 percent. The drainage area for the upstream gage (_{g}^{2}. The weighted average estimate of the 1-percent AEP flood discharge for the upstream gage (_{P}_{(}_{g}_{)}_{w}_{P}_{(}_{g}_{)}_{r}^{3}/s and 740 ft^{3}/s, respectively (see section “Example 1”). The regression estimate for the 1-percent AEP flood discharge for the ungaged location (_{P}_{(}_{u}_{)}_{r}^{3}/s, can be computed using the basin characteristics at the ungaged location and the appropriate equation from

Finally, the regression-weighted estimate of flood discharge for the 1-percent AEP at the ungaged location (in ft^{3}/s) can be computed using

Flood discharge estimates at ungaged locations that are not near a streamgage are calculated using the appropriate regional regression equations from

This example illustrates the calculation of the 1-percent AEP flood discharge and 90-percent confidence intervals at an ungaged stream in Wisconsin. For this example, an ungaged location was selected in flood frequency region 3 and basin characteristics were computed with the StreamStats web application (^{2}, 0 percent open water, and 6.44 percent wetlands. First, the estimate of the 1-percent AEP flood discharge is estimated using the appropriate equation in

Next, the 90-percent confidence interval for the estimate is computed using

Compute the vector (_{i}

Find the covariance matrix for the regression coefficients (

Variable
Intercept
Intercept
0.00549
−0.00008
−0.00065
−0.00167
−0.00008
0.00002
0.00004
−0.00010
−0.00065
0.00004
0.01713
−0.00289
−0.00167
−0.00010
−0.00289
0.00204

To compute the _{i}_{i}U_{i}U

Obtain the model error variance (_{p}_{,}_{i}

Compute

The 90-percent prediction interval is computed from

The USGS StreamStats web application incorporates the new peak flow frequency regression equations for Wisconsin and provides flood discharge estimates for unregulated streams in the basin (

This study updates the regional regression equations that are used to estimate the magnitude of annual peak streamflows corresponding to the 50-, 20-, 10-, 4-, 2-, 1-, 0.5-, and 0.2-percent annual exceedance probabilities for nonurbanized, unregulated streams in Wisconsin. Estimates of flood discharge were computed at 299 streamgages in Wisconsin using the expected moments algorithm (EMA) to fit a log-Pearson type III frequency distribution and regional skew values that were developed previously. The EMA method addresses several methodological concerns identified with the previous procedures for determining flood frequency outlined in Bulletin 17B. Specifically, the EMA method can accommodate censored values, which are common at crest-stage gages, and has improved statistical treatment of potentially influential low floods.

A cluster analysis, using basin characteristics at streamgage locations, was used to delineate four new flood frequency regions in Wisconsin. The flood frequency regions developed from the clustering analysis were affected by north-south gradients of precipitation, snowfall, and patterns in land cover such as percent forest, wetlands, and open water and divide Wisconsin roughly into central, northern, southeastern, and southwestern regions. Regions were selected such that homogeneity of basin characteristics within the groups was maximized while retaining a minimum of at least 40 streamgages in each region.

Regression equations were developed for each flood frequency region by relating basin characteristics at streamgages in the region to the log-Pearson Type III distribution flood discharge estimates using generalized least-squares regression. Redundancy and trend analyses were performed to identify and remove streamages from the analysis that may violate assumptions of the GLS regression. Basin characteristics that were statistically significant in the equations included drainage area (

Director, USGS Upper Midwest Water Science Center

1 Gifford Pinchot Drive

Madison, WI 53726

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