Runoff Estimation for an Ungauged Catchment Using Geomorphological Instantaneous Unit Hydrograph (GIUH) and Copulas

Abstract

A methodology is proposed to apply Geomorphological Instantaneous Unit Hydrograph to ungauged basins using Monte Carlo Simulations and copulas. The effective rainfall, input of GIUH is assumed to be unknown; it is estimated with infiltration index method (ϕ-index). Correlations are detected between this index and the characteristics of rainfall. They are modeled with copulas, and are used to derive effective rainfall hyetographs. The generated hydrographs from GIUH are analyzed and give statistically the same results: dispersion and variability for all studied characteristics (volume, peak discharge, peak time and base time). However, only these hydrographs derived from ϕ conditioned to maximum intensity distribution allow reconstituting the observed hydrographs. Moreover comparing the series of order statistics of interest output and observed series, leads to decide on the representative hydrograph of the catchment behavior.

Appendices

Appendix 1

Table 5 Several references published copula models. One famous reference: Nelsen (1999) who presented among these models, the Archimedean ones and particularly those of one parameter. The following table shows the copula generators and the relation between the copula parameter and Kendall’s τ

Appendix 2

K ( z ) function: is the distribution function of the copula C(U,V). Genest and Rivest (1993) showed that this distribution function is related to the generator φ of an Archimedean copula through the expression of K(z):

$$ K(z) = z - \phi (z)/\phi '(z) $$

(A2.1)

An empirical K(z) can be calculated for any z as the proportion of empirical values of C(u,v) that is less than z:

$$ {K_{\text{emp}}}(z) = \left\{ {{\text{number}}\,\,{\text{o}}f\,\,{z_i} \leqslant \,\,z} \right\}/n $$

(A2.2)

J ( z ) function or cumulative τ : tau is related to a copula through the expression:

$$ \tau = - 1 + 4\int\limits_0^1 {\int\limits_0^1 {C(u,v)c(u,v){\text{d}}u{\text{d}}v} } $$

(A2.3)

J(z) function is expressed by:

$$ J(z) = - 1 + 4\left( {\int\limits_0^z {\int\limits_0^z {C(u,v)c(u,v){\text{dudv}}} } } \right)/C{(z,z)^2} $$

(A2.4)

The full double integral is a probability weighted average of C(u,v). To compare this, the partial integral has to be divided by the weights, thus the first power of C(z, z) is the denominator. This quotient gives the average value of C(u,v), which increases as a function of z. The second C(z, z) divisor expresses this average relative to C(z, z). It should be that $J(1) = \tau$ _.

An empirical cumulative tau can also be calculated, expressed by:

$$ J(z) = - 1 + 4I(z)/C{(z,z)^2} $$

(A2.5)

Where I(z) is defined by $I(z) = \frac{1}{n}\,\,\sum\limits_{i = 1}^n {{z_i}\, \times }$I$\left\{ {{u_i} < z\,et\,{v_i}\, < \,z} \right\}$ , with I is the indicator function.

M ( z ) function is the cumulative conditional mean defined by:

$$ M(z) = E(V\left| U \right. < z) = \left( {\int\limits_{u = 0}^z {\int\limits_{v = 0}^1 {v \cdot c(u,v){\text{d}}u{\text{d}}v} } } \right)/z $$

(A2.6)

Verifying M(1) = 1/2.

Let D(z) $D(z) = \sum\limits_{i = 1}^n {\text{I}} \left\{ {{u_i} < z} \right\}$ and $N(z) = \sum\limits_{i = 1}^n {{v_i}{\text{I}}\left\{ {{u_i} < z} \right\}}$, the empirical version of M(z) is expressed by:

$$ M(z) = N(z)/D(z) $$

(A2.7)

With D(1) = n and N(1) = n/2.

L ( z ) and R ( z ) functions are Left and Right tail concentration functions. The two functions L(z) and R(z) are:

$$ L(z) = P(U < z,\,V < z)/z = C(z,z)/z $$

(A2.8)

$$ R(z) = P(U > z\,\,{,}\,\,V > z)/(1 - z) = \left( {1 - 2z + C(z,z)} \right)/(1 - z) $$

(A2.9)

Joe (1997) defined lower tail dependence parameter for L(0)=${\lambda_{\min }} = \mathop {{\lim }}\limits_{u \to 1} P(Y \leqslant F_Y^{ - 1}(u)\left| {X \leqslant F_X^{ - 1}(u)} \right.) = \mathop {{\lim }}\limits_{z \to 0} L(z)$ (left tail), and upper tail dependence parameter for R(1)=${\lambda_{\max }} = \mathop {{\lim }}\limits_{u \to 1} P(Y > F_Y^{ - 1}(u)\left| {X > F_X^{ - 1}(u)} \right.) = \mathop {{\lim }}\limits_{z - 1} R(z)$ (right tail).

L function is analyzed for z $\in \left[ {0,\frac{1}{2}} \right]$ and R function for all z $\in \left[ {\frac{1}{2},1} \right]$.

Appendix 3

Let (X,Y) be a sample of size n.

The X _i and Y _i are regrouped into 6 classes respectively (v ₀ ; v ₁ ]; (v ₁ ; v ₂ ]; …; (v ₅ ; v ₆ ], and (w ₀ ; w ₁ ]; (w ₁ ; w ₂ ]; …; (w ₅ ; w ₆ ], where the boundaries v _i ’s (w _i ) are chosen such that the number of observations λ₁, λ₂ …, λ₆ respectively η₁, η₂ …, η_6, in the corresponding classes are as symmetrically distributed as possible. We thus obtain 36 two-dimensional intervals (v _i-1 ; v _i ] x (w _i-1 ; v _i ], i,j = 1….6. Then we regroup these intervals in k larger rectangular interval classes, such that an expected frequency of at least 1% in each class and a 5% expected frequency in 80% of the classes. The fitted number of observations f _i,j in each 36 two-dimensional intervals (v _i-1 ; v _i ] x (w _i-1 ; v _i ], is given by:

$$ \begin{array}{*{20}l} {{f_{{i,j}} = n{\left[ {{\left( {F{\left( {v_{i} } \right)},F{\left( {w_{j} } \right)}} \right)} - {\left( {F{\left( {v_{{i - 1}} } \right)},F{\left( {w_{j} } \right)}} \right)} - {\left( {F{\left( {v_{i} } \right)},F{\left( {w_{{j - 1}} } \right)}} \right)} + {\left( {F{\left( {v_{{i - 1}} } \right)},F{\left( {w_{{j - 1}} } \right)}} \right)}} \right]}} \hfill} \\ {{i,j = 1.....6,\quad F{\left( {x,y} \right)} = C{\left( {F_{X} {\left( x \right)},F_{Y} {\left( y \right)}} \right)}} \hfill} \\ \end{array} $$

(A3.1)

Let z _i,j be the number of observations in the 36 two-dimensional intervals. Through summation of z _{i j} ’s respectively f _i;j ’s, one obtains the number of observations O _k , respectively, the expected number of observations E _k , in each rectangular interval class k. The bivariate Chi-square statistic is then defined by:

$$ {\chi^2} = \sum\limits_{k = 1}^m {{{\left( {{O_k} - {E_k}} \right)}^2}/{E_k}} $$

(A3.2)