Regressions Relating Watershed Physical Characteristics to Instantaneous Unit
Hydrograph Parameters for Rainfall-Runoff Modeling in Central Texas
by Theodore G. Cleveland, and Xin He
ABSTRACT
This poster presents the results of on-going study to evaluate regionalized unit
hydrograph methods for Texas watersheds in the 200-acre to 10 square mile range. The
research was conducted as part of a four-institution team (Texas Tech University, Lamar
University, University of Houston, and the U.S. Geological Survey) to develop
regionalized methods for use in watersheds with limited stream gage data for use by the
Texas Department of Transportation for drainage areas in the specified size range.
Currently the department uses the NRCS unit hydrograph as implemented in HEC-HMS.
Our research explored an alternate method where instantaneous unit hydrographs are
synthesized from a two-parameter Rayleigh distribution, and excess precipitation is
synthesized from an initial-abstraction, constant proportion (runoff coefficient) model.
These two components are combined to simulate runoff hydrographs from a precipitation
event.
The study has four fundamental steps:
1. Determine the underlying Rayleigh unit hydrograph for several events at each
watershed.
2. Determine a median unit hydrograph for each watershed
3. Develop regional regression equations for the unit hydrograph and excess rainfall
model in terms of watershed physical characteristics.
4. Evaluate the performance of this approach.
5. Compare the results to current NRCS methodology.
In this research a database for 90 watersheds was constructed containing paired rainfallrunoff events for 1600 storms. Each member of the research team then subjected these
data to various analyses.
The University of Houston team created psuedo 1-minute data for instantaneous unit
hydrograph development then performed a simple baseflow separation procedure. Next
storm-optimum unit hydrographs were developed by pattern search for timing
parameters, shape parameters, initial abstraction depths, and runoff coefficients. This
step was accomplished using a purpose-built psuedo-parallel computer. Once the stormoptimum results were obtained, the storms were screened using an acceptance algorithm
to automatically remove pathologically poor data (e.g. runoff arrives before precipitation
begins, etc.). The remaining data are then correlated to selected watershed parameters
(area, basin length, slope along main channel, etc.) to develop regression equations to
predict unit hydrograph parameters given these simple measures. Lastly, the regression
equations are applied to a handful stations that were omitted from the original analysis as
a test of method performance.
Acknowledgements
The research described in this poster is a joint project conducted by Texas Tech
University, Lamar University, the U.S. Geological Survey, and the University of Houston
in cooperation with the Texas Department of Transportation.
Purpose and Scope
The use of NRCS or other rainfall-runoff models to simulate storm hydrographs for the
design of transportation drainage infrastructure requires (1) a user defined precipitation
amount and a rainfall distribution over the duration of a storm, (2) procedures for
estimating excess precipitation (a loss model), and (3) procedures for distributing the
excess precipitation over time to produce a direct runoff hydrograph. The research
team’s goal was to address these three issues from a variety of approaches, one of which
is the use of empirical instantaneous unit hydrographs (relevant to items 2 and 3).
In this poster we present techniques used to estimate instantaneous unit hydrograph
(IUH) characteristics for small (200 acres – 20 mi2) watersheds in Central Texas.
Statistical (regression) relations were developed for estimating runoff hydrographs for an
arbitrary storm event based on selected basin characteristics and a unit-hydrograph
distribution.
Description of the Study Area
The study area is comprised of 91 selected watersheds in Central Texas. Figure 1 is a
map illustrating the locations of the watersheds used in the study. The obvious urban
areas are displayed, and the small rural watersheds (many of which are in the urban
clusters) comprise the remainder of the stations. The distances between stations are
apparent from the map scale.
Figure 1. Study Area Map - circles are gaging station locations.
(From Asquith, 2003. Used with permission)
Basin Characteristics
Selected basin characteristics were complied for use as explanatory variables in
developing statistical relations to estimate unit hydrograph distribution parameters and
rainfall-loss model parameters. The physical characteristics are of importance in this
poster; descriptive characteristics (land-use, soil-type, etc.) are ignored in the IUH work
to date. The University of Houston team compiled selected physical characteristics
manually, and later additional characteristics were compiled by the U.S.G.S. using a
geographic information system (GIS). The two approaches produced practically identical
results for the common characteristics, and all correlations are based on the U.S.G.S.
physical characteristics. Table 1 lists the physical characteristics for the study watersheds
depicted in Figure 1. Of the 91 watersheds listed in Table 1, 58 are smaller than 10
square miles in drainage area, and 72 are smaller than 20 square miles, and thus well
within the project scope for small watersheds (as defined above).
Maximum basin elevation (ft)
Average basin elevation (ft)
Headwater elevation (ft);
Pourpoint (outlet) elevation (ft)
Effective basin width (mi)
Basin shape factor
Elongation ratio
Rotundity of basin
Compactness ratio
Relative relief (ft/mi)
Basin factor (MCL^2/A)
Main channel length (mi)
Main channel slope (ft/mi)
Main channel sinuosity ratio
Slope ratio of main channel slope to basin slope
Alternate Main channel slope (ft/mi)
67.7
91.2
19.7
37.1
29.1
20.8
12.5
31.6
13.0
78.8
106.1
10.2
15.2
16.6
29.3
17.9
34.2
10.1
14.8
23.9
32.9
14.2
22.2
53.7
10.2
12.1
14.8
30.3
42.0
16.6
Minimum basin elevation (ft)
18.2
26.4
5.0
10.5
8.8
6.0
3.8
8.5
3.4
20.8
31.0
2.1
3.8
4.2
8.8
4.8
10.6
3.8
4.7
3.7
8.0
4.0
7.2
14.4
3.0
2.9
4.4
9.5
13.8
5.4
372.3 752.2
416.8 983.0
318.8 374.1
310.5 590.6
221.0 443.6
171.5 309.8
200.2 265.7
591.5 568.3
125.5 155.8
330.6 794.6
301.8 1013.7
160.1 183.2
165.4 242.0
170.4 257.9
178.7 438.7
286.6 313.8
211.3 534.2
125.2 212.6
145.3 256.9
156.2 292.9
193.0 401.7
129.6 167.1
184.0 315.1
215.5 528.9
235.4 243.2
107.2 169.9
336.3 315.4
217.0 492.8
207.4 607.4
116.3 174.9
750
520
868
651
690
429
606
540
645
880
661
710
671
655
474
880
660
560
516
675
566
634
486
439
461
673
801
623
509
430
1503
1503
1242
1242
1134
739
871
1108
801
1674
1674
893
913
913
913
1194
1194
773
773
968
968
801
801
968
704
843
1116
1116
1116
605
1109
1039
1053
980
891
567
713
869
734
1223
1138
792
778
774
716
1028
911
670
641
824
776
730
687
702
616
755
959
839
776
525
1479
1479
1236
1236
1130
737
868
1107
793
1673
1673
892
892
892
892
1192
1192
773
773
948
948
793
793
948
704
834
1109
1109
1109
590
750
520
868
651
690
429
606
540
645
880
661
710
671
655
474
880
660
560
516
675
566
634
486
439
461
673
801
623
509
430
4.9
4.4
2.5
2.3
2.4
2.1
0.9
2.7
1.5
6.0
5.4
1.3
1.7
1.6
1.4
1.8
2.2
0.6
0.9
3.4
3.3
1.4
1.7
3.7
0.9
1.5
1.4
2.0
2.0
1.3
3.70
5.98
2.00
4.52
3.66
2.86
4.08
3.15
2.24
3.48
5.75
1.64
2.22
2.60
6.07
2.65
4.80
6.54
5.37
1.10
2.42
2.87
4.32
3.87
3.32
1.93
3.02
4.81
6.96
4.12
0.59
0.46
0.80
0.53
0.59
0.67
0.56
0.64
0.75
0.60
0.47
0.88
0.76
0.70
0.46
0.69
0.51
0.44
0.49
1.08
0.73
0.67
0.54
0.57
0.62
0.81
0.65
0.51
0.43
0.56
2.91
4.70
1.57
3.55
2.87
2.24
3.20
2.47
1.76
2.73
4.51
1.29
1.74
2.04
4.76
2.08
3.77
5.14
4.22
0.87
1.90
2.25
3.39
3.04
2.61
1.51
2.37
3.78
5.46
3.24
2.0
2.4
1.6
2.1
1.8
1.6
1.9
1.9
1.6
2.0
2.3
1.7
1.7
1.8
2.3
1.7
2.0
1.9
2.0
1.9
1.8
1.7
1.8
2.1
1.8
1.6
1.7
2.0
2.3
1.8
11.1
10.8
19.0
15.9
15.3
14.9
21.2
18.0
12.0
10.1
9.6
18.0
15.9
15.5
15.0
17.6
15.6
21.0
17.4
12.2
12.2
11.8
14.2
9.9
23.9
14.1
21.3
16.2
14.5
10.5
9.06
17.42
3.21
9.00
7.47
4.29
5.43
4.43
3.06
8.95
14.32
3.32
3.23
3.85
8.79
2.81
7.05
7.67
6.39
2.52
4.51
3.51
6.08
7.08
5.02
3.13
3.93
5.78
11.32
4.09
28.5
45.1
6.3
14.8
12.5
7.4
4.4
10.0
4.0
33.3
48.9
3.0
4.5
5.1
10.6
5.0
12.8
4.1
5.2
5.7
10.9
4.5
8.6
19.5
3.7
3.7
5.0
10.4
17.6
5.4
19.1
15.3
49.1
28.5
27.4
39.3
43.2
36.3
32.2
16.3
13.9
48.0
34.0
30.2
30.5
48.7
32.0
47.2
45.6
47.1
30.1
32.2
33.8
20.5
69.9
42.4
51.3
37.5
27.1
34.4
1.56
1.71
1.27
1.41
1.43
1.23
1.15
1.19
1.17
1.60
1.58
1.42
1.21
1.22
1.20
1.03
1.21
1.08
1.09
1.51
1.36
1.11
1.19
1.35
1.23
1.28
1.14
1.10
1.28
1.00
0.05
0.04
0.15
0.09
0.12
0.23
0.22
0.06
0.26
0.05
0.05
0.30
0.21
0.18
0.17
0.17
0.15
0.38
0.31
0.30
0.16
0.25
0.18
0.10
0.30
0.40
0.15
0.17
0.13
0.30
25.6
21.3
58.5
39.4
35.1
41.8
59.5
56.5
36.9
23.8
20.7
60.7
48.8
46.2
39.5
62.9
41.6
51.7
49.8
48.2
35.0
35.5
35.7
26.1
66.4
43.1
61.9
46.7
34.1
29.5
Average basin slope (ft/mi)
89.6
116.6
12.3
24.5
21.0
12.6
3.6
22.8
5.3
123.7
167.3
2.7
6.3
6.8
12.7
8.8
23.2
2.2
4.2
12.7
26.4
5.7
12.1
53.6
2.7
4.5
6.3
18.7
27.4
7.2
Basin relief (ft)
08155200
08155300
08158810
08158820
08158825
08158050
08158880
08154700
08158380
08158700
08158800
08156650
08156700
08156750
08156800
08158840
08158860
08157000
08157500
08158100
08158200
08158400
08158500
08158600
08155550
08159150
08158920
08158930
08158970
08057320
Basin perimeter (mi)
Station_ID
SubBasin
none
none
none
none
none
none
none
none
none
none
none
none
none
none
none
none
none
none
none
none
none
none
none
none
none
none
none
none
none
none
Basin length (mi)
BartonCreek
BartonCreek
BearCreek
BearCreek
BearCreek
BoggyCreek
BoggySouthCreek
BullCreek
LittleWalnutCreek
OnionCreek
OnionCreek
ShoalCreek
ShoalCreek
ShoalCreek
ShoalCreek
SlaughterCreek
SlaughterCreek
WallerCreek
WallerCreek
WalnutCreek
WalnutCreek
WalnutCreek
WalnutCreek
WalnutCreek
WestBouldinCreek
WilbargerCreek
WilliamsonCreek
WilliamsonCreek
WilliamsonCreek
AshCreek
Total drainage area (mi2)
Austin
Austin
Austin
Austin
Austin
Austin
Austin
Austin
Austin
Austin
Austin
Austin
Austin
Austin
Austin
Austin
Austin
Austin
Austin
Austin
Austin
Austin
Austin
Austin
Austin
Austin
Austin
Austin
Austin
Dallas
Basin
Module
Table 1 – Physical Characteristics for 91 Study Watersheds (1 of 3).
Elongation ratio
Rotundity of basin
Compactness ratio
Relative relief (ft/mi)
Basin factor (MCL^2/A)
Main channel length (mi)
Main channel slope (ft/mi)
Main channel sinuosity ratio
Slope ratio of main channel slope to basin slope
Alternate Main channel slope (ft/mi)
570
548
577
629
623
498
650
626
612
584
532
559
508
649
520
496
673
558
528
631
617
595
712
665
761
714
703
660
818
1213
Basin shape factor
440 654
401 660
423 685
499 730
560 683
468 540
520 756
471 756
503 683
520 635
420 635
435 643
412 582
591 718
440 597
392 597
560 766
424 642
430 589
475 746
590 639
542 639
621 812
560 812
640 860
664 770
630 770
590 730
691 1008
1001 1465
Effective basin width (mi)
213.2
258.5
261.8
231.0
122.7
71.8
235.8
285.1
180.4
115.0
215.1
208.4
170.3
126.6
156.5
205.0
206.0
218.0
159.3
270.2
49.6
97.7
190.9
251.4
219.6
106.3
140.5
---316.3
463.2
647 440 1.7
657 401 1.8
684 423 1.0
726 499 1.4
673 560 1.8
531 468 0.6
755 520 1.5
755 471 2.0
683 503 0.9
634 520 0.7
634 420 1.0
624 435 1.6
573 412 1.2
717 591 0.5
597 440 1.9
597 392 2.1
763 560 1.4
637 424 1.1
585 430 0.8
731 475 2.0
638 590 0.6
638 542 0.7
811 621 1.0
811 560 1.6
842 640 3.0
770 664 0.7
770 630 0.7
------- ---1006 691 1.0
1450 1001 1.3
3.98
3.08
4.30
4.31
2.49
2.89
3.43
3.56
5.30
3.43
6.29
2.20
5.98
4.41
3.62
5.40
3.33
4.82
4.03
2.47
3.10
5.23
5.14
5.16
2.01
1.79
2.86
---3.03
3.17
0.57
0.64
0.54
0.54
0.71
0.66
0.61
0.60
0.49
0.61
0.45
0.76
0.46
0.54
0.59
0.49
0.62
0.51
0.56
0.72
0.65
0.49
0.50
0.50
0.80
0.84
0.67
---0.65
0.63
3.12
2.42
3.38
3.38
1.96
2.27
2.71
2.79
4.16
2.66
4.91
1.73
4.70
3.43
2.85
4.24
2.62
3.79
3.16
1.95
2.40
4.11
4.04
4.05
1.57
1.40
2.24
---2.38
2.49
1.8
1.7
2.0
1.9
1.7
1.6
1.8
1.8
2.1
1.6
2.1
1.6
2.1
1.8
1.9
2.2
1.7
1.9
1.7
1.7
1.8
2.0
2.0
2.1
1.6
1.4
1.6
---1.6
1.7
9.9
14.0
17.3
11.5
7.2
12.5
12.8
11.7
11.6
14.6
12.3
15.3
7.8
17.3
6.4
5.5
13.3
12.7
16.4
14.1
7.6
8.6
11.3
9.4
9.0
21.4
22.3
---30.0
32.0
5.46
4.06
5.72
6.45
3.97
3.67
3.96
4.83
6.21
4.72
7.99
2.87
7.93
5.35
4.54
6.86
4.10
6.37
4.72
3.67
3.68
5.75
6.42
6.87
3.22
2.97
4.35
---3.93
5.30
7.8
6.2
5.1
7.5
5.5
1.9
5.6
8.3
5.3
3.0
6.7
4.1
8.4
2.6
7.6
12.6
5.2
6.4
3.5
6.2
2.0
3.8
6.0
9.4
7.5
1.7
2.4
1.3
3.6
5.4
27.4
38.3
49.4
30.0
18.3
31.4
38.0
30.6
32.6
38.3
31.1
46.0
18.6
41.2
18.5
13.9
34.3
32.0
45.3
37.4
23.6
23.7
30.5
25.5
25.6
65.9
49.9
---81.1
52.3
1.17
1.15
1.15
1.22
1.26
1.13
1.07
1.17
1.08
1.18
1.13
1.14
1.15
1.11
1.12
1.13
1.11
1.15
1.08
1.21
1.10
1.05
1.12
1.15
1.27
1.29
1.23
---1.14
1.29
0.24
0.21
0.23
0.25
0.29
0.64
0.22
0.16
0.28
0.36
0.32
0.30
0.19
0.51
0.15
0.10
0.28
0.29
0.25
0.19
0.46
0.41
0.46
0.33
0.24
0.50
0.39
---0.34
0.08
26.7
41.2
51.3
30.4
20.5
33.4
41.6
34.1
33.7
38.0
31.7
45.8
19.1
47.9
20.5
16.2
39.1
33.5
44.1
41.6
23.8
25.0
31.5
26.7
26.8
62.4
59.1
---87.9
82.8
Pourpoint (outlet) elevation (ft)
112.0
179.2
218.1
122.1
63.9
48.8
174.5
196.2
116.0
105.1
96.6
152.8
99.5
81.3
120.0
136.0
122.2
112.1
180.2
201.2
50.8
57.8
65.7
76.5
106.1
131.3
127.2
---238.4
685.7
Headwater elevation (ft);
21.5
18.4
15.1
20.1
17.0
5.7
18.5
24.3
15.6
7.9
17.5
13.6
21.9
7.3
24.6
37.0
15.4
17.2
9.7
19.2
6.6
11.4
16.9
26.9
24.3
5.0
6.3
3.3
10.6
14.5
Average basin elevation (ft)
6.6
5.4
4.4
6.1
4.4
1.7
5.3
7.1
4.9
2.5
6.0
3.6
7.3
2.4
6.8
11.2
4.7
5.5
3.2
5.1
1.8
3.7
5.4
8.1
5.9
1.3
1.9
1.6
3.1
4.2
Maximum basin elevation (ft)
Basin relief (ft)
11.0
9.5
4.5
8.6
7.7
1.0
8.1
14.4
4.6
1.9
5.7
5.9
8.9
1.3
12.9
23.3
6.6
6.4
2.6
10.3
1.1
2.6
5.7
12.9
17.6
1.0
1.3
0.4
3.3
5.5
Minimum basin elevation (ft)
Average basin slope (ft/mi)
08055700
08057050
08057020
08057140
08061620
08057415
08057418
08057420
08057160
08055580
08055600
08057435
08057445
08057130
08061920
08061950
08057120
08056500
08057440
08057425
08048550
08048600
08048820
08048850
08048520
08048530
08048540
SSSC*
08178300
08181000
Basin perimeter (mi)
Station_ID
SubBasin
none
none
none
none
none
none
none
none
none
none
none
none
none
none
none
none
none
none
none
none
none
none
none
none
none
none
none
none
none
none
Basin length (mi)
BachmanBranch
CedarCreek
CoombsCreek
CottonWoodCreek
DuckCreek
ElamCreek
FiveMileCreek
FiveMileCreek
FloydBranch
JoesCreek
JoesCreek
NewtonCreek
PrairieCreek
RushBranch
SouthMesquite
SouthMesquite
SpankyCreek
TurtleCreek
WhitesBranch
WoodyBranch
DryBranch
DryBranch
LittleFossil
LittleFossil
Sycamore
Sycamore
Sycamore
Sycamore
AlazanCreek
LeonCreek
Total drainage area (mi2)
Dallas
Dallas
Dallas
Dallas
Dallas
Dallas
Dallas
Dallas
Dallas
Dallas
Dallas
Dallas
Dallas
Dallas
Dallas
Dallas
Dallas
Dallas
Dallas
Dallas
FortWorth
FortWorth
FortWorth
FortWorth
FortWorth
FortWorth
FortWorth
FortWorth
SanAntonio
SanAntonio
Basin
Module
Table 1 – Physical Characteristics for 91 Study Watersheds (2 of 3).
Minimum basin elevation (ft)
Maximum basin elevation (ft)
Average basin elevation (ft)
Headwater elevation (ft);
Pourpoint (outlet) elevation (ft)
Effective basin width (mi)
Basin shape factor
Elongation ratio
Rotundity of basin
Compactness ratio
Relative relief (ft/mi)
Basin factor (MCL^2/A)
Main channel length (mi)
Main channel slope (ft/mi)
Main channel sinuosity ratio
8.5
2.8
1.2
9.4
3.5
6.7
3.3
2.9
3.4
0.0
1.5
3.6
3.0
11.7
17.6
3.0
5.4
10.2
4.0
16.9
4.5
2.8
4.4
2.6
2.3
2.0
3.1
18.7
3.8
7.7
7.4
31.2
8.1
3.6
30.7
10.4
21.4
11.3
9.3
10.6
4.3
5.2
14.1
9.9
40.8
60.7
11.0
18.8
37.5
13.8
61.0
17.0
9.9
16.5
6.9
8.3
6.1
9.3
66.9
15.0
36.3
27.3
693.5
32.9
129.0
207.5
24.3
345.8
166.6
319.9
267.5
31.2
78.9
308.2
193.7
84.2
115.3
254.8
191.4
121.0
119.8
106.8
143.6
189.8
158.8
194.8
249.8
170.9
146.3
121.8
334.7
288.1
109.4
691.4
53.1
101.5
410.3
51.9
489.5
227.9
328.2
340.2
46.1
82.5
265.2
158.7
191.6
274.1
269.3
319.7
502.7
121.7
568.5
146.5
143.9
145.2
149.0
120.6
113.3
114.0
297.5
338.3
416.6
192.6
1017
661
915
740
608
1007
832
903
891
745
709
610
1410
401
331
1504
1352
1457
1520
1391
546
371
308
970
653
640
630
550
1092
1023
468
1709
714
1020
1150
660
1496
1060
1231
1232
791
792
875
1569
592
605
1773
1672
1959
1641
1959
693
515
453
1119
774
753
744
847
1430
1439
660
1327
688
965
921
633
1208
941
1048
1024
763
756
733
1486
475
455
1621
1461
1586
1569
1545
615
447
375
1048
717
698
683
671
1234
1217
559
1647
714
1017
1121
660
1474
1060
1218
1231
770
792
875
1564
592
597
1773
1641
1959
1630
1959
693
514
443
1119
769
753
743
820
1429
1391
653
1017
661
918
740
608
1007
832
903
891
745
709
610
1410
401
331
1504
1352
1457
1520
1391
546
371
308
970
653
640
630
550
1092
1023
468
1.8
0.4
0.3
2.2
0.5
1.4
1.2
0.8
0.7
0.0
0.5
1.4
0.8
2.0
2.6
1.1
1.4
2.1
1.0
4.1
1.6
1.1
2.0
0.3
0.9
0.6
0.7
3.9
1.7
3.1
2.5
4.80
6.45
4.81
4.26
6.34
4.65
2.63
3.47
4.60
0.00
3.18
2.59
3.70
5.94
6.67
2.80
3.96
4.78
3.94
4.13
2.87
2.59
2.16
7.50
2.56
3.16
4.75
4.81
2.25
2.46
3.00
0.52
0.44
0.51
0.55
0.45
0.52
0.70
0.61
0.53
0.00
0.63
0.70
0.59
0.46
0.44
0.67
0.57
0.52
0.57
0.56
0.67
0.70
0.77
0.41
0.71
0.63
0.52
0.51
0.75
0.72
0.65
3.77
5.07
3.78
3.34
4.98
3.65
2.07
2.72
3.61
0.00
2.50
2.03
2.90
4.67
5.24
2.20
3.11
3.75
3.10
3.24
2.26
2.03
1.70
5.89
2.01
2.49
3.73
3.77
1.76
1.93
2.35
2.3
2.0
1.8
1.9
2.1
2.0
1.6
1.7
1.9
1.9
1.8
1.8
1.8
2.4
2.5
1.8
2.0
2.3
1.9
2.1
1.8
1.6
1.6
2.1
1.6
1.6
1.8
2.2
1.7
2.1
1.8
22.2
6.6
28.4
13.4
5.0
22.8
20.2
35.4
32.1
10.7
15.8
18.8
16.1
4.7
4.5
24.4
17.0
13.4
8.8
9.3
8.6
14.5
8.8
21.6
14.6
18.5
12.3
4.4
22.6
11.5
7.1
6.47
7.93
5.38
5.76
8.61
5.17
3.21
3.77
6.38
3.23
4.05
3.98
4.72
8.20
8.59
3.60
4.77
7.09
4.74
5.43
3.31
2.52
2.70
8.00
2.05
3.61
5.30
7.38
3.27
5.58
4.19
9.8
3.1
1.3
11.0
4.1
7.1
3.6
3.0
4.0
1.2
1.7
4.5
3.4
13.7
20.0
3.4
5.9
12.4
4.4
19.4
4.9
2.8
4.9
2.6
2.1
2.1
3.3
23.2
4.6
11.6
8.7
48.1
16.6
69.7
25.4
13.3
43.2
51.9
80.6
58.4
18.7
46.3
52.5
45.8
10.3
11.0
83.6
31.4
22.4
18.3
15.7
33.4
48.1
21.7
58.9
54.7
52.7
34.7
8.7
48.2
26.5
15.8
1.16
1.11
1.06
1.16
1.17
1.05
1.11
1.04
1.18
0.00
1.13
1.24
1.13
1.18
1.13
1.13
1.10
1.22
1.10
1.15
1.07
0.99
1.12
1.03
0.90
1.07
1.06
1.24
1.21
1.51
1.18
Alternate Main channel slope (ft/mi)
Basin relief (ft)
14.9
1.2
0.3
20.8
1.9
9.6
4.1
2.5
2.5
0.4
0.7
5.1
2.4
23.0
46.4
3.1
7.3
21.7
4.1
69.2
7.2
3.1
8.8
0.9
2.1
1.2
2.1
73.1
6.6
24.0
18.2
Slope ratio of main channel slope to basin slope
Average basin slope (ft/mi)
08181400
08181450
08177600
08177700
08178555
08178600
08178620
08178640
08178645
08178690
08178736
08096800
08094000
08098300
08108200
08139000
08140000
08136900
08137000
08137500
08182400
08187000
08187900
08050200
08057500
08058000
08052630
08052700
08042650
08042700
08063200
Basin perimeter (mi)
Station_ID
SubBasin
none
none
none
none
none
none
none
none
none
none
none
CowBayou
Green
Pond-Elm
Pond-Elm
Deep
Deep
Mukewater
Mukewater
Mukewater
Calaveras
Escondido
Escondido
ElmFork
Honey
Honey
LittleElm
LittleElm
North
North
PinOak
Basin length (mi)
LeonCreek
LeonCreek
OlmosCreek
OlmosCreek
OlmosCreek
SaladoCreek
SaladoCreek
SaladoCreek
SaladoCreek
SaladoCreek
SaladoCreek
BrasosBasin
BrasosBasin
BrasosBasin
BrasosBasin
ColoradoBasin
ColoradoBasin
ColoradoBasin
ColoradoBasin
ColoradoBasin
SanAntonioBasin
SanAntonioBasin
SanAntonioBasin
TrinityBasin
TrinityBasin
TrinityBasin
TrinityBasin
TrinityBasin
TrinityBasin
TrinityBasin
TrinityBasin
Total drainage area (mi2)
SanAntonio
SanAntonio
SanAntonio
SanAntonio
SanAntonio
SanAntonio
SanAntonio
SanAntonio
SanAntonio
SanAntonio
SanAntonio
SmallRuralSheds
SmallRuralSheds
SmallRuralSheds
SmallRuralSheds
SmallRuralSheds
SmallRuralSheds
SmallRuralSheds
SmallRuralSheds
SmallRuralSheds
SmallRuralSheds
SmallRuralSheds
SmallRuralSheds
SmallRuralSheds
SmallRuralSheds
SmallRuralSheds
SmallRuralSheds
SmallRuralSheds
SmallRuralSheds
SmallRuralSheds
SmallRuralSheds
Basin
Module
Table 1 – Physical Characteristics for 91 Study Watersheds (3 of 3).
0.07 64.1
0.50 16.9
0.54 75.9
0.12 34.8
0.55 12.8
0.12 66.2
0.31 63.2
0.25 103.5
0.22 85.9
0.60 21.3
0.59 49.7
0.17 59.0
0.24 46.0
0.12 13.9
0.10 13.3
0.33 80.1
0.16 48.9
0.19 40.4
0.15 25.0
0.15 29.3
0.23 30.2
0.25 51.4
0.14 27.7
0.30 56.4
0.22 56.0
0.31 54.1
0.24 34.3
0.07 11.6
0.14 72.7
0.09 31.8
0.14 21.2
Precipitation and Discharge Data
These 91 watersheds have paired precipitation and discharge data that were recorded
during USGS small watershed studies in Texas from the 1960's to the middle 1970's. The
data were not digitally available until this study and the printed reports represented the
sole data source. Asquith et. al. (2004) describes the data entry effort and the current
electronic database. The resulting database has about 1600 storms over the entire set of
gaging stations with a minimum of two storms at each station and some stations having
over 30 storms. All the files are ASCII files so the data should be forward compatible for
many years.
The watershed characteristics and rainfall-runoff data comprise the database used for this
IUH study.
Data Preparation
The file pairs (~1600 storms) were parsed to extract the date/time, accumulated runoff,
and accumulated weighted precipitation. In all cases an artificial record was added so
that all the data start at 00:00:00 of the day the records began. Once these files were
constructed, the date/time information column was converted into elapsed time, in
minutes, from the start of the record day. When the elapsed times were completed, these
files were then interpolated using linear interpolation between elapsed times to produce
interpolated values of cumulative precipitation and runoff. For the purpose of this study
we used one-minute time increments in the interpolation because a one-minute interval is
convenient and consistent with resolution of the original data, however the underlying
data for rainfall and runoff are from larger time intervals, thus the one-minute resolution
in this work is artificial.
Base flow separation using the constant discharge method was applied because it is
simple to automate and apply to multiple peaked hydrographs. Prior researchers (e.g.
Laurenson and O’Donell, 1969; Bates and Davies, 1988) have demonstrated that unit
hydrograph derivation is insensitive to base flow separation method when the base flow
is a small fraction of the flood hydrograph – a situation satisfied in this work.
Effective precipitation was modeled using an initial abstraction constant proportion
model (McCuen, 1998), where some constant ratio of precipitation becomes runoff. This
approach was selected, in part, for simplicity with regards to automated analysis, and
because one does not require the total precipitation depth a-priori to generate a
hydrograph. This approach implicitly assumes that the rainfall loss model is a watershed
property and independent of storm behavior. Additional details of the data preparation,
separation techniques, and rainfall loss models are reported in He (2004).
Unit Hydrograph Analysis
The instantaneous unit hydrograph (IUH) is a direct runoff hydrograph (DRH) resulting
from a unit depth of an effective precipitation hyetograph (EPH) applied uniformly over a
watershed. A major advantage of an IUH over a unit hydrograph is that the IUH does not
require the effective precipitation hyetograph to have a specific duration. The direct
runoff hydrograph is computed as the convolution of the effective precipitation
hyetograph and the IUH kernel function as described by Equation 1.
t
Q(t )   i( )u (t   )d
[Eqn. 1]
0
where,
i(t) is the EPH (precipitation rate as a function of time)
u(t) is the IUH (unit response rate as a function of time)
Q(t) is the DRH (direct runoff rate as a function of time).
The function, u(t), is required to exhibit linearity with respect to effective precipitation
and integrate to unity; properties shared by probability distributions. This similarity is not
coincidental, and one interpretation is that u(t) is a residence time distribution of
precipitation on the watershed.
Nash (1958), Leinhard (1972), Dooge (1973), and others, through conceptual approaches
ranging from cascade of linear reservoirs to statistical-mechanical methods, have derived
candidate IUH functions from observed DRH and EPH. Many of these IUH functions are
gamma-family probability distributions. Singh (2000) developed methods to represent
the Natural Resources Conservation Service (NRCS) dimensionless unit hydrograph as a
gamma distribution (Singh, 2000). Cleveland et. al. (2003) analyzed the 1600 storms in
the database and found that several Gamma-family distributions were suitable as IUH
functions and produced nearly identical results. A Rayleigh-distribution model was
eventually selected that upon close inspection is identical to the hydrograph distribution
derived using statistical-mechanical arguments by Leinhard (19##). Equation 2 is the
current working model for the Central Texas data.
2 1  (t   ) 2 N 1 
 (t   ) 

 exp( 
Q(t ) m   {i(t )}A
 )d
2 N 1
t ( N )  t
 t 

0
2
t
[Eqn 2]
t
{i (t )}
0
if
{i (t )}  C r p(t ) if
 p( )d  I
a
0
t
 p( )d  I
a
0
where
{i(t)}
p(t)
Cr, Ia
Q(t)m
t
N
is the effective precipitation rate as a function of time.
is the observed precipitation rate as a function of time
rainfall loss model parameters; Ia has dimensions of depth.
is the direct runoff rate as a function of time (modeled).
is the mean residence time of precipitation in the watershed.
is the reservoir number (a shape factor, not necessarily an integer).
This model has a total of four parameters; two parameters are associated with the rainfall
loss model (Cr, Ia), and two are associated with the unit hydrograph ( t ,N) (redistribution
in time of the effective precipitation). The two hydrograph parameters can also be
transformed into the traditional (Qp,Tp) parameters.
The IUH parameters for each storm are estimated by calculating the DRH from the
effective rainfall signal (Equation 2) and adjusting values until some merit function is
minimized. A modified direct-search technique (Hooke and Jeeves, 1961) was used
where the parameter space was represented as discrete values, and possible combinations
of that space were tested as candidate parameter values for each storm. This method,
while requiring a great number of function evaluations (in our case convolutions), is
robust, reliable, and relatively simple to implement. To increase the computational
throughput for the large number of convolutions, a cluster computer was constructed
from discarded PCs (Wallace, 2004). Information on this cluster computer is available at
(http://cleveland1.cive.uh.edu/).
Two different merit functions considered were the sum of squared errors (SSE) and a
maximum absolute deviation at peak discharge (QpMAD). Mathematically these merit
functions are represented as
SSE 
NOBS
 (Q
i 1
s
 Qo )i2
[Eqn 3]
and
Q p MAD  Qs (t peak )  Qo (t peak )
[Eqn 4]
where Q is the discharge (L3/T); the subscripts O and S represent observed and simulated
discharge; NOBS is the total number of values in a particular storm event; tpeak is the
actual time in the observations when the peak observed discharge occurs. The first merit
function produces results that sacrifice exact peak matching in favor of matching the
general shape of the discharge hydrograph, while the second merit function is designed to
favor matching the peak discharge magnitude with little regard for the rest of the
hydrograph.
Results of this fitting exercise are parameter values for the Raleigh IUH for each storm in
the database. Each storm after analysis produced a set of four parameters that we call the
storm-optimum values for the distribution.
Figure 2 is a representative plot of typical results for analysis of a single storm. The left
panel displays cumulative hyetographs and hydrographs (this is how the data are actually
collected, from cumulative recording), and the right panel displays the incremental values
(the derivative of the left panel). This particular data is for Ash Creek in the Dallas
module and is typical of the complexity of the actual storms (multiple precipitation pulses
of differing magnitude resulting in multiple peaked hydrographs).
Figure 2. Observed and Model Hydrographs after Pattern Search Analysis
Statistical Model(s) to Estimate Direct Runoff Hydrographs
These storm-optimum values (1600 storms, 2 merit functions, 4 model parameters) are
used to develop correlation (regression) equations that allow for the prediction of
watershed response for watersheds in Central Texas based on the measurable physical
characteristics.
Gray (1962) used power-law models and correlation methods to develop a synthetic
hydrograph procedure for 46 watersheds, mostly in Iowa and Missouri. Wu (1963) used
power-law models based on selected watershed properties as predictor equations for unit
hydrographs in Indiana. Graf et. al. (1982) used a power-law model to relate
(TC+R)(Used in HEC 1 and HEC-HMS models) to basin slope and main channel length
for unit hydrographs in Illinois. Wilson and Brown (1992) used physiographic
correlations in an attempt to predict a single constant in the NRCS hydrograph with some
success. Other similar studies adopted a similar approach; Meadows and Ramsey (1991)
used power-law models to develop regional synthetic unit hydrographs for South
Carolina. Weaver (2003) also used a power-law regression for estimating unit
hydrograph behavior in North Carolina. Common to all these researchers is a set of
physical (and in some cases descriptive characteristics) and a need to estimate model
parameter values for modeling rainfall-runoff behavior on un-gaged watersheds (or for
future storms on a gaged watershed). Like these prior researchers we also adopted this
well established approach (a power law model based on explanatory variables from the
watershed characteristics table).
Examination of scatter plots showing the storm-optimum parameter values and various
explanatory variables indicated the strongest correlations with drainage area, basin slope,
basin perimeter, main channel length, and basin length. These last three explanatory
variables are all strongly correlated with each other (all are characteristic lengths). Figure
3 is an example of a typical exploratory data analysis scatter plot. In the figure, there is
evidence that watershed area has some predictive value in determining the mean
residence time. Similar plots were produced using each predictor variable and
combinations of predictor variables for the distribution parameters. After considerable
exploratory analysis we found that only three characteristics in our selected watershed
characteristics produced meaningful correlations. These were watershed area, perimeter,
and slope along the primary channel.
Next we postulated power-law models and used log-linear regressions to determine
predictive equations of the model parameters. A trial effort assuming that Ia = 0, and
using values from the QpMAX merit function produces the following regression models
for estimating distribution and loss model parameters from watershed physical
characteristics.
Ia  0
C r  0.137 A 0.109 S 0..206
[Eqn 5]
A
t  138( ) 0.334 S 0..500
P
N  2.43P 0.102 S 0.064
where
A
P
S
is watershed area in square miles,
is perimeter in feet,
is slope (as a decimal).
N and Cr are dimensionless, the dimension on the residence time is minutes.
400
350
CTP (min^2)
300
250
200
150
100
50
0
0.1
1
10
100
1000
Area (sq.mi)
Figure 3. EDA Plot of t versus Area. Open circles are median values from stormoptimum solutions for each watershed. Solid Triangles are plot of t  20.2 A with A in
square miles.
Figures 4 – 9 are plots of the estimated parameters (from the equations above) and the
storm-optimum values for each storm. In the plots, there are 1600+ open circles (one for
each storm) and 96 closed circles (one for each watershed). The plot scales are
logarithmic so vertical variation is considerable. Despite this variation, the ability of the
regression model to produce parameters that predict the peak discharge is remarkable as
evidenced by Figure 10.
Figure 10 is a plot of the median maximum discharge for each station from the storm-bystorm analysis (median of maximum discharge rate factor fitted to observed peaks),
versus the maximum discharge rate factor as determined by the regression model. In this
figure the model and “observed” peak rate factors nearly fall along the diagonal line
suggesting that the regression does produce a tool that can reasonable estimate the peak
discharges.
1000
t_bar (minutes)
100
Model
Observed
10
1
0.1
1
10
100
1000
Area (square miles)
Figure 4. t versus watershed area . Solid markers are estimates of t from watershed characteristics, open markers are t_bar
determined from analysis of paired events.
1000
t_bar (minutes)
100
Model
Observed
10
1
0.001
0.01
0.1
Slope
Figure 5 t versus watershed slope for QpMAX critetion. Solid markers are estimates of t from watershed characteristics, open
markers are t determined from analysis of paired events.
1.2
1
Runoff_Coefficient
0.8
Model
0.6
Observed
0.4
0.2
0
0.1
1
10
100
1000
Area (square miles)
Figure 6 Cr versus watershed area for QpMAX critetion. Solid markers are estimates of Cr from watershed characteristics, open
markers are Cr determined from analysis of paired events.
1.2
1
Runoff_Coefficient
0.8
Model
0.6
Observed
0.4
0.2
0
0.001
0.01
0.1
Slope
Figure 7. Cr versus watershed slope for QpMAX critetion. Solid markers are estimates of Cr from watershed characteristics. open
markers are t_bar determined from analysis of paired events.
10
9
8
Reservoir Number (N)
7
6
Model
5
Observed
4
3
2
1
0
0.1
1
10
100
1000
Area (square miles)
Figure 8. N versus watershed area for QpMAX critetion. Solid markers are estimates of N from watershed characteristics, open
markers are N determined from analysis of paired events.
10
9
8
Reservoir Number (N)
7
6
Model
5
Observed
4
3
2
1
0
10000
100000
1000000
Perimeter (feet)
Figure 9. N versus watershed perimeter for QpMAX critetion. Solid markers are estimates of N from watershed characteristics, open
markers are N determined from analysis of paired events.
0.035
0.03
Qpeak - Observed
0.025
0.02
0.015
0.01
0.005
0
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
Qpeak - Model
Figure 10. Plot of peak discharge computed using the correlation model to generate IUH parameters and median peak discharges
observed for each watershed. The discharges are normalized by respective watershed area; units in the plot are inches per minute.
Performance Evaluation
To test the ability of the regression equations to predict watershed behavior, watersheds
that were excluded from the regression analysis (in this study purely by accident!) are
characterized (area, slope, etc.) then the model parameters are determined from the
regression equations. Actual observed precipitation is then applied and the direct runoff
hydrograph determined by Equation 2. This runoff hydrograph is compared to the
observed hydrograph for the same storm to evaluate the regression model’s ability to
predict behavior.
Figures 11 and 12 are examples of the regression equation applied to a test watershed. In
these figures, there were no storm optimum parameters, the parameters used to generate
the model runoff values are derived entirely from the watershed physical characteristics
(and the regression equations). Qualitatively, the regression models are fair. It is worth
noting that Austin and Dallas are quite far apart in hydrologic terms and in physiography.
The two samples shown also reflect an areal difference of tenfold, yet the regression
model produced qualitatively fair results and these kind of models (regression equations)
are simpler to apply than conventional NRCS methods that require the estimation of
overland flow distances and speeds, then shallow concentrated flow distances and speeds,
and channel flow distances and speeds for the 50% chance event.
Conclusions
The use of physical characteristics to predict watershed response is not novel. We have
presented an alternate method for Texas watersheds that produces qualitatively
reasonable results based on three simple to determine physical characteristics (the method
here can be performed manually). The approach is compatible with current modeling
methodology after some variable transformations (not presented) and offers the ability to
compare performance with actual events.
Future Effort
The results presented in this poster reflect work through December 2004. Since that time
the loss model has been adjusted to allow for up to 1-inch of initial abstraction. This
adjustment is based on post-analysis examination of the reservoir number which
introduces delay in to the model. In the initial work the reservoir number is too large –
Leinhard demonstrated that physically realistic values should not be much larger than 3.
The inclusion of some abstraction allows delay to be incorporated into the model by an
alternate reasonably acceptable process (abstraction never appears as runoff in a event
model) and keeps the resulting reservoir numbers closer to the physically realistic range
of values. Oddly enough, the preservation of model runoff volumes appears to be
improved by this approach.
The remaining work after this modified loss model is completed is to evaluate
quantitatively the performance of the regression model using the acceptance criteria
approach (He, 2004); to provide the necessary transformations and unit conversions so
the model parameters have the conventional terminology (even though the distribution is
different, the model can be transformed into Qp,Tp form) and conventional discharge,
area, and time units (CFS, sq.mi., and hours), to compare the methodology with
conventional NRCS methods, and write a guidance document to explain the use and
limitations of the regression model.
Figure 11. Observed and Model Hydrographs after Using Regression Model.
(Station08057440 is in Dallas; Area = 2.62 sq. mi.)
Figure 12. Observed and Model Hydrographs after Using Regression Model.
(Station08158810 is in Austin; Area = 12.2 sq. mi.)
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