Regional Analysis of the Dependence of Peak-Flow Quantiles on Climate with Application to Adjustment to Climate Trends
Abstract
1. Introduction
2. Materials and Methods
2.1. Single-Station Regression
2.2. Panel–Quantile Regression
- Compute the estimate of the location model coefficient by regressing the time-demeaned observations , where is the number of observations at the ith basin, on the time-demeaned explanatory (climate) variables, by least squares. Time-demeaning eliminates the location effects and the differences in means between the demeaned variables, so the regression only considers the variations “within” the data associated with each individual (i.e., station) i as they vary through time to compute .
- Compute the estimates of the location effects as , and define the residuals , which estimate , according to Equation (9).
- 3.
- Compute the estimate of the scale model coefficient by regressing the time-demeaned absolute residuals on the time-demeaned explanatory variables by least squares.
- 4.
- Compute the estimates of the scale effects as .
2.3. Goodness-of-Fit Statistics
2.4. Computation of Temporal Trends
2.5. Station Selection and Peak-Flow Data
2.6. Computation of the Climate Predictors
3. Results
3.1. Temporal Trends
3.2. Goodness of Fit and the Selection of Climate Predictors and Regression Transformations
3.3. Regression Coefficients and Fixed Effects
3.4. Goodness of Fit and Comparison of Modeling Approaches
3.5. Results Summary
4. Applications
4.1. General Considerations
4.2. Estimation via Adjustment
4.2.1. Method and Study-Wide Results
- Because the climate coefficients depend on , the next step is to estimate the value associated with each observed peak. This was achieved by interpolating among the conditional quantiles (Equation (5) for the single-station regressions and Equation (10) for the MMQR model); this interpolation was performed linearly in the space of log-transformed peaks versus the unit Gaussian quantile function with as the argument (equivalent to a set of straight-line segments on lognormal probability paper).
- Given the estimated value for each peak, estimate the coefficient value by interpolation among the fitted quantile coefficients and their values, which are given by (Equations (5) and (7)) for the single-station regressions and for the MMQR model (Equation (11)). This interpolation was performed linearly. Both interpolations are limited by the range of values, which are from to for the MMQR regressions, which is sufficient for all but the most extreme peaks, and from to for the single-station regressions. When the estimated value was beyond the most extreme value for which a coefficient estimate was available, the value at the nearest extreme was used.
- Given the coefficient value , the adjusted peak was computed as:
4.2.2. Example Basin: Vermilion River near Danville, Illinois
4.2.3. Example Basin: Sugar River near Brodhead, Wisconsin
5. Discussion and Conclusions
Supplementary Materials
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
Appendix A
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Quantile | Years of Record | Drainage Area (sq. km.) | Mean Annual Precipitation (MAP) (mm) | Aridity = PET/MAP | Mean Elevation (m) |
---|---|---|---|---|---|
0% (min.) | 38 | 28.0 | 317 | 0.249 | 145 |
5% | 45 | 107 | 419 | 0.559 | 197 |
10% | 47 | 163 | 496 | 0.657 | 217 |
25% | 48 | 485 | 649 | 0.727 | 274 |
50% | 69 | 1116 | 824 | 0.791 | 349 |
75% | 73 | 2501 | 913 | 0.917 | 503 |
90% | 93.1 | 4688 | 1062 | 1.281 | 1661 |
95% | 97 | 6808 | 1123 | 1.497 | 2182 |
100% (max.) | 98 | 38,732 | 1755 | 1.964 | 3137 |
Parameter Name | Definition | Range of Values Considered in This Study | Minimum Calibrated Value | Median Calibrated Value | Maximum Calibrated Value |
---|---|---|---|---|---|
Train | Temperature above which all precipitation is rain | 0 to 10 degC | 0 | 4.26 | 10 |
Tsnow | Temperature below which all precipitation is snow | −10 to 0 degC | −10 | −1.60 | 0 |
Melt factor | Upper bound of fraction of snow storage that can melt each day | 0.01 to 0.2 | 0.01 | 0.154 | 0.2 |
STC | Soil water storage capacity | 20 to 500 mm | 20 | 329 | 500 |
Drofrac | Fraction of rain that runs off directly, i.e., during the same day as it occurs. | 0 to 0.50 | 0 | 0.0132 | 0.282 |
Rofrac | Fraction of excess soil moisture that runs off each day. | 0.01 to 0.3 | 0.01 | 0.0851 | 0.3 |
Peak Flow vs. Water Year| trend ≤ 0 | Peak Flow vs. Water Year| trend > 0 | ann_max_Q vs. Water Year| trend ≤ 0 | ann_max_Q vs. Water Year| trend > 0 | ann_max_prcp vs. Water Year| trend ≤ 0 | ann_max_prcp vs. Water Year| trend > 0 | ann_tot_Q vs. Water Year| trend ≤ 0 | ann_tot_Q vs. Water Year| trend > 0 | ann_tot_prcp vs. Water Year| trend ≤ 0 | ann_tot_prcp vs. Water Year| trend > 0 | |
---|---|---|---|---|---|---|---|---|---|---|
p ≤ 0.05 | 0.079 | 0.161 | 0.012 | 0.348 | 0.009 | 0.139 | 0.012 | 0.412 | 0.012 | 0.418 |
p > 0.05 | 0.352 | 0.409 | 0.133 | 0.506 | 0.267 | 0.585 | 0.109 | 0.467 | 0.079 | 0.491 |
Total | 0.430 | 0.570 | 0.145 | 0.855 | 0.276 | 0.724 | 0.121 | 0.879 | 0.091 | 0.909 |
Peak Flow | ann_max_Q | ann_max_prcp | ann_tot_Q | ann_tot_prcp | |
---|---|---|---|---|---|
peak flow | 1 | 0.284 | 0.153 | 0.160 | 0.148 |
ann_max_Q | 0.284 | 1 | 0.440 | 0.606 | 0.550 |
ann_max_prcp | 0.153 | 0.440 | 1 | 0.331 | 0.337 |
ann_tot_Q | 0.160 | 0.606 | 0.331 | 1 | 0.769 |
ann_tot_prcp | 0.148 | 0.550 | 0.337 | 0.769 | 1 |
tau (1) | Single-Station Mean | Single-Station Q1 | Single-Station Median | Single-Station Median SE | Single-Station Median t Statistic | Single-Station Q3 | Single-Station Coeff .v0_SE_ frac2 (2) | Single-Station Coeff. vMMQR_SE_ frac2 (3) | MMQR Coeff | MMQR Coeff Robust SE | MMQR Coeff t Statistic | q (tau) |
---|---|---|---|---|---|---|---|---|---|---|---|---|
lm/plm | 0.848 | 0.548 | 0.751 | 0.118 | 6.4 | 1.017 | 0.985 | 0.488 | 0.668 | 0.017 | 38.2 | NA |
MMQR scale | NA | NA | NA | NA | NA | NA | NA | NA | −0.0782 | 0.0066 | NA | NA |
0.002 | NA | NA | NA | NA | NA | NA | NA | NA | 1.134 | 0.049 | 23.0 | −5.33 |
0.01 | NA | NA | NA | NA | NA | NA | NA | NA | 0.964 | 0.030 | 31.8 | −3.51 |
0.04 | 1.074 | 0.651 | 0.911 | 0.223 | 4.1 | 1.311 | 0.800 | 0.636 | 0.863 | 0.022 | 39.4 | −2.35 |
0.1 | 1.000 | 0.619 | 0.872 | 0.215 | 4.1 | 1.212 | 0.867 | 0.579 | 0.802 | 0.019 | 42.2 | −1.63 |
0.25 | 0.907 | 0.559 | 0.819 | 0.165 | 5.0 | 1.115 | 0.936 | 0.555 | 0.735 | 0.018 | 41.8 | −0.83 |
0.5 | 0.838 | 0.524 | 0.769 | 0.134 | 5.8 | 1.015 | 0.955 | 0.530 | 0.665 | 0.017 | 38.6 | 0.04 |
0.75 | 0.760 | 0.469 | 0.667 | 0.133 | 5.0 | 0.934 | 0.921 | 0.464 | 0.600 | 0.017 | 34.5 | 0.84 |
0.9 | 0.686 | 0.403 | 0.584 | 0.184 | 3.2 | 0.836 | 0.773 | 0.488 | 0.542 | 0.019 | 29.3 | 1.54 |
0.96 | 0.666 | 0.362 | 0.564 | 0.196 | 2.9 | 0.852 | 0.664 | 0.506 | 0.494 | 0.021 | 23.6 | 2.13 |
0.99 | NA | NA | NA | NA | NA | NA | NA | NA | 0.423 | 0.027 | 15.6 | 2.93 |
0.998 | NA | NA | NA | NA | NA | NA | NA | NA | 0.333 | 0.038 | 8.8 | 4.00 |
Statistic | tau0.04 | tau0.1 | tau0.25 | tau0.5 | tau0.75 | tau0.9 | tau0.96 | |
---|---|---|---|---|---|---|---|---|
Single-station | Min. | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
MMQR | Min. | 6.89 × 10−7 | 2.18 × 10−6 | 4.46 × 10−6 | 1.93 × 10−5 | 5.73 × 10−6 | 7.70 × 10−8 | 2.18 × 10−6 |
MMQR-LOOCV | Min. | 2.89 × 10−6 | 3.09 × 10−5 | 3.76 × 10−5 | 3.82 × 10−5 | 3.31 × 10−5 | 5.33 × 10−6 | 1.65 × 10−6 |
Single-station | Q1 | 0.0178 | 0.0318 | 0.0504 | 0.0581 | 0.0473 | 0.0303 | 0.0166 |
MMQR | Q1 | 0.0214 | 0.0373 | 0.0612 | 0.0695 | 0.0563 | 0.0346 | 0.0190 |
MMQR-LOOCV | Q1 | 0.0360 | 0.0719 | 0.1327 | 0.1599 | 0.1130 | 0.0542 | 0.0241 |
Single-station | Median | 0.0316 | 0.0616 | 0.1110 | 0.1388 | 0.1080 | 0.0596 | 0.0297 |
MMQR | Median | 0.0358 | 0.0692 | 0.1243 | 0.1552 | 0.1177 | 0.0637 | 0.0318 |
MMQR-LOOCV | Median | 0.0619 | 0.1312 | 0.2541 | 0.3295 | 0.2467 | 0.1181 | 0.0506 |
Single-station | Mean | 0.0454 | 0.0943 | 0.1694 | 0.2073 | 0.1605 | 0.0861 | 0.0405 |
MMQR | Mean | 0.0553 | 0.1065 | 0.1852 | 0.2238 | 0.1717 | 0.0939 | 0.0468 |
MMQR-LOOCV | Mean | 0.0990 | 0.1812 | 0.3096 | 0.3931 | 0.3464 | 0.2355 | 0.1493 |
Single-station | Q3 | 0.0517 | 0.1095 | 0.2108 | 0.2704 | 0.2055 | 0.1047 | 0.0491 |
MMQR | Q3 | 0.0583 | 0.1208 | 0.2308 | 0.2930 | 0.2210 | 0.1113 | 0.0521 |
MMQR-LOOCV | Q3 | 0.0908 | 0.1994 | 0.4030 | 0.5450 | 0.4742 | 0.2661 | 0.1093 |
Single-station | Max. | 2.62 | 3.49 | 4.97 | 4.28 | 2.49 | 2.16 | 1.40 |
MMQR | Max. | 4.05 | 4.72 | 4.86 | 4.21 | 2.65 | 2.92 | 2.88 |
MMQR-LOOCV | Max. | 7.47 | 7.75 | 7.14 | 5.26 | 2.86 | 2.99 | 2.93 |
By Quantile (tau) | By Observation | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
0.002 | 0.01 | 0.04 | 0.1 | 0.25 | 0.5 | 0.75 | 0.9 | 0.96 | 0.99 | 0.998 | Minimum | Mean | |
Single-station | NA | NA | 1 | 0.979 | 0.993 | 0.996 | 0.996 | 0.991 | 0.980 | NA | NA | 0.286 | 0.9907 |
MMQR | 1 | 0.9985 | 0.9989 | 0.9990 | 0.9992 | 0.9993 | 0.9993 | 0.9993 | 0.9993 | 0.9992 | 0.9993 | 0.0909 | 0.9992 |
MMQR-LOOCV | 1 | 0.9982 | 0.9988 | 0.9989 | 0.9992 | 0.9992 | 0.9993 | 0.9993 | 0.9993 | 0.9992 | 0.9993 | 0.0909 | 0.9991 |
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Over, T.; Marti, M.; Podzorski, H. Regional Analysis of the Dependence of Peak-Flow Quantiles on Climate with Application to Adjustment to Climate Trends. Hydrology 2025, 12, 119. https://doi.org/10.3390/hydrology12050119
Over T, Marti M, Podzorski H. Regional Analysis of the Dependence of Peak-Flow Quantiles on Climate with Application to Adjustment to Climate Trends. Hydrology. 2025; 12(5):119. https://doi.org/10.3390/hydrology12050119
Chicago/Turabian StyleOver, Thomas, Mackenzie Marti, and Hannah Podzorski. 2025. "Regional Analysis of the Dependence of Peak-Flow Quantiles on Climate with Application to Adjustment to Climate Trends" Hydrology 12, no. 5: 119. https://doi.org/10.3390/hydrology12050119
APA StyleOver, T., Marti, M., & Podzorski, H. (2025). Regional Analysis of the Dependence of Peak-Flow Quantiles on Climate with Application to Adjustment to Climate Trends. Hydrology, 12(5), 119. https://doi.org/10.3390/hydrology12050119