Revision History for U.S. Geological Survey Scientific Investigations Report 2016–5050
By Thomas Over, Riki Saito, Andrea Veilleux, Jennifer Sharpe, David Soong, and
Audrey Ishii
Estimation of Peak Discharge Quantiles for Selected Annual Exceedance Probabilities in Northeastern Illinois
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Version 1.0: Posted online June 28, 2016
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Version 2.0: Revised and reposted November 27, 2017
The text version was modified.
The covers, title page, and the back of the title page were revised to show the new version number for SIR 2016-5050. Two graphs were removed from the front cover.
Subsequent to publication of SIR 2016–5050, an error was discovered in the reported methodology. It was discovered that the at-site flood frequencies computed using so-called “weighted” skew coefficient values were not actually computed using the weighted skew; that is, as a weighted average of the station and regional skew values. Instead, the flood frequencies were computed using the regional skew only.
The preferred skew coefficients to use for computing at-site flood frequencies are the weighted skews; however, it was determined that the flood frequencies obtained using the regional skew have only modest differences from those obtained using the weighted skews. As a result, it was decided to correct the report to reflect the methodology actually implemented. An update of the analysis and report are planned, which will use the weighted skew values and include additional years of record.
The following revisions to SIR 2016–5050 reflect what was actually done in the analysis and correct a few typographical and grammatical errors.
Corrections related to skewness coefficients:
On p. 1, in the second paragraph of the abstract, "...generalized skew coefficients..." was changed to "...regional skew coefficients..."
On p. 8, in the first paragraph in the left column, "...that is weighted with the at-streamgage skewness values in the frequency analysis...” was revised to read, "...for use in the frequency analysis..."
On p. 12, in the last sentence that begins in the left column, "The skewness estimate from this relation was combined with the at-site skew according to their associated uncertainties to obtain a weighted skew estimate for use in obtaining the at-site peak discharge quantile estimation..." was revised to read, "The regional skewness estimate from equation 3, without weighting with the at-site skewness, was used to obtain at-site peak discharge quantile estimates..."
On p. 12, in the last paragraph, including equation 4 and following "where" entries (now deleted), the entire paragraph was revised to read, "For use in the frequency analysis, the skewness values for each streamgage were taken as equal to the regional skewness values developed as described in the “Regional Skew Analysis” section and appendix 1."
On p. 22, in the first paragraph, "...including weighting with the regional skew model...” was revised to read, "...including application of the regional skew model..."
On p. 22, in the first "where" paragraph after equation 8 (formerly eq. 9), "...with weighted skew in cubic..." was revised to read, "...with regional skew in cubic..."
On p. 33, in the second "where" paragraph after equation 16 (formerly eq. 17), "...by using EMA with weighted skew as..." was revised to read, "...by using EMA with regional skew as..."
On p. 33, in the first paragraph after the "where" paragraphs after equation 16 (formerly eq. 17), '…in the rows labeled "EMA (weighted skew"), "Regional regression", and "Weighted EMA (weighted skew) and regional regression", respectively.' has been revised to read "…in the third, fourth, and fifth rows, respectively, for each streamgage."
On p. 39, in the left column, in the first complete paragraph, "To improve the skew estimates used for the peak discharge quantile estimation, weighted skew coefficients were computed as the variance-weighted average of at-site and generalized skew coefficients. The generalized skew coefficients..." was revised to read, "For the peak discharge quantile estimation, regional skew coefficients were used in place of the at-site skew. The regional skew coefficients..."
On various pages, equation numbers and references to equations 5 and larger have been reduced by 1 because of the deletion of equation 4.
Downloadable file of table 2, in the third of five rows for each streamgage, in the "Frequency analysis method" column, "At-site, adjusted for urbanization, EMA, weighted skew: (Qp)g,s" has been revised to read, "At-site, adjusted for urbanization, EMA, regional skew: (Qp)g,s"
Corrections of typographical and grammatical errors:
On title page and p. 1, middle initials were added with author names.
On p. x, water year definition was added.
On p. 2, first paragraph, "whereas for simulation models, regional calibration is a 'difficult, unsolved problem' (Vogel, 2006)." has been revised to read, "whereas significant challenges remain for regional calibration of simulation models (Vogel, 2006)."
On p. 2, second paragraph under "Description of Study Area" heading, "...vegetation type, primarily alfisols, which form under forest, nearer to Lake Michigan and mollisols, which form under grasslands, over..." has been revised to read, "...vegetation type. Alfisols, which form under forests, dominate nearer to Lake Michigan, whereas mollisols, which form under grasslands, cover..."
On p. 7, added a reference for the quote in the second paragraph.
On p. 8, in the x-axis for figure 4, "Fraction of basins" has been revised to read, "Urbanized fraction (Theobald, 2005, classes 7–10) of basin at end of record used in Soong and others (2004)."
On p. 8, in the y-axis for figure 4, "Empirical distribution function of fraction of Theobald (2005) classes 7-10 at end of the streamgage record" has been revised to read, "Fraction of basins having urbanized fraction less than value on x axis."
On p. 8, in the caption for figure 4, "...(from Soong and others, 2004)" has been revised to read, "...of Soong and others (2004)."
On p. 9, in the first line of text, "The list streamgages with at least 10 years of record..." has been revised to read, "The list of streamgages with at least 10 years of record..."
On p. 11, in the last line of the caption for figure 6, "greater than...” has been revised to read, “fewer than..."
On p. 14, added a reference for the National Inventory of Dams in the first full paragraph of the second column.
On p. 20, in the middle line of the caption for figure 10, "...urban fraction with least squares with quantile regression..." has been revised to read,"...urban fraction with quantile regression..."
On p. 47, table 1-3, fixed several values in the "Average regional skew" column
Throughout the report:
Corrected capitalization and formatting of text and headings, and the use of punctuation, where applicable.
Corrected some usage of abbreviations to ensure they were defined at first use and used thereafter (and redefined for every figure and table).
URLs were changed to "https," where applicable, and DOI URLs were changed from "http://dx.doi.org/..." to "https://doi.org/..."
Where necessary, changed singular verbs referring to "data" to plural verbs because "data" is the plural of "datum."
Corrected out-of-order figure callouts by adding callouts to sections in the report where the figures are located.
In parentheses, changed "equation" to "eq." and "figure" to "fig."
Downloadable file of table 2, column K in the group of columns with the header "Discharge for a given annual exceedance probability, ft3/s," the subheader "0.01" has been revised to read "0.005"
Downloadable file of table 13, changed "DEM_P_1_0" to "DEM_1_0_P"
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Version 3.0: Revised and reposted June 30, 2021
The text version was modified.
The covers, title page, and the back of the title page were revised to show the new version number for SIR 2016-5050 and to add Padraic S. O’Shea as a report author.
This version of the SIR revises the report to use weighted skew coefficients, to update the perception thresholds for the at-site flood frequency computations, and to implement updates to the software package WREG for regional regression analysis. At-site and regional flood frequencies have been recalculated. The following revisions to SIR 2016–5050 reflect these new calculations and wording to indicate the use of weighted skew.
Corrections to figures:
On p. 32, Figure 14 was replaced with an updated figure to reflect new peak discharge quantiles.
Corrections in the manuscript text:
On p. 1, in the second paragraph of the abstract, the following sentence was added: “The skew coefficient values for each streamgage were then computed as the variance-weighted average of at-site and regional skew coefficients.”
On p. 8, the first paragraph of the left column, “…for use in the frequency analysis” was replaced with “that is weighted with the at-streamgage skewness values in the frequency analysis”
On p. 9, in the final paragraph of the right column, values were changed from “…117 nonredundant streamgages, of which 56 were CSGs throughout their peak discharge record, 10 were CSGs during part of their peak discharge record, and 51 had continuous record throughout their peak discharge records” to “117 nonredundant streamgages, of which 57 were CSGs throughout their peak discharge record, 13 were CSGs during part of their peak discharge record, and 47 had continuous record throughout their peak discharge records.”
On p. 12, in the final paragraph of the left column and the top of the right column, “…without weighting with the at-site skewnesss, was used to obtain at-site discharge quantile estimates…” was replaced with “…was combined with the at-site skewness according to their associated uncertainties to obtain a weighted skew estimate for use in computing the at-site peak discharge quantile estimates…”
On p. 12, in the first paragraph under Frequency Analysis, “remove” was replaced with “censor.”
On p. 12, in the first paragraph under Frequency Analysis, “PeakFQ version 7.1” was replaced with “PeakFQ version 7.3, https://water.usgs.gov/software/PeakFQ/”
On p. 12, the second paragraph under Frequency Analysis was replaced with the following text and equation:
“As part of the frequency analysis, the skewness values for each streamgage were computed as a weighted average Y^_W of the station and regional skew values following the method suggested in Bulletin 17B (U.S. Interagency Advisory Com¬mittee on Water Data, 1982):
y^_W = (AVP_new * y^_S + MSE[y^_S] * y^_R) / (AVP_new + MSE[y^_S]), eq.(4)
where
y^_W is the weighted skewness,
y^_R is the regional skewness computed as described in the “Regional Skew Analysis” section and in appendix 1,
y^_S is the station skewness computed by using the EMA approach as implemented in PeakFQ for each streamgage,
AVP_new = 0.19 is the average variance of prediction at a new site computed as described in appendix 1, and
MSE[y^_S] is the mean square error of the at-site skewness, which also is computed by using the EMA approach as implemented in PeakFQ.”
**As a result of adding this equation to the document, all subsequent equation numbers and in-text references are increased by one for a total of 32 equations.**
On p. 18, in the last paragraph of the right column, “(table 8)” was added after “exceedance probability of 0.002”
On p. 22, in the first paragraph under the ‘Regional Spatial Regression Analyses’ section, “…including application for the regional skew model developed for this study.” was replaced with “including weighting with the regional skew model.”
On p. 22, in the first paragraph under the “Selection of Basin Characteristics” section after “area always included,” the following sentence was added: “For these exploratory regressions, a preliminary set of peak-flow quantiles developed using regional skew were used.”
On p. 27, in the first paragraph of the right column, “…the computer program WREG (Eng and others, 2009), version 1.05, which was downloaded from https://water.usgs.gov/software/WREG/.” was replaced with “…..the function WREG.GLS from the package WREG (Farmer, 2017) in R version 3.6.3 (R Core Team, 2020).”
On p. 27, in the first paragraph of the right column, “WREG” was replaced with “WREG.GLS”
On p. 27, in the first paragraph of the right column after “distribution using techniques”, the following text was added: “…described in Eng and others (2009)…”
On p. 28, in the where list for now-equation 11, the variable definition for Q_p was changed to: “is the peak discharge quantile with weighted skew, in cubic feet per second, with annual exceedance probability p,”
On page 28, for Table 11 the coefficients and associated standard error for all "Annual exceedance probability" rows have been revised.
On page 29, for Table 12 the "Pseudo R^2 (percent)", "Average variance of prediction (log units)", "Average standard error of prediction (percent)", "Model error variance Y^2 (log units)", "Standard model error (percent)", and "Standard model error (percent), WLS fit" columns for all "Annual exceedance probability" rows have been revised.
On p. 33, in the where list for now-equation 17 for the variable definition for (Q_p)_g,s, “regional” was changed to “weighted”
On p. 35, now-equation 24 was updated to the following:
Q_0.01 = 10^((b_0)_0.01) A^((b_A)_0.01 ) 10^((b_U)_0.01 U^(1/2)) 10^((b_w)_0.01 W^(1/2)) S^((b_S)_0.01) eq.(24)
Q_0.01 = 10^1.968 50^0.771 10^(0.207 * 0.20^(1/2)) 10^(-1.025 * 0.15^(1/2)) 3.67^0.494
Q_0.01 = 1790 ft3/s,
On p. 35, in the paragraph above table 15, “y^2 = 0.0527” was replaced with “y^2 = 0.0528”
On p. 36, the two matrix values were replaced with the following:
“and the matrix (X^t L^(-1) X)^(-1) = 0.001 [(31.003, -8.3921, -7.6988, -4.5311, -20.598),]
(-8.3921, 3.2522, 1.1394, -2.6149, 5.9206),]
(-7.6988, 1.1394, 10.542, -1.2200, 2.4035),]
(-4.5311, -2.6149, -1.2200, 40.324, -1.2044),]
(-20.598, 5.9206, 2.4035, -1.2044, 18.346)]
….
0.001 [1,1.699,0.4472,0.3873,0.5647] [(31.003, -8.3921, -7.6988, -4.5311, -20.598),] [1,]
(-8.3921, 3.2522, 1.1394, -2.6149, 5.9206),] [1.699,]
(-7.6988, 1.1394, 10.542, -1.2200, 2.4035),] [0.4472,]
(-4.5311, -2.6149, -1.2200, 40.324, -1.2044),] [0.3873,]
(-20.598, 5.9206, 2.4035, -1.2044, 18.346)] [0.5647]
= 0.00214.
Therefore V_i = 0.0528 + 0.00214 = 0.0549.
On p. 36, now-equation 25 was replaced with:
S_pi = 100 {exp[(ln10)^2 V_i ] -1}^(1/2) eq.(25)
S_pi = 100 {exp[(ln10)^2 0.0549] -1}^(1/2)
S_pi = 58.1 percent.On p. 36, now-equation 26 was replaced with:
log_10(Q_i) ± t_(a/2,n-p) S_i = log_10(1790) ± 1.659*0.234
log_10(Q_i) ± t_(a/2,n-p) S_i=[2.864,3.641]
On p. 36, now-equation 27 was replaced with:
[Q_i 10^(-t_(a/2,n-p) S_i ),Q_i 10^(t_(a/2,n-p) S_i ) ]=[1790*10^(-1.659*0.234),1790*10^(1.659*0.234) ] eq.(27)
[Q_i 10^(-t_(a/2,n-p) S_i ),Q_i 10^(t_(a/2,n-p) S_i ) ]=10^[2.864,3.641]
[Q_i 10^(-t_(a/2,n-p) S_i ),Q_i 10^(t_(a/2,n-p) S_i ) ]=[730,4,380]
On p. 37, in the paragraph above Example 2: Ungaged Location in the Study Region, Near a Streamgage “….is 1,890 ft3/s with a standard error of prediction of 58.0 percent and a 90-percent confidence interval of [772, 4,610] ft3/s.” was changed to “…is 1,790 ft3/s with a standard error of prediction of 58.1 percent and a 90-percent confidence interval of [730, 4,380] ft3/s.”
On p. 37, in the first paragraph under Example 2: Ungaged Location in the Study Region, Near a Streamgage, “…at streamgage 05527950 is 2,160 ft3/s, and as previously computed the regional regression estimate at the ungaged site of interest (Q_0.01)_u,r is 1,890 ft3/s.” was replaced with “…at streamgage 05527950 is 2,050 ft3/s, and as previously computed the regional regression estimate at the ungaged site of interest (Q_0.01)_u,r is 1.790 ft3/s.”
On p. 37, now-equation 28 was replace with:
(Q_0.01 )_(u,w) = (0.330*1,790) + (0.670*2,050*0.835) eq.(28)
(Q_0.01 )_(u,w) = 1,740 ft3/s
On p. 37, below now-equation 28, “…is 1,830 ft3/s, compared to 1,890 ft3/s without…” was replaced with “…is 1,740 ft3/s, compared to 1,790 ft3/s without…”
On p. 37, in the final paragraph of the left column, “2,160 ft3/s.” was replaced with “2,050 ft3/s.”
On p. 37, now-equation 30 was replaced with:
V_U = 0.0549 + 0.134^2 (0.4 - 0.0)^2 eq.(30)
V_U = 0.05777,
On p. 38, in the first paragraph of the left column, “S_U = (V_U)^1/2 = 0.242.” was replaced with “S_U = (V_U)^1/2 = 0.240.”
On p. 38, now-equation 31 was replaced with:
S_pU = 100 {exp[(ln10)^2 0.05777] -1}^(1/2)
S_pU = 59.9 percent.
On p. 38, under now-equation 31, “…log10 units is [3.0275, 3.8241] and in cubic feet per second, [1,070, 6,670].” was replaced with “…log10 units is [3.0269, 3.8247] and in cubic feet per second, [1,060, 6,680].”
On p. 38, in the paragraph above Example 5: Adjustment for Effects of Future Urbanization, “…prediction of 59.8 percent and a 90-percent confidence interval of [1,070, 6,670] ft3/s.” was replace with “…prediction of 59.9 percent and a 90-percent confidence interval of [1,060, 6,680] ft3/s.”
On p. 38, now-equation 32 was replaced with:
Q_0.01(U) = 1,790 * 10^(0.312*(1.0 - 0.4) )
Q_0.01(U) = 1,790 * 1.539
Q_0.01(U) = 2,750 ft3/s,
where
Q_0.01(U_0) = 1,790
On p. 38, in the first paragraph below now-equation 32, “…is 61.9 percent, and the 90-percent confidence interval is [3.0275, 3.8241] in log_10 units and [1,130, 7,480]…” was replaced with“…is 62.0 percent, and the 90-percent confidence interval is [3.0286, 3.8507] in log_10 units and [1,170, 7,090]…”
On p. 38, in the second paragraph below now-equation 32, “1,890 ft3/s to 2,910 ft3/s” was replaced with “1,790 ft3/s to 2,750 ft3/s”
On p. 38, in the second paragraph below now-equation 32, “…a 90-percent confidence interval of [1,130, 7,480]…” was replaced with “…a 90-percent confidence interval of [1,070, 7,090]…”
On p. 39, in the first full paragraph in the left column, “removal” was replaced with “the censoring.”
On p. 39, in the first full paragraph in the left column, “defined” was replaced with “identified”
On p. 39, in the first full paragraph in the left column, “For the peak discharge quantile estimation, regional skew coefficients were used in place of the at-site skew.” was replaced with “To improve the skew estimates used for the peak discharge quantile estimation, weighted skew coefficients were computed as the variance-weighted average of at-site and regional skew coefficients.”
In the references cited of the main text, entries for Farmer, 2017 and R Core Team, 2020 were added.
On p. 46, in the first paragraph of the right column, “0.019” was replaced with “0.19”
Downloadable file of table 1, column H with the header "Crest-stage gage (CSG)" has been revised to "Partial" for U.S. Geological Survey streamgage number (column A) 05438250, 05519500, 05528000, 05542000, and 05549000; and to "Yes" for 05551030.
Downloadable file of table 2, column D (“Frequency analysis method”), third of five rows for each streamgage, was revised from “At-site, adjusted for urbanization, EMA, regional skew: (Qp)g,s” to “At-site, adjusted for urbanization, EMA, weighted skew: (Qp)g,s”.
Downloadable file of table 2, columns E-L in the group of columns with the header "Discharge for a given annual exceedance probability, ft3/s", have been revised for each "USGS streamgage number" (column A) and all "Frequency analysis method" (column D) rows as follows:
- The first two rows (“At-site unadjusted for urbanization, EMA, at-site skew” and “At-site, adjusted for urbanization, EMA, at-site skew”) were updated for any sites which included crest-stage gage years of operation.
- The third rows (“At-site, adjusted for urbanization, EMA, weighted skew: (Qp)g,s”) were updated based on the use of weighted skew, instead of regional skew.
- The fourth rows (“Regional regression: (Qp)g,r”) were updated for all sites based on the use of the updated WREG software.
- The fifth rows (“(Qp)g,w: Weighted at-site (Qp)g,s and regional (Qp)g,r”) are the weighted average of the third and fourth rows and were also updated.
Downloadable file of table 13, columns D-H with the headers "Intercept", "log10(Drainage area (mi^2))", NLCD_22_23_24^(1/2)", "DrainageClass1a^(1/2)", and "log10(DEM_1_0_P (feet per mile))" have been revised for all rows of each instance of "Exceedance probability" (column A).