# Model Calibration Criteria for Estimating Ecological Flow Characteristics

^{1}

^{2}

^{3}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Study Catchments

^{2}) (range 104–4799 km

^{2}) and average elevation of 491 m above the North American Vertical Datum of 1988 (NAVD 88) (range 174–937 m) (Table 1). Hardwood forest and pasture are the dominant land cover in the study area. Soils are deep in the Blue Ridge ecoregion which leads to increased baseflow in comparison to the relatively thinner soils of the Appalachian Plateau and Ridge and Valley ecoregions [20] Generally, topographic slope and regolith thickness decreases from east to west, while karst development is most prominent in the Ridge and Valley [21]. Combined, these catchment characteristics produce noticeable and documented regional variations in hydrologic response and streamflow regimes [21,22,23,24].

**Figure 1.**Catchment outlet locations for 27 basins modelled using 7 calibration schemes for HBV-light.

**Table 1.**U.S. Geological Survey (USGS) stream gaging sites used for model calibration and error evaluation. Latitude and longitude represent the basin outlet; ecoregion defined as the Level 3 ecoregion with the majority of the basin area; km

^{2}, square kilometers; horizontal reference is North American Datum 1983; vertical reference is North American Vertical Datum 1988.

Map Number (Figure 1) | USGS Station Number | Latitude | Longitude | Average Elevation (m) | Primary Ecoregion (Omernik, 1987) | Basin Area (km^{2}) |
---|---|---|---|---|---|---|

1 | 03441000 | 35.2731 | −82.7058 | 645 | Blue Ridge | 104 |

2 | 03443000 | 35.2992 | −82.6239 | 628 | Blue Ridge | 766 |

3 | 03446000 | 35.3981 | −82.5950 | 637 | Blue Ridge | 173 |

4 | 03455000 | 35.9816 | −83.1611 | 308 | Blue Ridge | 4799 |

5 | 03459500 | 35.6350 | −82.9900 | 712 | Blue Ridge | 906 |

6 | 03460000 | 35.6675 | −83.0736 | 749 | Blue Ridge | 127 |

7 | 03463300 | 35.8314 | −82.1842 | 810 | Blue Ridge | 112 |

8 | 03465500 | 36.1765 | −82.4574 | 463 | Blue Ridge | 2082 |

9 | 03471500 | 36.7604 | −81.6312 | 642 | Blue Ridge | 198 |

10 | 03473000 | 36.6518 | −81.8440 | 546 | Blue Ridge | 785 |

11 | 03475000 | 36.7132 | −81.8187 | 555 | Ridge and Valley | 534 |

12 | 03479000 | 36.2392 | −81.8222 | 795 | Blue Ridge | 236 |

13 | 03488000 | 36.8968 | −81.7462 | 519 | Ridge and Valley | 578 |

14 | 03497300 | 35.6645 | −83.7113 | 337 | Blue Ridge | 271 |

15 | 03498500 | 35.7856 | −83.8846 | 259 | Blue Ridge | 697 |

16 | 03500000 | 35.1500 | −83.3797 | 612 | Blue Ridge | 361 |

17 | 03500240 | 35.1589 | −83.3942 | 615 | Blue Ridge | 146 |

18 | 03503000 | 35.3364 | −83.5269 | 537 | Blue Ridge | 1130 |

19 | 03504000 | 35.1275 | −83.6186 | 937 | Blue Ridge | 135 |

20 | 03512000 | 35.4614 | −83.3536 | 562 | Blue Ridge | 476 |

21 | 03524000 | 36.9448 | −82.1549 | 457 | Ridge and Valley | 1382 |

22 | 03528000 | 36.4251 | −83.3982 | 323 | Ridge and Valley | 3816 |

23 | 03531500 | 36.6620 | −83.0949 | 384 | Central Appalachians | 828 |

24 | 03540500 | 35.9831 | −84.5580 | 232 | Cumberland Plateau | 1815 |

25 | 03550000 | 35.1389 | −83.9806 | 474 | Blue Ridge | 268 |

26 | 03568933 | 34.8975 | −85.4631 | 202 | Ridge and Valley | 379 |

27 | 03574500 | 34.6243 | −86.3064 | 174 | Cumberland Plateau | 814 |

#### 2.2. HBV Model

#### 2.3. Calibration

**Table 2.**Parameter ranges used during the Genetic Algorithm and Powell optimization (GAP) calibrations within HBV-light. (°C, degrees Celsius; mm, millimeter; D, day).

Parameter | Explanation | Minimum | Maximum | Unit |
---|---|---|---|---|

Snow Routine | ||||

TT | Threshold temperature | −2 | 2.5 | °C |

CFMAX | Degree-day factor | 0.5 | 10 | mm·°C^{−1}·D^{−1} |

SFCF | Snowfall correction factor | 0.5 | 1.2 | - |

CFR | Refreezing coefficient | 0 | 0.1 | - |

CWH | Water holding capacity | 0 | 0.2 | - |

Soil Routine | ||||

FC | Maximum storage in soil box | 100 | 550 | mm |

LP | Threshold for reduction of evaporation (relative storage in the soil box) | 0.3 | 1 | - |

BETA | Shape coefficient | 1 | 5 | - |

Response Routine | ||||

PERC | Maximal flow from upper to lower box | 0 | 4 | mm·D^{−1} |

UZL | Maximal storage in the soil upper zone | 0 | 70 | mm |

K0 | Recession coefficient (upper box, upper outflow) | 0.1 | 0.5 | D^{−1} |

K1 | Recession coefficient (upper box, lower outflow) | 0.01 | 0.2 | D^{−1} |

K2 | Recession coefficient (lower box) | 0.00005 | 0.1 | D^{−1} |

Routing Routine | ||||

MAXBAS | Routing, length of weighting function | 1 | 5 | D |

**Table 3.**Definitions criteria used in objective functions for the automatic calibration trials using the Genetic Algorithm and Powell optimization (GAP) algorithm.

Criterion | Description | Definition |
---|---|---|

Reff | Model efficiency | $1-\frac{{\displaystyle \sum}{({Q}_{\text{obs}}-{Q}_{\text{sim}})}^{2}}{{\displaystyle \sum}{({Q}_{\text{obs}}-\overline{{Q}_{\text{obs}}})}^{2}}$ |

LogReff | Efficiency for log(Q) | $1-\frac{{\displaystyle \sum}{(\mathrm{ln}{Q}_{\text{obs}}-\mathrm{ln}{Q}_{\text{sim}})}^{2}}{{\displaystyle \sum}{(\mathrm{ln}{Q}_{\text{obs}}-\mathrm{ln}\overline{{Q}_{\text{obs}}})}^{2}}$ |

Lindström | Lindström measure | $Reff-0.1\frac{\left|{\displaystyle \sum}({Q}_{\text{obs}}-{Q}_{\text{sim}})\right|}{{\displaystyle \sum}({Q}_{\text{obs}})}$ |

MARE | Measure based on the Mean Absolute Relative Error ^{(1)} | $1-\frac{1}{n}{\displaystyle \sum}\frac{\left|{Q}_{\text{obs}}-{Q}_{\text{sim}}\right|}{{Q}_{\text{obs}}}$ |

Spearman | Spearman rank correlation ^{(2)} | $\frac{{\displaystyle \sum}\left({R}_{\text{obs}}-\overline{{R}_{\text{obs}}}\right)({S}_{\text{sim}}-\overline{{S}_{\text{sim}}})}{\sqrt{{\displaystyle \sum}{({R}_{\text{obs}}-\overline{{R}_{\text{obs}}})}^{2}}\sqrt{{\displaystyle \sum}{({S}_{\text{sim}}-\overline{{S}_{\text{sim}}})}^{2}}}$ |

VolumeError | Volume error | $1-\frac{\left|{\displaystyle \sum}({Q}_{\text{obs}}-{Q}_{\text{sim}})\right|}{{\displaystyle \sum}({Q}_{\text{obs}})}$ |

^{(1)}Where n is the number of days;

^{(2)}Where R

_{obs}and S

_{sim}are the ranks of Q

_{obs}and Q

_{sim}, respectively.

**Table 4.**The three combination objective functions used during the Genetic Algorithm and Powell optimization (GAP) calibrations within HBV-light. The criteria were weighted equally in each case. See Table 3 for a more detailed specification of each of the criteria.

Combined Objective Function | Criteria |
---|---|

C1 | Reff, LogReff, VolumeError |

C2 | Reff, MARE, Spearman, VolumeError |

C3 | Spearman, VolumeError |

#### 2.4. Evaluation

**Table 5.**Definition of streamflow characteristics used in this study (adapted and modified from Knight et al., 2014 and Thomson and Archfield, 2014) (mm/day, millimeters per day; -, no units; %, percent).

Streamflow Characteristic | Abbreviation | Description | Units |
---|---|---|---|

Magnitude | |||

Mean annual runoff | MA41 | Annual mean daily streamflow | mm/day |

Maximum October runoff | MH10 | Mean maximum October streamflow across the period of record | mm/day |

Lowest 15% of daily runoff | Flowperc | 85% exceedance of daily mean streamflow for the period of record | mm/day |

Rate of streamflow recession | RA7 | Median change in log of streamflow for days in which the change is negative across the period of record | mm/day |

Ratio | |||

Average 30-day maximum runoff | DH13 | Mean annual maximum of a 30-day moving average streamflow divided by the median for the entire record | – |

Stability of runoff | TA1 | Measure of the constancy of a flow regime by dividing daily flows into predetermined flow classes | – |

Frequency | |||

Frequency of moderate floods | FH6 | Average number of high-flow events per year that are equal to or greater than three times the median annual flow for the period of record | number/year |

Frequency of moderate floods | FH7 | Average number of high-flow events per year that are equal to or greater than three times the median annual flow for the period of record | number/year |

Variability | |||

Variability of March runoff | MA26 | Standard deviation for March streamflow divided by the mean streamflow for March | – |

Variability in high-flow pulse duration | DH16 | 100 times the standard deviation for the yearly average high-flow pulse durations (daily flow greater than the 75th percentile) divided by the mean of the yearly average high pulse durations | % |

Variability of low-flow pulse count | FL2 | 100 times the standard deviation for the average number of yearly low-flow pulses (daily flow less than the 25th percentile) divided by the mean low-flow pulse counts | % |

Date | |||

Timing of annual minimum runoff | TL1 | Julian date of annual minimum flow occurrence | Julian day |

## 3. Results

**Figure 2.**Boxplots for catchment 4 (03455000) and (

**a**) streamflow characteristic DH16 (Variability in high-flow pulse duration); (

**b**) streamflow characteristic MA41 (Mean annual runoff). Cal1 and Cal2 are calibration of period 1, respectively period 2, whereas Val1 and Val2 are validation of period 1, respectively period 2.

**Figure 3.**Normalized median flow characteristic values for five different flow characteristics: (

**a**) DH16 (Variability in high-flow pulse duration); (

**b**) FL2 (Variability of low-flow pulse count); (

**c**) MA41 (Mean annual runoff); (

**d**) MH10 (Maximum October runoff) and (

**e**) TA1 (Stability of runoff). Each color corresponds to an objective function. Per objective function, the four boxplots represent (from left to right) calibration period 1 (Cal1), validation period 1 (Val1), calibration period 2 (Cal2) and validation period 2 (Val2). Each boxplot is based on 27 normalized median flow characteristic values, one value for each of the 27 catchments. Medians were computed over 100 runs per catchment. Normalization was carried out by dividing the median values by the corresponding observed flow characteristic value.

**Figure 4.**Relative ranges as a measure for parameter uncertainty for streamflow characteristics (

**a**) DH16 (Variability in high-flow pulse duration); (

**b**) FL2 (Variability of low-flow pulse count); (

**c**) MA41 (Mean annual runoff); (

**d**) MH10 (Maximum October runoff) and (

**e**) TA1 (Stability of runoff). Each color corresponds to an objective function. Per objective function, the four boxplots represent (from left to right) calibration period 1 (Cal1), validation period 1 (Val1), calibration period 2 (Cal2) and validation period 2 (Val2). Each boxplot is based on 27 values, one value for each of the 27 catchments. Relative ranges were computed by dividing the range over the 100 runs per catchment by the range over the 27 median catchment values. Note that the Mean annual runoff (MA41) has been plotted on a different scale.

**Figure 5.**Scatterplots for the streamflow characteristics (

**a**) DH16 (Variability in high-flow pulse duration); (

**b**) FL2 (Variability of low-flow pulse count); (

**c**) MA41 (Mean annual runoff); (

**d**) MH10 (Maximum October runoff) and (

**e**) TA1 (Stability of runoff) for calibration period 1. The points represent the median value of all 100 calibration trials in each catchment based on single criteria objective functions (

**left column**) and multi-criteria objective functions (

**right column**).

**Table 6.**Spearman rank correlation coefficients between objective functions (horizontal) and streamflow characteristics (vertical) based on observed respective simulated streamflow (for each group of four values: upper − left = calibration period 1 (Cal1), upper − right = validation period 2 (Val2), lower − left = validation period 1 (Val1), lower − right = calibration period 2 (Cal2)). Colors are ranging from white (for a Spearman rank correlation of 0) to dark green (for a Spearman rank correlation of 1).

Reff | LogReff | Lindström | MARE | C1 | C2 | C3 | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

MA41 | 0.973 | 0.978 | 0.930 | 0.927 | 0.980 | 0.983 | 0.919 | 0.918 | 0.980 | 0.981 | 0.947 | 0.928 | 0.981 | 0.986 |

0.957 | 0.991 | 0.929 | 0.947 | 0.961 | 0.998 | 0.926 | 0.950 | 0.961 | 1.000 | 0.952 | 0.979 | 0.962 | 1.000 | |

MH10 | 0.930 | 0.831 | 0.874 | 0.853 | 0.916 | 0.837 | 0.834 | 0.829 | 0.941 | 0.837 | 0.958 | 0.874 | 0.918 | 0.898 |

0.960 | 0.940 | 0.862 | 0.868 | 0.958 | 0.934 | 0.822 | 0.829 | 0.957 | 0.918 | 0.942 | 0.903 | 0.885 | 0.933 | |

Flowperc | 0.796 | 0.978 | 0.810 | 0.986 | 0.790 | 0.961 | 0.814 | 0.979 | 0.808 | 0.980 | 0.810 | 0.983 | 0.685 | 0.867 |

0.778 | 0.985 | 0.808 | 0.996 | 0.781 | 0.980 | 0.804 | 0.996 | 0.803 | 0.995 | 0.806 | 0.996 | 0.683 | 0.897 | |

RA7 | 0.736 | 0.724 | 0.877 | 0.885 | 0.726 | 0.735 | 0.888 | 0.896 | 0.870 | 0.873 | 0.851 | 0.892 | 0.696 | 0.797 |

0.756 | 0.836 | 0.930 | 0.930 | 0.719 | 0.775 | 0.848 | 0.902 | 0.878 | 0.919 | 0.880 | 0.917 | 0.744 | 0.789 | |

DH13 | 0.977 | 0.938 | 0.974 | 0.948 | 0.971 | 0.908 | 0.960 | 0.960 | 0.981 | 0.945 | 0.976 | 0.945 | 0.926 | 0.691 |

0.955 | 0.866 | 0.976 | 0.937 | 0.955 | 0.877 | 0.964 | 0.957 | 0.971 | 0.910 | 0.978 | 0.885 | 0.871 | 0.573 | |

TA1 | 0.972 | 0.929 | 0.968 | 0.943 | 0.977 | 0.906 | 0.947 | 0.974 | 0.968 | 0.884 | 0.960 | 0.899 | 0.875 | 0.766 |

0.936 | 0.956 | 0.933 | 0.966 | 0.952 | 0.942 | 0.884 | 0.936 | 0.958 | 0.948 | 0.942 | 0.964 | 0.904 | 0.924 | |

FH6 | 0.943 | 0.851 | 0.916 | 0.906 | 0.935 | 0.875 | 0.728 | 0.863 | 0.953 | 0.916 | 0.900 | 0.921 | 0.569 | 0.663 |

0.926 | 0.888 | 0.853 | 0.931 | 0.931 | 0.898 | 0.634 | 0.855 | 0.942 | 0.930 | 0.901 | 0.919 | 0.498 | 0.613 | |

FH7 | 0.948 | 0.933 | 0.881 | 0.889 | 0.949 | 0.935 | 0.810 | 0.887 | 0.967 | 0.945 | 0.965 | 0.952 | 0.688 | 0.563 |

0.927 | 0.951 | 0.842 | 0.889 | 0.941 | 0.960 | 0.763 | 0.805 | 0.945 | 0.967 | 0.944 | 0.967 | 0.480 | 0.520 | |

MA26 | 0.849 | 0.917 | 0.789 | 0.906 | 0.855 | 0.920 | 0.704 | 0.858 | 0.894 | 0.923 | 0.903 | 0.915 | 0.631 | 0.856 |

0.752 | 0.932 | 0.699 | 0.894 | 0.782 | 0.935 | 0.672 | 0.829 | 0.821 | 0.933 | 0.831 | 0.928 | 0.381 | 0.769 | |

DH16 | 0.534 | 0.645 | 0.443 | 0.662 | 0.503 | 0.673 | 0.402 | 0.471 | 0.510 | 0.745 | 0.525 | 0.683 | 0.145 | 0.482 |

0.429 | 0.549 | 0.421 | 0.654 | 0.410 | 0.514 | 0.346 | 0.645 | 0.526 | 0.659 | 0.511 | 0.650 | 0.094 | 0.518 | |

FL2 | 0.521 | 0.443 | 0.740 | 0.628 | 0.609 | 0.449 | 0.734 | 0.703 | 0.709 | 0.602 | 0.684 | 0.668 | 0.755 | 0.594 |

0.548 | 0.617 | 0.659 | 0.604 | 0.579 | 0.659 | 0.641 | 0.626 | 0.672 | 0.711 | 0.620 | 0.695 | 0.616 | 0.628 | |

TL1 | 0.477 | 0.394 | 0.643 | 0.520 | 0.471 | 0.347 | 0.612 | 0.753 | 0.603 | 0.330 | 0.531 | 0.428 | 0.574 | 0.418 |

0.407 | 0.112 | 0.646 | 0.546 | 0.418 | 0.065 | 0.623 | 0.777 | 0.497 | 0.362 | 0.531 | 0.201 | 0.600 | 0.280 |

**Table 7.**Nash-Sutcliffe efficiencies between objective functions (horizontal) and streamflow characteristics (vertical) based on observed respective simulated streamflow (for each group of four values: upper − left = calibration period 1 (Cal1), upper − right = validation period 2 (Val2), lower − left = validation period 1 (Val1), lower − right = calibration period 2 (Cal2)). Colors are ranging from white (for Nash-Sutcliffe efficiencies of 0 or lower) to dark green (for a Nash-Sutcliffe efficiency of 1).

Reff | LogReff | Lindström | MARE | C1 | C2 | C3 | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

MA41 | 0.917 | 0.936 | 0.840 | 0.881 | 0.936 | 0.933 | 0.584 | 0.626 | 0.946 | 0.939 | 0.922 | 0.927 | 0.949 | 0.930 |

0.858 | 0.967 | 0.746 | 0.835 | 0.900 | 0.993 | 0.490 | 0.554 | 0.914 | 0.999 | 0.875 | 0.965 | 0.916 | 1.000 | |

MH10 | 0.848 | 0.820 | −0.627 | 0.570 | 0.841 | 0.796 | −3.942 | −1.220 | 0.820 | 0.871 | 0.796 | 0.879 | −1.630 | 0.663 |

0.859 | 0.934 | −0.931 | 0.332 | 0.874 | 0.926 | −5.692 | −2.258 | 0.848 | 0.926 | 0.756 | 0.850 | −1.367 | 0.667 | |

Flowperc | 0.416 | 0.749 | 0.611 | 0.837 | 0.356 | 0.660 | 0.647 | 0.960 | 0.463 | 0.680 | 0.614 | 0.804 | 0.170 | 0.477 |

0.484 | 0.868 | 0.569 | 0.967 | 0.491 | 0.820 | 0.465 | 0.966 | 0.538 | 0.848 | 0.591 | 0.939 | 0.373 | 0.669 | |

RA7 | 0.209 | 0.281 | 0.071 | 0.193 | 0.279 | 0.370 | −0.420 | −0.284 | −0.043 | −0.063 | −0.229 | −0.197 | −9.226 | −7.224 |

−0.628 | −0.230 | 0.369 | 0.385 | −0.608 | −0.277 | 0.156 | 0.186 | 0.276 | 0.252 | 0.190 | 0.231 | −5.173 | −4.088 | |

DH13 | 0.372 | −0.164 | 0.884 | 0.472 | −0.601 | −1.895 | 0.910 | 0.858 | 0.797 | 0.522 | 0.770 | 0.874 | −7.603 | −20.044 |

0.638 | 0.427 | 0.919 | 0.748 | 0.437 | −0.030 | 0.814 | 0.914 | 0.902 | 0.813 | 0.672 | 0.817 | −4.235 | −14.891 | |

TA1 | 0.898 | 0.432 | 0.856 | 0.882 | 0.829 | 0.108 | 0.672 | 0.803 | 0.918 | 0.477 | 0.886 | 0.749 | 0.502 | -1.020 |

0.863 | 0.912 | 0.718 | 0.845 | 0.892 | 0.926 | 0.548 | 0.685 | 0.881 | 0.974 | 0.839 | 0.953 | 0.806 | 0.705 | |

FH6 | 0.709 | 0.628 | −1.354 | −0.967 | 0.660 | 0.559 | −7.331 | −4.461 | 0.513 | 0.502 | 0.210 | 0.282 | −3.781 | −5.629 |

0.714 | 0.622 | −0.788 | −0.465 | 0.717 | 0.612 | −4.768 | −3.426 | 0.736 | 0.680 | 0.533 | 0.522 | −2.536 | −4.020 | |

FH7 | 0.746 | 0.756 | −0.440 | −1.246 | 0.585 | 0.600 | −0.752 | −1.837 | 0.769 | 0.725 | 0.842 | 0.820 | −13.413 | −22.837 |

0.813 | 0.826 | 0.290 | −0.242 | 0.801 | 0.820 | −0.260 | −0.612 | 0.912 | 0.930 | 0.932 | 0.954 | −9.425 | −11.728 | |

MA26 | 0.618 | 0.849 | 0.080 | 0.033 | 0.582 | 0.832 | −0.418 | −1.114 | 0.789 | 0.882 | 0.848 | 0.872 | −4.116 | −4.256 |

0.331 | 0.862 | 0.184 | 0.320 | 0.324 | 0.886 | 0.178 | −0.513 | 0.500 | 0.894 | 0.564 | 0.878 | −1.898 | −2.343 | |

DH16 | −3.044 | −0.329 | −3.375 | 0.050 | −3.323 | −0.307 | −0.463 | −0.371 | −3.727 | −0.006 | −2.768 | 0.192 | −3.474 | −0.562 |

−0.937 | −0.182 | −2.056 | 0.186 | −1.012 | −0.234 | −1.025 | 0.006 | −1.535 | −0.092 | −1.562 | 0.119 | −2.785 | −0.309 | |

FL2 | 0.118 | −1.176 | −0.469 | −1.557 | 0.201 | −0.931 | −0.556 | −1.448 | −0.266 | −0.827 | −0.167 | −1.773 | 0.139 | −0.948 |

−0.040 | −1.198 | −0.530 | −1.841 | 0.056 | −1.123 | −0.759 | −1.703 | −0.203 | −0.409 | −0.132 | −1.246 | -0.104 | −1.018 | |

TL1 | −0.376 | −4.676 | −0.211 | −3.016 | −0.310 | −5.502 | −0.361 | −2.672 | −0.017 | −4.483 | −0.196 | −4.053 | −0.023 | −2.708 |

−0.505 | −4.322 | −0.250 | −3.892 | −0.518 | −4.338 | −0.557 | −2.218 | −0.400 | −4.503 | −0.489 | −5.932 | 0.021 | −3.529 |

**Table 8.**Median percent error for streamflow characteristics by model objective function for calibration period 1 (Cal1).

Objective Function | MA41 | MH10 | RA7 | TA1 | DH13 | FH7 | FH6 | FL2 | MA26 | DH16 | TL1 | E85 | Average Median Error (Percent) |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Lindström | −0.6 | −1.8 | −25.0 | −15.2 | −18.1 | −23.0 | −12.0 | 16.8 | 9.1 | −20.8 | 3.7 | 19.1 | −5.6 |

LogReff | −9.5 | −20.0 | −50.0 | 7.7 | −9.5 | −37.5 | −27.0 | 26.9 | −7.3 | −10.0 | 4.8 | 15.2 | −9.7 |

MARE | −18.9 | −44.0 | −57.1 | 25.0 | −7.4 | −44.4 | −41.4 | 28.2 | −19.6 | 9.9 | 5.5 | −7.3 | −14.3 |

Reff | −2.5 | −2.1 | −18.2 | −10.8 | −14.7 | −20.0 | −12.0 | 17.5 | 9.8 | −20.2 | 4.2 | 9.8 | −4.9 |

C1 | 0.0 | −4.8 | −50.0 | −7.7 | −13.1 | −19.0 | −14.1 | 28.6 | 4.9 | −19.7 | 3.4 | 29.9 | −5.1 |

C2 | −0.8 | −10.6 | −42.9 | 0.0 | −7.5 | −14.0 | −18.2 | 17.7 | 2.2 | −16.4 | 4.0 | 13.2 | −6.1 |

C3 | 0.0 | −24.5 | −44.4 | −18.9 | −18.9 | −69.3 | −37.6 | 23.6 | −28.1 | −12.5 | 3.4 | 24.1 | −16.9 |

Average Median Percent Error | −4.6 | −15.4 | −41.1 | −2.8 | −12.7 | −32.5 | −23.2 | 22.8 | −4.2 | −12.8 | 4.1 | 14.9 | – |

## 4. Discussion

**Table 9.**Comparison of selected streamflow characteristics based on simulated and observed streamflow time series for a single model location (site 13 (03488000)) and calibration period 1 (Cal1). (TA1, RA7, and FH6, defined in Table 5; values in parentheses represent the specialized insectivore score using the associated streamflow characteristic value based on linear equations presented in Knight et al. [41], Figure 2; hydro, percent error for streamflow characteristic derived from simulated and observed streamflow time series; eco, percent error for specialized insectivore score based on streamflow characteristic derived from simulated and observed streamflow time series).

Objective Function (see Table 3 for Definitions) | RA7 | Percent Error | TA1 | Percent Error | FH6 | Percent Error | |||
---|---|---|---|---|---|---|---|---|---|

Simulated | Observed | Hydro/Eco | Simulated | Observed | Hydro/Eco | Simulated | Observed | Hydro/Eco | |

Lindström | 0.14 (0.49) | 27.3/−19.7 | 0.4 (0.55) | −16.7/−9.8 | 13 (0.59) | 13.4/−9.2 | |||

LogReff | 0.1 (0.66) | −9.1/8.2 | 0.67 (0.75) | 39.6/23 | 10.08 (0.7) | −12/7.7 | |||

MARE | 0.06 (0.83) | 0.11 (0.61) | −45.5/36.1 | 0.73 (0.8) | 0.48 (0.61) | 52.1/31.1 | 6.62 (0.84) | 11.46 (0.65) | −42.2/29.2 |

Reff | 0.125 (0.55) | 13.6/−9.8 | 0.41 (0.56) | −14.6/−8.2 | 13.38 (0.58) | 16.8/−10.8 | |||

C1 | 0.12 (0.57) | 9.1/−6.6 | 0.43 (0.57) | −10.4/−6.6 | 12.92 (0.59) | 12.7/−9.2 | |||

C2 | 0.09 (0.7) | −18.2/14.8 | 0.57 (0.68) | 18.8/11.5 | 12.38 (0.62) | 8/−4.6 | |||

C3 | 0.05 (0.87) | −54.5/42.6 | 0.38 (0.53) | −20.8/−13.1 | 6.54 (0.84) | −42.9/29.2 |

**Figure 6.**Example of an ecological flow application by comparison of estimated values for three streamflow characteristics for site 13 (03488000) (Table 1, Figure 1) and calibration period 1 (Cal1). (

**a**) Constancy; (

**b**) Frequency of moderate flooding (number per year) and (

**c**) Rate of streamflow recession (log of flow units per day). Black triangles represent model estimated values based on the seven objective functions. Green triangle represents streamflow characteristics based on observed data. Values for RA7 (Rate of streamflow recession) were multiplied by negative 1 to convert values to those in the original analysis. Thin black lines represent 80th percentile quantile regression lines based on the 33 data point (grayed) in the background used by Knight et al. [41]. (Figure modified from Knight et al. [41]).

**Figure 7.**Minimum, maximum, and median percent errors according to objective function and streamflow characteristic for calibration period 1 (Cal1). Each vertical bar is based on the median error for the 27 catchments. The gray band in the center of the figure represents ±30 percent difference [46] Vertical bars with arrows indicate the maximum percent error exceeded the axis scale.

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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Vis, M.; Knight, R.; Pool, S.; Wolfe, W.; Seibert, J. Model Calibration Criteria for Estimating Ecological Flow Characteristics. *Water* **2015**, *7*, 2358-2381.
https://doi.org/10.3390/w7052358

**AMA Style**

Vis M, Knight R, Pool S, Wolfe W, Seibert J. Model Calibration Criteria for Estimating Ecological Flow Characteristics. *Water*. 2015; 7(5):2358-2381.
https://doi.org/10.3390/w7052358

**Chicago/Turabian Style**

Vis, Marc, Rodney Knight, Sandra Pool, William Wolfe, and Jan Seibert. 2015. "Model Calibration Criteria for Estimating Ecological Flow Characteristics" *Water* 7, no. 5: 2358-2381.
https://doi.org/10.3390/w7052358