Nonstationary statistical approach for designing LNWLs in inland waterways: a case study in the downstream of the Lancang River

Abstract

Conventional methods to design the lowest navigable water level (LNWL) in inland waterways are usually based on stationary time series. However, these methods are not applicable when nonstationarity is encountered, and new methods should be developed for designing the LNWL under nonstationary conditions. Accordingly, this article proposes an approach to design the LNWL in nonstationary conditions, with a case study at the Yunjinghong station in the Lancang River basin in Southwest China. Both deterministic (trends, jumps and periodicities) and stochastic components in the hydrological time series are considered and distinguished, and the rank version of the von Neumann’s ratio (RVN) test is used to detect the stationarity of observed data and its residue after the deterministic components are removed. The stationary water level series under different environments are then generated by adding the corresponding deterministic component to the stationary stochastic component. The LNWL at the Yunjinghong station was estimated by this method using the synthetic duration curve. The results showed that the annual water level series at the Yunjinghong station presented a significant jump in 2004 with an average magnitude decline of − 0.63 m afterwards. Furthermore, the difference of the LNWL at certain guaranteed rate (90%, 95% and 98%) was nearly − 0.63 m between the current and past environments, while the estimated LNWL under the current environment had a difference of − 0.60 m depending on nonstationarity impacts. Overall, the results clearly confirmed the influence of hydrological nonstationarity on the estimation of LNWL, which should be carefully considered and evaluated for channel planning and design, as well as for navigation risk assessment.

Appendices

Appendix 1: Description of special terms

Term	Description
Water level	Elevation of the free surface of a stream, lake or reservoir relative to a specified datum
Low water level	Water level is relatively lower than average level for specific time scale
Lowest navigable water level (LNWL)	Minimum required water level for ships to navigate under the certain guaranteed rate
Lowest safe channel depth	Minimum required channel depth for ships to navigate under the certain guaranteed rate
Ship draft	Vertical distance between the waterline and the bottom of the hull
Under keel clearance (UKC)	Minimum clearance available between the deepest point on a vessel and the bottom
Water depth	Elevation of the free surface of a stream, lake or reservoir to the corresponding bottom
Changing environment	As a result of climate change and human activities, and will lead to a change in the statistical distribution of hydrological process
Additive model	Assumption of the time series composed of various components
Deterministic component	Consists of trend, jump and periodic components, which is considered to be due to changing environment with nonstationary characters
Trend component	A trend exists when there is an increasing or decreasing direction in the time series. The trend component does not have to be linear
Jump component	A jump exists when there is a change point in the time series
Periodic component	A period pattern exists when a time series is influenced by seasonal factors (e.g., the quarter of the year, the month, or day of the week)
Stochastic component	Remainder of the time series after the deterministic components have been removed, which can be described by a random process
Annual water level series	Composed of annual average water level for continuous years
Daily water level series	Composed of daily average water level for continuous days

Appendix 2: Methods for detection of deterministic components

• 1. The Mann–Kendall test. For the hydrologic series $X_{t}$ ($t = 1,2 \ldots n$), the Mann–Kendall test has the basic idea of calculating the character S:

  $$S = \sum\limits_{i = 1}^{n - 1} {\sum\limits_{j = i + 1}^{n} {\text{sgn} (X_{j} - X_{i} )} }$$

  (18)

  where $n$ is the number of data points; $\text{sgn} (y) = 1$ if $y > 0$, and $\text{sgn} (y) = 0$ if $y = 0$.$S$ has a null expected value $E\left[ S \right] = 0$ and its variance is given by:

  $$Var\left[ S \right] = \frac{1}{18}\left[ {N(N - 1)(2N + 5) - \sum\limits_{m = 1}^{M} {t_{m} (t_{m} - 1)(2t_{m} + 5)} } \right]$$

  (19)

  where M is the number of sets of tied groups and $t_{m}$ is the size of the mth tied group. The standardized test statistic $U_{MK}$, which follows a standard Normal distribution, is computed as:

  $$U_{MK} = \left\{ {\begin{array}{*{20}l} {(S - 1)/\sqrt {Var(S)} } \hfill & {S > 0} \hfill \\ 0 \hfill & {S = 0} \hfill \\ {(S + 1)/\sqrt {Var(S)} } \hfill & {S < 0} \hfill \\ \end{array} } \right.$$

  (20)

  The null hypothesis of no trend is rejected if the absolute value of $U_{MK}$ is bigger than the theoretical value $U_{1 - \alpha /2}$ with the specification of the significance level $\alpha$.

• 2. The Brown–Forsythe test. If hydrologic series $X_{t}$ ($t = 1,2 \ldots n$) has a jump point, with the sub-series $\left\{ {X_{1i} |i = 1,2 \ldots n_{1} } \right\}$ and $\left\{ {X_{2i} |i = 1,2 \ldots n_{2} } \right\}$.The test statistic $F$ is given as:

  $$F = \frac{{n_{1} (\bar{X}_{1} - E)^{2} + n_{2} (\bar{X}_{2} - E)^{2} }}{{\left( {1 - \frac{{n_{1} }}{{n_{2} }}} \right)S_{1}^{2} + \left( {1 - \frac{{n_{2} }}{{n_{1} }}} \right)S_{2}^{2} }} \sim F(1,f)$$

  (21)

  with $\bar{X}_{1} = \frac{1}{{n_{1} }}\sum\nolimits_{i = 1}^{{n_{1} }} {X_{1i} }$, $\bar{X}_{2} = \frac{1}{{n_{2} }}\sum\nolimits_{i = 1}^{{n_{2} }} {X_{2i} }$, $S_{1}^{2} = \frac{1}{{n_{1} - 1}}\sum\nolimits_{i = 1}^{{n_{1} }} {(X_{1i} - \bar{X}_{1} )^{2} }$, $S_{2}^{2} = \frac{1}{{n_{2} - 1}}\sum\nolimits_{i = 1}^{{n_{2} }} {(X_{2i} - \bar{X}_{2} )^{2} }$, $E = \frac{1}{{n_{1} }}(\sum\nolimits_{i = 1}^{{n_{1} }} {X_{1i} + } \sum\nolimits_{i = 1}^{{n_{2} }} {X_{2i} } )$, $n = n_{1} + n_{2}$, $f = 1/\sum\nolimits_{i = 1}^{2} {(c_{i}^{2} /(n_{2} - 1))}$, $c_{i} = (1 - n_{i} /n)S_{i}^{2} /(\sum\nolimits_{i = 1}^{2} {(1 - n_{i} /n)S_{i}^{2} } )$.

  The Brown–Forsythe test has the null hypothesis H₀: no jump point in the series $X_{t}$. If $F > F_{\alpha }$ $F$ is bigger than the theoretical value $F_{\alpha }$ with the specification of the significance level $\alpha$, H0 is rejected, meaning significant jump.

• 3. The Fourier Series Method. If there exists periodicity in a time series free of trend and jump, it can be represented by a Fourier series, which is expressed as follows:

  $$Y(t) = A_{0} + \sum\limits_{k = 1}^{l} {[(A_{k} \sin (2\pi kt/T) + B_{k} \cos (2\pi kt/T)]}$$

  (22)

  where $Y(t)$ is the harmonically fitted means at period $t(t = 1,2, \ldots ,T)$, $A_{0}$ is the population mean, $l$ is the total number of harmonics ($l = T/2$ for even T and $(T + 1)/2$ for odd T). T is the base period or period of the function and $A_{k}$ and $B_{k}$ are sine and cosine Fourier coefficients, respectively, and here they are computed as:

  $$\begin{aligned} A_{0} & = (1/T)\sum\limits_{t = 1}^{T} {\bar{x}_{t} } \\ A_{k} & = (2/T)\sum\limits_{t = 1}^{T} {\bar{x}_{t} } \sin (2\pi kt/T) \\ B_{k} & = (2/T)\sum\limits_{t = 1}^{T} {\bar{x}_{t} } \cos (2\pi kt/T) \\ \end{aligned}$$

  (23)

  where $k = 1,2, \ldots ,T/2 - 1$.

The variations caused by a periodic component $I_{k}$, say kth harmonic, is computed as:

$$I_{k} = (1/2)(A_{k}^{2} + B_{k}^{2} )$$

(24)

Then, the significance of different harmonics is tested using the Fisher’s g-statistic, which is given as:

$$g_{k} = {{I_{k} } \mathord{\left/ {\vphantom {{I_{k} } {\sum\limits_{k = 1}^{T/2} {(A_{k}^{2} + B_{k}^{2} )/2} }}} \right. \kern-0pt} {\sum\limits_{k = 1}^{T/2} {(A_{k}^{2} + B_{k}^{2} )/2} }}$$

(25)

The cumulative periodogram $T_{j}$ is calculated as:

$$T_{j} = \frac{{\sum\nolimits_{k = 1}^{j} {(A_{k}^{2} + B_{k}^{2} )/2} }}{{\sum\nolimits_{t = 1}^{T} {(\bar{x}_{t} - \mu )^{2} /T} }}$$

(26)

where $\mu$ is the mean of $Y_{t}$.