Comparison of Frequency Calculation Methods for Precipitation Series Containing Zero Values

Abstract

The Muskingum-based (MK-based) distributions and their probability weighted moments (PWMs) have been used for frequency calculation of hydrological data that contain zero values. However, the performance of different MK-based distributions have not been compared and evaluated. Moreover, the partial L-moments (PLMs), which are used for analyzing censored samples, have not been used for frequency calculation of such hydrological data. To obtain the most effective method, this study compares and evaluates the performance of four MK-based distributions by fitting 64 monthly precipitation series and using the ordinary least square (OLS) criterion, Akaike information criterion (AIC), residual square sum criterion (RSS), and the Quasi-optimal Deterministic coefficient (QD). The distributions include ​exponential distribution combines with Dirac delta function (M-like), two-parameter gamma distribution (GA2) combines with Dirac delta function (DGA2), two-parameter generalized Pareto distribution combines with Dirac delta function (DGP2), and two-parameter Weibull distribution (WB2) combines with Dirac delta function (DWB2). The applicability of PLMs were also tested and PLMs of four traditional distributions, including GA2, WB2, generalized extreme value distribution (GEV) and three-parameter generalized Pareto distribution (GP3) were used in application. Results showed that the PLMs are feasible for frequency calculation of hydrological data with zeros. The DGP2 and GP3 are superior to the other MK-based distributions and traditional distributions, respectively. The DGP2 distribution is the optimal choice in most cases and is more universal than the other distributions.

Appendices

Appendix

1.1 Pdfs of GA2, GP2, WB2, GEV and GP3 Distributions

The pdf of GA2 distribution is given by

$$f\left( x \right) = \frac{1}{{{a^b}\Gamma \left( b \right)}}{x^{b - 1}}{e^{ - \frac{x}{a}}}$$

(23)

where a and b are the scale and shape parameters of GA2, respectively; $\Gamma \left( b \right)$ is gamma function.

The pdf of GP2 distribution is given by

$$f\left( x \right) = \frac{1}{b}{\left( {1 - \frac{a}{b}x} \right)^{1/a - 1}}{\kern 1pt} {\kern 1pt} {\kern 1pt} ;{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} a \ne 0$$

(24)

where a and b are the scale parameter and shape parameter of GP2 distribution, respectively.

The pdf of WB2 distribution is given by

$$f\left( x \right) = \frac{a}{b}{\left( \frac{x}{b} \right)^{a - 1}}\exp {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \left[ { - {{\left( \frac{x}{b} \right)}^a}} \right]$$

(25)

where a and b are the scale parameter and shape parameter of WB2 distribution, respectively.

The distribution function of GP3 distribution is given by

$$F\left( x \right) = 1 - {\left[ {1 - \frac{k}{\alpha }\left( {x - \xi } \right)} \right]^{1/k}}$$

(26)

where ξ, α, and k are the location, scale and shape parameters of the GP3 distribution, respectively.

The pdf of GEV distribution is given by

$$f\left( x \right) = \frac{1}{\alpha }{\left[ {1 - \frac{k}{\alpha }\left( {x - \xi } \right)} \right]^{1/k - 1}}\exp \left\{ { - {{\left[ {1 - \frac{k}{\alpha }\left( {x - \xi } \right)} \right]}^{1/k}}} \right\}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} ;{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} k \ne 0$$

(27)

where ξ, α, and k are the location, scale and shape parameters of the GEV distribution, respectively.

1.2 PWMs of Muskingum-based Distributions

According to the definition of PWM given by (Greenwood et al. 1979), the PWMs for the distribution shown in Eq. (2) can be written as

$$\begin{aligned}{M_{1,j,0}} &= \int\limits_0^\infty {x{{\left[ {\int\limits_0^x {\left[ {\beta \delta \left( t \right) + \left( {1 - \beta } \right){f_1}\left( t \right) \times 1(t)} \right]dt} } \right]\,}^j}} f\left( x \right)\; dx \\ & = \int\limits_0^\infty {x{{\left[ {\beta + \left( {1 - \beta } \right)\int\limits_0^x {{f_1}\left( t \right)} dt} \right]\,}^j}\left[ {\beta \delta \left( x \right) + \left( {1 - \beta } \right){f_1}\left( x \right) 1(x)} \right]} \; dx \\ &= \left( {1 - \beta } \right)\sum\limits_{i = 0}^j {C_j^i{\beta^i}{{\left( {1 - \beta } \right)}^{j - i}}\int\limits_0^\infty {x{{\left[ {\int\limits_0^x {{f_1}\left( t \right)} dt} \right]\,}^{^{j - i}}} {f_1}\left( x \right)}\; dx{\kern 1pt} } \\ & = \left( {1 - \beta } \right)\sum\limits_{i = 0}^j {C_j^i{\beta^i}{{\left( {1 - \beta } \right)}^{j - i}}{W_{1,j - i,0}}{\kern 1pt} } \end{aligned}$$

(28)

For the M-like distribution, the first two order PWMs are derived as

$${M_{1,0,0}}{\kern 1pt} {\kern 1pt} = \left( {1 - \beta } \right)\gamma$$

(29)

$${M_{1,1,0}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} = \left( {1 - \beta } \right)\left( {\frac{3\gamma }{4} - \frac{\beta \gamma }{4}} \right)$$

(30)

For the DGA2 distribution, the first three order PWMs are derived as

$${M_{1,0,0}}{\kern 1pt} {\kern 1pt} = \left( {1 - \beta } \right)ab$$

(31)

$${M_{1,1,0}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} = a\left( {1 - \beta } \right)\left[ {\beta b + \left( {1 - \beta } \right)\left( {\frac{b}{2} + \frac{1}{{2B\left( {b,{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} \right)}}} \right)} \right]$$

(32)

$${M_{1,2,0}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} = \left( {1 - \beta } \right)\left[ {{\beta^2}ab + \beta \left( {1 - \beta } \right)a\left( {b + \frac{1}{{B\left( {b,{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} \right)}}} \right) + {{\left( {1 - \beta } \right)}^2}a\left( {\frac{b}{3} + \frac{{\Gamma \left( {2b} \right)}}{{\Gamma \left( b \right){2^{2b - 1}}}}{I_\frac{1}{3}}\left( {b,2b} \right)} \right)} \right]$$

(33)

where $B\left( {x,y} \right)$ is the beta function and ${I_z}\left( {x,y} \right)$ is the incomplete beta function.

For the DGP2 distribution, the first three order PWMs are derived as

$${M_{1,0,0}}{\kern 1pt} {\kern 1pt} = \frac{{b\left( {1 - \beta } \right)}}{a}\left( {1 - B\left( {1,a + 1} \right)} \right)$$

(34)

$${M_{1,1,0}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} = \frac{{b\left( {1 - \beta } \right)}}{a}\left[ {\beta \left( {1 - B\left( {1,a + 1} \right)} \right) + {\kern 1pt} \left( {1 - \beta } \right)\left( {\frac{1}{2} - B\left( {2,a + 1} \right)} \right)} \right]$$

(35)

$${M_{1,2,0}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} = \frac{{b\left( {1 - \beta } \right)}}{a}\left[ {{\beta^2}\left( {1 - B\left( {1,a + 1} \right)} \right) + 2\beta \left( {1 - \beta } \right)\left( {\frac{1}{2} - B\left( {2,a + 1} \right)} \right) + {{\left( {1 - \beta } \right)}^2}\left( {\frac{1}{3} - B\left( {3,a + 1} \right)} \right)} \right]$$

(36)

For the DWB2 distribution, the first three order PWMs are derived as:

$${M_{1,0,0}}{\kern 1pt} {\kern 1pt} = \left( {1 - \beta } \right)\frac{b}{a}\Gamma \left( \frac{1}{a} \right)$$

(37)

$${M_{1,1,0}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} = \frac{{b\left( {1 - \beta } \right)}}{a}\Gamma \left( \frac{1}{a} \right)\left[ {\beta + {\kern 1pt} \left( {1 - \beta } \right)\left( {1 - \frac{1}{{{2^{\left( {{1 \mathord{\left/ {\vphantom {1 a}} \right. \kern-\nulldelimiterspace} a} + 1} \right)}}}}} \right)} \right]$$

(38)

$${M_{1,2,0}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} = \left( {1 - \beta } \right)\frac{b}{a}\Gamma \left( \frac{1}{a} \right)\left[ {{\beta^2} + 2\beta \left( {1 - \beta } \right)\left( {1 - \frac{1}{{{2^{\left( {{1 \mathord{\left/ {\vphantom {1 a}} \right. \kern-\nulldelimiterspace} a} + 1} \right)}}}}} \right) + {{\left( {1 - \beta } \right)}^2}\left( {1 - \frac{1}{{{2^{{1 \mathord{\left/ {\vphantom {1 a}} \right. \kern-\nulldelimiterspace} a}}}}} - \frac{1}{{{3^{\left( {{1 \mathord{\left/ {\vphantom {1 a}} \right. \kern-\nulldelimiterspace} a} + 1} \right)}}}}} \right)} \right]$$

(39)

1.3 Parameters Estimation for GA2 Distribution Using PLMs

Substituting Eq. (23) into Eq. (13), one obtains the lower censored PPWMs of GA2 distribution as

$$\begin{gathered} {{\beta ^{\prime}}_r} = \int_{x_0}^\infty {x{{\left[ {\int_0^x {f\left( t \right)dt} } \right]}^r}f\left( x \right)dx} \hfill \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} = \int_{x_0}^\infty {x{{\left[ {\int_0^x {\frac{1}{{{a^b}\Gamma \left( b \right)}}{t^{b - 1}}{e^{ - \frac{t}{a}}}dt} } \right]}^r}\frac{1}{{{a^b}\Gamma \left( b \right)}}{x^{b - 1}}{e^{ - \frac{x}{a}}}dx} \hfill \\ \end{gathered}$$

(40)

where x₀ being the lower censoring threshold.

Let $y = {x \mathord{\left/ {\vphantom {x a}} \right. \kern-\nulldelimiterspace} a}$ and $u = {t \mathord{\left/ {\vphantom {t a}} \right. \kern-\nulldelimiterspace} a}$, Eq. (40) can be written as

$$\begin{gathered} {{\beta {'}}_r} = \int_{\frac{{x_0}}{a}}^\infty {{{\left[ {\int_0^y {\frac{1}{{{a^b}\Gamma \left( b \right)}}{{\left( {au} \right)}^{b - 1}}{e^{ - u}}adu} } \right]}^r}\frac{1}{{{a^b}\Gamma \left( b \right)}}{{\left( {ay} \right)}^b}{e^{ - y}}ady} \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} = \int_{\frac{{x_0}}{a}}^\infty {{{\left[ {\int_0^y {\frac{1}{\Gamma \left( b \right)}{u^{b - 1}}{e^{ - u}}du} } \right]}^r}\frac{a}{\Gamma \left( b \right)}{y^b}{e^{ - y}}dy} \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} = aS{'}_r^1\left( b \right) \\ \end{gathered}$$

(41)

where

$$S{'}_r^1\left( b \right){\kern 1pt} = \int_{\frac{{x_0}}{a}}^\infty {{{\left[ {\int_0^y {\frac{1}{\Gamma \left( b \right)}{u^{b - 1}}{e^{ - u}}du} } \right]}^r}\frac{1}{\Gamma \left( b \right)}{y^b}{e^{ - y}}dy}$$

(42)

Let r = 0 and 1, one gets

$${\beta {'}_0} = aS{'}_0^1\left( b \right) = a\int_{\frac{{x_0}}{a}}^\infty {\frac{1}{\Gamma \left( b \right)}{y^b}{e^{ - y}}dy} = ab\left[ {1 - P\left( {b + 1,\frac{{x_0}}{a}} \right)} \right]$$

(43)

$$\begin{gathered} {{\beta {'}}_1} = \int_{\frac{{x_0}}{a}}^\infty {\left[ {\int_0^y {\frac{1}{\Gamma \left( b \right)}{u^{b - 1}}{e^{ - u}}du} } \right]\frac{1}{\Gamma \left( b \right)}{y^b}{e^{ - y}}dy} \\ = a\int_0^\infty {\left[ {\int_0^y {\frac{1}{\Gamma \left( b \right)}{u^{b - 1}}{e^{ - u}}du} } \right]\frac{1}{\Gamma \left( b \right)}{y^b}{e^{ - y}}dy} - a\int_0^{\frac{{x_0}}{a}} {\left[ {\int_0^y {\frac{1}{\Gamma \left( b \right)}{u^{b - 1}}{e^{ - u}}du} } \right]\frac{1}{\Gamma \left( b \right)}{y^b}{e^{ - y}}dy} {\kern 1pt} {\kern 1pt} \\ = {\kern 1pt} {\kern 1pt} a\left[ {S{'}_{11}^1\left( b \right){\kern 1pt} - {\kern 1pt} S{'}_{12}^1\left( b \right)} \right] \\ \end{gathered}$$

(44)

where $P\left( {a,x} \right)$ is the incomplete gamma function, and

$$S{'}_{11}^1\left( b \right) = \int_0^\infty {\left[ {\int_0^y {\frac{1}{\Gamma \left( b \right)}{u^{b - 1}}{e^{ - u}}du} } \right]\frac{1}{\Gamma \left( b \right)}{y^b}{e^{ - y}}dy}$$

(45)

$$S{'}_{12}^1\left( b \right) = \int_0^{\frac{{x_0}}{a}} {\left[ {\int_0^y {\frac{1}{\Gamma \left( b \right)}{u^{b - 1}}{e^{ - u}}du} } \right]\frac{1}{\Gamma \left( b \right)}{y^b}{e^{ - y}}dy}$$

(46)

With the use of the properties of both the gamma function and the incomplete gamma function, Eqs. (45) and (46) are derived as

$$S{'}_{11}^1\left( b \right) = \frac{b}{2} + \frac{1}{2}{B^{ - 1}}\left( {b,\frac{1}{2}} \right)$$

(47)

$$S{'}_{12}^1\left( b \right) = bP\left( {b,\frac{{x_0}}{a}} \right)P\left( {b + 1,\frac{{x_0}}{a}} \right) - \frac{b}{2}{\left[ {P\left( {b,\frac{{x_0}}{a}} \right)} \right]^2} + \frac{1}{2}{B^{ - 1}}\left( {b,\frac{1}{2}} \right)P\left( {2b,\frac{{2{x_0}}}{a}} \right)$$

(48)

where $B\left( {x,y} \right)$ is the beta function.

Substituting Eqs. (47) and (48) into Eq. (44), we obtain

$$\begin{gathered} {{\beta ^{\prime}}_1} = a\left[ {\frac{b}{2} + \frac{1}{2}{B^{ - 1}}\left( {b,\frac{1}{2}} \right)} \right] \\ - a\left\{ {bP\left( {b,\frac{{x_0}}{a}} \right)P\left( {b + 1,\frac{{x_0}}{a}} \right) - \frac{b}{2}{{\left[ {P\left( {b,\frac{{x_0}}{a}} \right)} \right]}^2} + \frac{1}{2}{B^{ - 1}}\left( {b,\frac{1}{2}} \right)P\left( {2b,\frac{{2{x_0}}}{a}} \right)} \right\} \\ \end{gathered}$$

(49)

Combining Eqs. (14) and (15), one obtains

$${\lambda ^{\prime}_1} = ab\left[ {1 - P\left( {b + 1,\frac{{x_0}}{a}} \right)} \right]$$

(50)

$$\begin{gathered} \frac{{{{\lambda ^{\prime}}_1} + {{\lambda ^{\prime}}_2}}}{2} = a\left[ {\frac{b}{2} + \frac{1}{2}{B^{ - 1}}\left( {b,\frac{1}{2}} \right)} \right] \\ - a\left\{ {bP\left( {b,\frac{{x_0}}{a}} \right)P\left( {b + 1,\frac{{x_0}}{a}} \right) - \frac{b}{2}{{\left[ {P\left( {b,\frac{{x_0}}{a}} \right)} \right]}^2} + \frac{1}{2}{B^{ - 1}}\left( {b,\frac{1}{2}} \right)P\left( {2b,\frac{{2{x_0}}}{a}} \right)} \right\} \\ \end{gathered}$$

(51)

1.4 Parameters Estimation for WB2 Distribution Using PLMs

According to Eqs. (12) and (25), the PPWMs of WB2 distribution are derived as

$${\alpha ^{\prime}_0} = b\Gamma \left( {\frac{1}{a} + 1} \right)\left\{ {1 - P\left[ {\frac{1}{a} + 1, - \ln \left( {1 - {F_0}} \right)} \right]} \right\}$$

(52)

$${\alpha ^{\prime}_1} = \frac{{b\Gamma \left( {\frac{1}{a} + 1} \right)}}{{{2^{1/a + 1}}}}\left\{ {1 - P\left[ {\frac{1}{a} + 1, - 2\ln \left( {1 - {F_0}} \right)} \right]} \right\}$$

(53)

Using Eqs. (52) and (53) one can write

$$\frac{{{{\alpha ^{\prime}}_1}}}{{{{\alpha ^{\prime}}_0}}} = \frac{{1 - P\left[ {\frac{1}{a} + 1, - 2\ln \left( {1 - {F_0}} \right)} \right]}}{{{2^{1/a + 1}}\left\{ {1 - P\left[ {\frac{1}{a} + 1, - \ln \left( {1 - {F_0}} \right)} \right]} \right\}}} = z$$

(54)

The curve of z vs. a can be approximated by a quadratic function of the form

$$a = {a_0} + {a_1}z + {a_2}{z^2}$$

(55)

For a fixed F0, five z values can be calculated by the right side of Eq. (54), corresponding to five given a values, e.g. a = 0.1–0.5. Substituting these calculated z values into Eq. (55) yields a set of linear equations. One can then get the solutions for a₀, a₁ and a₂ corresponding to that fixed value of F0.

Replacing the values of the PPWM ${\alpha ^{\prime}_0}$ and ${\alpha ^{\prime}_1}$ in Eq. (54) by the PLMs ${\lambda ^{\prime}_1}$ and ${\lambda ^{\prime}_2}$ shown in Eqs. (14)-(15), one can write

$$z = \frac{{{{\alpha ^{\prime}}_1}}}{{{{\alpha ^{\prime}}_0}}} = \frac{{\left( {{{\lambda ^{\prime}}_1} - {{\lambda ^{\prime}}_2}} \right)}}{{2{{\lambda ^{\prime}}_1}}}$$

(56)

The sample estimate $\hat z$ can then be obtained by replacing ${\lambda ^{\prime}_1}$ and ${\lambda ^{\prime}_2}$ by their sample estimate ${l^{\prime}_1}$ and ${l^{\prime}_2}$.

$$\hat z = \frac{{\left( {{{l^{\prime}}_1} - {{l^{\prime}}_2}} \right)}}{{2{{l^{\prime}}_1}}}$$

(57)

Substituting Eq. (57) and the estimates of a₀, a₁ and a₂ in Eq. (55), one can get the estimate for $\hat a$. The parameter b can then be obtained successively from Eq. (58).

$$\hat b = \frac{{{{l^{\prime}}_1}}}{{\Gamma \left( {\frac{1}{\hat a} + 1} \right)\left\{ {1 - P\left[ {\frac{1}{\hat a} + 1, - \ln \left( {1 - {F_0}} \right)} \right]} \right\}}}$$

(58)

The Optimal Distribution for Each Station

Table 7 The optimal distribution for each station