Use of Rational and Modified Rational Method for ... - CTR Library

10. The intercepts of β and exponents on A and S of the equations by recurrence interval were
tabulated by county and are listed in tables 3.2, 3.3, and 3.4, respectively (located at end
of Section 3 and starting on page 34). The 1,524 β values are listed (table 3.2) after 10 B
conversion, and the 1,524 S-exponent values were multiplied by −1 (table 3.4).
The enumerated steps produce the equation ensemble for T ⋆ . The equations show that the effects
of recurrence interval on time are contained in the three regression coefficients and are spatially
dependent. The representation of the time ensemble by county and by recurrence interval is:
β 2 A
⎧⎪ α 2
S −κ 2
for 2-year recurrence interval
β 5 A α 5
S −κ 5
for 5-year recurrence interval
⎨
county = β 10 A α 10
S −κ 10
for 10-year recurrence interval
β 25 A α 25
S −κ 25
for 25-year recurrence interval
⎪ ⎩
T ⋆[·]
β 50 A α 50
S −κ 50
for 50-year recurrence interval
β 100 A α 100
S −κ 100
for 100-year recurrence interval
(3.5)
where T ⋆[·]
county is in minutes for an optionally identified annual recurrence interval shown by [·], A is
drainage area in acres, and S is dimensionless main-channel slope. The regression coefficients β,
α, and κ are listed in tables 3.2, 3.3, and 3.4, respectively. The subscripts (2, 5, 10, 25, 50, and 100)
represent annual recurrence interval. The values listed in table 3.2 are multiplied by 60 to yield a
time estimate in minutes for eq. 3.5 (the source algorithms used hours). The values listed in table 3.4
are negated; therefore multiplication by −1 (re-negation) is shown in eq. 3.5. (The negation was
made to avoid using the “−” sign in table 3.4.)
Graphical depiction of the β T , α T , and κ T values is shown in figures 3.1, 3.2, and 3.3, respectively
(located at end of Section 3 and starting on page 55). The figures show that systematic spatial
variation in the coefficients for T ⋆ estimation exists. Precise interpretation of the magnitudes and
patterns is difficult to provide. However and in particular, the systematic variation of the α T , and
κ T shown in figures 3.2 and 3.3 emphasizes the semi-independent interplay between the DDF, P ,
and OmegaEM across the state.
Finally, an expression for a statewide T ⋆ generalization based on the median values 1 of the by-county
regression coefficients is:
⎧
0.228 A 0.519 S −0.430 for 2-year recurrence interval
0.113 A 0.384 S −0.628 for 5-year recurrence interval
⎪⎨
TTEXAS ⋆ 0.099 A 0.315 S −0.693 for 10-year recurrence interval
=
0.104 A 0.218 S −0.757 for 25-year recurrence interval
0.137 A 0.150 S −0.780 for 50-year recurrence interval
⎪⎩
0.150 A 0.095 S −0.837 for 100-year recurrence interval
where TTEXAS ⋆ is in minutes, A is drainage area in acres, and S is dimensionless main-channel slope.
These equations are intended for generalized description of watershed time or critical storm duration
for comparison to county-specific values.
1 The sample size for each median computation is 254 because there are 254 counties in Texas.
(3.6)
22