Sensitivity analysis of the probability distribution of groundwater level series based on information entropy

Abstract

Information entropy is an effective method to analyze uncertainty in various processes. The principle of maximum entropy (POME) provides a guide line for the parameter estimation of probability density function (PDF). Mutual entropy analysis is well qualified for delineating the nonlinear and complex multivariable relationship. The probability distribution of model output is the element of model uncertainty analysis. In this paper, a synthetic groundwater flow field is build to produce groundwater level series (GLS). The probability distribution of GLS is obtained by the frequency analysis method based on POME and Chi-Squared test. The important uncertainty factors that affect the parameters of PDF of GLS are assessed by the sensitivity analysis methods, which include stepwise regression analysis and mutual entropy analysis. Results of this analysis indicate that most of the GLS follow normal distribution (or log-normal distribution), while a few obey others. The mean and variance of normal GLS are affected differently by the input variables of groundwater model. Mutual entropy analysis is more competitive and appropriate for delineating the nonlinear and nonmonotonic multivariable relationship than stepwise regression analysis.

Appendix: Methods for parameters estimation

Supposing a number of samples X = (x ₁, x ₂, …, x _n ), n is the length of sample series.

1.1 Normal distribution

The probability density function (PDF) of normal distribution can be expressed as:

$$ f(x) = \frac{1}{{\sqrt {2\pi } \sigma }}\exp \left[ { - \frac{{(x - \mu )^{2} }}{{2\sigma^{2} }}} \right] $$

(9)

The parameters can be estimated by:

$$ \mu = E[x],\quad \sigma^{2} = Var[x] $$

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E[x], Var[x] are mean and variance of samples.

1.2 Gamma-2 distribution

The PDF of gamma-2 distribution is given as:

$$ \left\{ {\begin{array}{*{20}c} {f(\alpha ,\beta ,x) = \frac{{x^{\alpha - 1} \beta^{ - \alpha } e^{( - x/\beta )} }}{\Upgamma (\alpha )}} \hfill \\ {\Upgamma (\alpha ) = \int\limits_{0}^{\infty } {y^{\alpha - 1} \exp ( - y)dy} } \hfill \\ \end{array} } \right. $$

(11)

where α is shape parameter, β is scale parameter. When the parameters are estimated by traditional moment method, α and β can be expressed as follows:

$$ E[x] = \alpha /\beta ,\quad Var[x] = \alpha^{2} /\beta $$

(12)

The parameters are estimated as the function of the second moment of samples, and the estimation error cannot be ignored if the length of data series is not long enough. According to the principle of maximum entropy (POME), the equation of constrains can be written in the following form (Singh and Singh 1985a):

$$ \left\{ {\begin{array}{*{20}c} {{\text{Maximize }}H = - \int\limits_{ - \infty }^{\infty } {f(x)\ln f(x)dx} } \hfill \\ {\int\limits_{0}^{\infty } {f(x)dx = 1} } \hfill \\ {\int\limits_{0}^{\infty } {xf(x)dx = E[x]} } \hfill \\ {\int\limits_{0}^{\infty } {\ln x \cdot f(x)dx = E[\ln x]} } \hfill \\ \end{array} } \right. $$

(13)

And the parameters are estimated by following equations:

$$ \left\{ {\begin{array}{*{20}c} {E[x] = \alpha /\beta } \hfill \\ {E[\ln x] = \ln (1/\beta ) + \psi (\alpha )} \hfill \\ {\psi (x) = d[\ln \Upgamma (x)]/dx = \ln x - \frac{1}{2x} - \frac{1}{{12x^{2} }} + \frac{1}{{120x^{4} }} - \frac{1}{{252x^{6} }}} \hfill \\ {\Upgamma (x) = \int\limits_{0}^{\infty } {y^{x - 1} \exp ( - y)dy} } \hfill \\ \end{array} } \right. $$

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1.3 P-III distribution

The PDF of p-III distribution can be expressed as:

$$ f(x) = \frac{{\beta^{\alpha } }}{\Upgamma (\alpha )}(x - c)^{\alpha - 1} e^{ - \beta (x - c)} ,\quad \alpha > 0,\;x \ge c $$

(15)

where α is shape parameter, β is scale parameter, c is location parameter. When the parameters are estimated by traditional moment method, the parameters can be given by:

$$ \left\{ {\begin{array}{*{20}c} {\alpha = \frac{4}{{C_{s}^{2} }}} \hfill \\ {\beta = \frac{2}{{\bar{x}C_{v} C_{s} }}\bar{x} = E[x]} \hfill \\ {c = \bar{x}\left( {1 - \frac{{2C_{v} }}{{C_{s} }}} \right)C_{v} = \frac{{\sigma_{x} }}{E[x]}} \hfill \\ \end{array} } \right. $$

(16)

C _s is skewness coefficient, C _v is deviation coefficient. Hereinto, the C _v is the third order moment of samples, a large estimation error will be inevitable if the length of data series is not long enough. Based on POME, the equation of constrains can be written in the following form (Singh and Singh 1985b; Singh 1987):

$$ \left\{ {\begin{array}{l} {{\text{Maximize}}\;H = - \int_{{ - \infty }}^{\infty } {f(x)\ln f(x)dx} } \\ {\int_{0}^{\infty } {f(x)dx = 1} } \\ {\int_{0}^{\infty } {xf(x)dx = E[x]} } \\ {\int_{c}^{\infty } {\ln (x - c) \cdot f(x)dx = E[\ln (x - c)]} } \\ \end{array} } \right.$$

(17)

Based on Eq. 17, a formula can be derived as:

$$ \frac{1}{n}\sum\limits_{i = 1}^{n} {\ln \left[ {x_{i} - \bar{x} + \sigma_{x} \alpha^{1/2} } \right]} = \frac{1}{2}\ln \alpha + \ln \sigma_{x} - \frac{1}{2\alpha } - \frac{1}{{12\alpha^{2} }} + \frac{1}{{120\alpha^{4} }} - \frac{1}{{252\alpha^{6} }} $$

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After that, Usually, the curve-fitting method (Chen et al. 2002) is adopted to solve the Eqs. 16,18.

1.4 Uniform distribution

The PDF of uniform distribution can be expressed as:

$$ f(x) = \left\langle {\begin{array}{*{20}c} {\frac{1}{{\theta_{2} - \theta_{1} }}} & {\theta_{1} \le x \le \theta_{2} } \\ 0 & {\text{else}} \\ \end{array} } \right. $$

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The parameters can be estimated by (Sun and Zheng 2006):

$$ \theta_{1} = \frac{1}{n - 1}\left( {nX_{1}^{*} - X_{n}^{*} } \right),\quad \theta_{2} = \frac{1}{n - 1}\left( {nX_{n}^{*} - X_{1}^{*} } \right) $$

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$$ X_{1}^{*} = \min (x_{1} , \ldots ,x_{n} ),\quad X_{n}^{*} = \max (x_{1} , \ldots ,x_{n} ) $$

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