The Monthly Distribution selection provides 9 distributions shown in the table below.  A user specified option is also available.  This method currently only works in conjunction with another outside software (i.e. HEC-WAT with the Hydrologic Sampler plugin).

Once the distribution has been selected, parameters must be specified for each month (January to December).  When used within HEC-WAT with the Hydrologic Sampler Plugin, the date selection from the Hydrologic Sampler is passed to HEC-HMS which then uses the distribution parameters for the selected month to create the distribution.  The selection of a model parameter value follows the sample process described in the Simple Distribution. HEC-HMS generates the seed value associated with the time the Uncertainty Compute Type was created. However, this initial seed value can be modified if required.


Function

Formula

Beta


f(x)=\frac{1}{B(\alpha,\beta)}\frac{(x-lower)^{\alpha-1}(upper-x)^{\beta-1}}{(upper-lower)^{\alpha+\beta-1}}

B(\alpha,\beta) is the Beta function

Exponential

f(x)=\frac{1}{\mu}\exp(-\frac{x-shift}{\mu})

Gamma



f(x)=\frac{(x-shift)^{\alpha-1}}{\beta^{\alpha}}\frac{1}{\Gamma(\alpha)}\exp(-\frac{x-min}{\beta})

Shape=α, Scale=β

\Gamma(\alpha) is the Gamma function

Gumbel

f(x)=\frac{1}{\alpha}\exp(-(\frac{x-\xi}{\alpha}+\exp(-\frac{x-\xi}{\alpha})))

Location = ξ, Scale = α

Log-normal

f(x)=\frac{1}{x\sqrt{2\pi\sigma^{2}}}\exp(-\frac{(\log(x)-\mu)^{2}}{2\sigma^{2}})

Normal

f(x)=\frac{1}{\sqrt{2\pi\sigma^{2}}}\exp(-\frac{(x-\mu)^{2}}{2\sigma^{2}})

Triangular


f(x)=\begin{cases} 0, & x < a,\\ \frac{2(x-a)}{(b-a)(c-a)}, & a \le x < c,\\ \frac{2}{b-a}, & x = c,\\ \frac{2(b-x)}{(b-a)(b-c)}, & c < x \le b,\\ 0, & b < x. \end{cases}

Lower = a, Upper = b, Mode = c

Uniform


f(x)=\begin{cases} \frac{1}{b-a}, & x \in [a,b] \\ 0, & \text{otherwise}. \end{cases}

Lower=a, Upper=b

Weibull


f(x)=\begin{cases} \frac{k}{\lambda}(\frac{x-shift}{\lambda})^{k-1}\exp(-(\frac{x-shift}{\lambda})^{k}), & x \ge 0 \\ 0, & x < 0 \end{cases}

Shape=k, Scale=λ