Nonstationary statistical approach for designing LNWLs in inland waterways: a case study in the downstream of the Lancang River
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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.
Keywords
Lowest navigable water level Inland waterways Synthetic duration curve Hydrological nonstationarity Lancang River1 Introduction
Inland waterways play a vital role in promoting economic development, due to their considerable capacity for the lowcost transport of goods. Especially in China, there are numerous rivers and lakes throughout the mainland, and they offer abundant water resources for the development of inland waterways (Paul et al. 2009; Yan et al. 2017b). However, navigation on these inland waterways has faced some troubles caused by the limited runoff and the low water levels (as described in “Appendix 1”) during dry seasons (Linde et al. 2016). Transport vessels have to reduce their loads and even stop navigating to avoid the risk of grounding (Samuelides et al. 2009; Hawkes et al. 2010; Mazaheri et al. 2016). These situations increase the costs of transport and will be unfavorable for the development of the navigation industry (Jonkeren et al. 2011).
To optimize the capacity of inland navigation, many researches have considered both ship loading variations and channel dredging to provide the largest ship draft and the best available navigable depth. Generally, declining water levels will reduce ship clearances in channels and increase the demand for dredging (Kling et al. 2003). While it was not always possible or desirable to increase channel depths, it may be necessary to either lighten the current ship loading maximum or use a new type of ships in order to meet the underkeel safety margin requirement (Grambsch 2002).
Among these methods, designing, building and outfitting new ships or dredging channels are both timeconsuming and costly, and are therefore insufficient to compensate for the losses resulting from recurrent low water levels. Eloot and Söhngen (2014) considered the impact of tide and water level changes as well as other factors on the design of channels to meet the requirement of navigable depth and to assess the restrictions on vessels caused by low water levels. In order to ensure continuous water levels and the minimum required navigable water level, Wagenpfeil et al. (2010) created a model to adjust the amount of pumping and discharge. However, navigable hydrological conditions were not thoroughly considered, especially in the context of a changing environment.
Because the available water depth is one of the most critical factors for efficient inland navigation, the concept of navigable depth (or nautical depth) has been accepted for defining a safe and effective channel bottom criterion. The requirement of water depth is that these vessels can safely navigate through the channel with a suitable under keel clearance (UKC) over the fluid mud (Herbich et al. 2015; Mcanally et al. 2015), where the UKC is the difference between the ship draft and the lowest safe channel depth, as defined by United States Army Corps of Engineers (USACE 2006). In China, the lowest navigable water level (LNWL) is designed using the synthetic duration curve, as the primary method, with the durationfrequency method utilized as a secondary method. Both the LNWL and the lowest safe channel depth can reflect the navigable capacity of a channel, and they can convert to each other with no essential differences. However, these methods for the design of the LNWL are mainly based on stationary hydrological time series; they are not applicable for nonstationary hydrological time series due to the change of environment (Parry et al. 2007; Blöschl et al. 2007; Sivapalan and Blöschl 2015) that can result in changes in the frequency distribution of hydrological events (e.g., floods, droughts, etc.) (Milly et al. 2008; Schiermeier 2011; Sang et al. 2017), as well as occurrences of different forms and degrees of hydrological variations (Villarini et al. 2009; Vogel et al. 2011). Therefore, an approach for dealing with the nonstationary hydrological time series is needed in determining LNWLs.
Generally, studies of nonstationary hydrological frequency analysis methods can be divided into two main classes (Liang et al. 2017). The first uses nonstationary frequency analysis models to estimate the hydrological design values (Strupczewski et al. 2001; Alila and Mtiraoui 2002; Singh et al. 2005; Villarini et al. 2009). However, these methods require the complex calculation of model parameters, generally associated with considerable estimation errors. In addition, the design standards (e.g., exceedance probability, return period, etc.) for stationary frequency analysis cannot be used when adopting distribution functions varying with time for this first class of methods. Thus, exceedance probability, return period or the other design standards need to be carefully defined in a changing environment (Yan et al. 2017a). The second class of methods involves converting the hydrological time series from nonstationary to stationary in order to apply conventional stationary frequency analysis methods (Xie et al. 2005; Hu et al. 2015; Gado and Nguyen 2016; Liang et al. 2017). These methods have been widely used in frequency analysis of nonstationary annual runoff or of extreme flow with the goals of estimating flood or drought risk. However, they have not been used in frequency analysis for nonstationary water levels, which could have a great influence on navigation safety.
Accordingly, a nonstationary approach is proposed for LNWL design. This method is developed based on the concept of detecting and decomposing different components in hydrological time series, and then composing these components in annual time series (Xie et al. 2005), and thus it is categorized with the second class of methods defined above. Specifically, the stationary daily water level series is composed by multiplying the observed daily water level by a ratio scaling index. The ratio scaling index is defined as the ratio of the reconstructed stationary annual water level and the observed annual water level for each year. This approach allows the use of conventional methods to design the LNWL. The synthetic duration curve is used here to estimate the LNWL based on the composed daily water level series. The article is organized as follows. Section 2 describes the study area and data, and then the methodology for designing nonstationary LNWLs is presented in Sect. 3. The results and discussion are presented in Sect. 4, followed by conclusions and suggestion for future work in Sect. 5. These sections are followed by two appendices that describe specific terminology and methods used for detecting deterministic components, respectively.
2 Study area and data
The Lancang River, with a length of 4880 km, is the upper section of the LancangMekong River, which is the largest river in Southeast Asia and has a drainage area of approximately 795 000 km^{2} (Campbell 2009). From its source to its outlet on the China–Myanmar border, the river plunges 4700 m through the high gorges of Tibet and Yunnan Province, which is more than 90% of its entire drop in elevation (Fan et al. 2015). The Lancang River is an important navigable river for Yunnan Province, and it also plays a significant role in international waterway transport linking China and Indochina Peninsula countries; the LancangMekong River is regarded as an international golden waterway. The river thus has special international meaning for the economic development of Yunnan Province and for the Association of Southeast Asian Nations (ASEAN). However, the physical conditions of hydrological cycles and of the water level profile in the Lancang River basin have changed under the influence of global climate change and especially highintensity human activities, including cascade hydropower development (Lauri et al. 2012; Tang et al. 2014; Räsänen et al. 2017). The observed water level series present variability and changes, which means that the assumption of stationary observations is not fulfilled. Especially at the Yunjinghong hydrological station, the main control station in the dowstream of the Lancang River, the streamflow regimes are affected not only by climate change but also by upstream dams.
Characteristics of the major dams in Lancang River basin
Dam  Year started  Year completed  Total storage (km^{3})  Catchment area (km^{2}) 

Manwan  1986  1992  0.92  114,500 
Dachaoshan  1992  2003  0.93  121,000 
Xiaowan  2002  2010  14.56  113,300 
Jinghong  2003  2009  1.23  149,100 
Nuozhadu  2004  2014  22.40  144,700 
Gongguoqiao  2007  2011  3.16  97,300 
For the design of the LNWL, water level variations should be considered, including its response to hydropower development and its influence on navigation safety. In this study, the daily water level data measured at the Yunjinghong station (national vertical datum 1985 in China) during the 1955–2014 are chosen to design the LNWL in the downstream section of the Lancang River.
3 Approach for designing nonstationary LNWLs
Hydrological time series include deterministic components, which show nonstationary characteristics (Papoulis and Pillai 2002; Salas 1993). Since conventional methods for the design of LNWLs in inland waterways are based on stationary data, they are not applicable when the requirement of stationarity cannot be satisfied. Thus, the initial problem for designing the nonstationary LNWL is to detect deterministic components in water level series. In this paper, the Mann–Kendall test, the Brown–Forsythe test and the Fourier series method (as described in “Appendix 2”) are chosen to detect trends, jumps and periodicities, respectively. The stationary series are then constructed by composing the stationary stochastic component and the deterministic component for any selected year. Since the deterministic component is a constant, the composed series are stationary with constant mean value (Milly et al. 2015). By utilizing this method, the LNWL can be determined using a synthetic duration curve.
3.1 Preliminary assumption
Hydrological time series can be decomposed into deterministic and stationary stochastic parts. Such decomposition allows the application of statistical tools based on stationarity in the characterization of the stationary stochastic components of observed time series (Milly et al. 2015).
3.2 Detection of hydrological components
Here the \(N(2,4/T)\) distribution can be used as an approximation of RVN (where \(N\) represents a normal distribution; 2 and \(4/T\) represent the mean and variance, respectively). The time series is nonstationary if the value of RVN is bigger than the value of \((2{ + }2U_{{1{  }\alpha /2}} /\sqrt n )\) or smaller than the value of \((2{  }2U_{{1{  }\alpha /2}} /\sqrt n )\), where \(U_{1  \alpha /2}\) is the critical value of standard normal distribution with the specification of the significance level \(\alpha\). Otherwise, the time series is stationary.
A and B can be calculated using the least squares method with each assumed \(t_{0}\), and their values are chosen by applying the Nash efficiency coefficient, which quantifies the similarity between \(Y_{t}\) and the residue obtained by removing the trend or jump component of the observed series.
3.3 Design of the LNWL

Step i: Detect the trend and jump components For an annual water level series, its trend and jump components are detected by using the Mann–Kendall test and the Brown–Forsythe test, respectively. Further, the Nash efficiency coefficient is used to make comprehensive judgement of trend and jump. The deterministic component with greater efficiency coefficient is chosen as the form of variation;

Step ii: Detect the periodic component If the residual component is nonstationary after removing the trend or jump in step i, the Fourier series method is used to detect the periodicity of the residual series, and then the periodic component can be determined;
 Step iii: Gain stationary stochastic component According to the additive model, the hydrological series \(X_{t} (t = 1,2, \ldots ,n)\) includes a deterministic component \(Y_{t} (t = 1,2, \ldots ,n)\) and stochastic component \(S_{t} (t = 1,2, \ldots ,n)\). The deterministic component is removed, so as to gain the stationary component \(S_{t}\):$$S_{t} = X_{t}  Y_{t}$$(8)
 Step iv: Compose stationary annual water level series The crux of gaining the stationary annual water level series is to determine the deterministic component \(Y_{{t_{0} }}\) for each year \(t_{0}\). The stationary annual water level series \(X_{{t,t_{0} }}\) corresponding to the environment of the year \(t_{0}\) is composed through the numerical formula as:where \(t = 1,2, \ldots ,n\); n is the length of the annual water level series.$$X_{{t,t_{0} }} = Y_{{t_{0} }} + S_{t} = Y_{{t_{0} }} + (X_{t}  Y_{t} )$$(9)
 Step v: Calculate ratio scaling index of water level for each year \(t_{0}\) The stationary annual water level series \(X_{{t,t_{0} }}\) for different environments are generated using the above steps. The ratio of the stationary annual water level and the observed annual water level for each year \(t_{0}\) is calculated as:where \(K_{t}\) is the ratio scaling index.$$K_{t} = X_{{t,t_{0} }} /X_{t}$$(10)

Step vi: Determine stationary daily water level series The observed daily water level in each year \(t_{0}\) is multiplied by \(K_{{t_{0} }}\), which is the ratio scaling index for the corresponding year. The stationary daily water level series is then generated with the above calculations.

Step vii: Draw synthetic duration curves for daily water levels All values of the daily average water level are first sorted from large to small in the statistical years, and then the cumulative guaranteed rate is calculated corresponding to the water level at each degree. The synthetic duration curve is then drawn based on those cumulative guaranteed rates.

Step viii: Design the LNWL According to the synthetic duration curve, the water level is estimated to satisfy the requirement of guaranteed rate. Thus, the designed LNWL can be determined by using the synthetic duration curve.
4 Results and discussion
4.1 Detecting the components of annual water level series
According to the RVN test, the residual component is stationary, which can be regarded as the stationary stochastic component. Thus, the periodicities would not need to be considered.
4.2 Design of the LNWL
Results of designing the lowest navigable water level (m)
LNWL for different environments and assumptions  Guaranteed rate  

90%  95%  98%  
Past environment  534.90  534.77  534.65 
Current environment  534.27  534.14  534.03 
Not considering nonstationarity  534.85  534.75  534.64 
Difference between current and past environment  − 0.63  − 0.63  − 0.62 
Difference between the current with and without considering nonstationarity  − 0.58  − 0.61  − 0.61 
When considering the nonstationarity of water levels, the stationary water level series can be constructed for different environments based on the corresponding deterministic components. The difference of the deterministic component is − 0.63 m after and before the jump point 2004 as presented in Eq. (15), which is consistent with the results in Table 2. It can be seen that the difference of the LNWL for the same guaranteed rate (90%, 95%, 98% respectively) between the current environment and the past one is nearly − 0.63 m. Moreover, when the observed water level series is directly used to calculate the LNWL using a synthetic duration curve, the difference of the LNWL under the current conditions between considering nonstationarity or not is approximately − 0.60 m. This may have a negative influence on navigation safety since the standard water depth of the channel is only 2.0 m.
4.3 Exploration of physical causes
The physical causes of the jump point in 2004 in the annual water levels at the Yunjinghong station and other characteristics of water level variation are analyzed from the following two aspects: regional precipitation change and cascade hydropower development.
The result of the RVN test shows that the annual precipitation series is stationary. The previous study showed that the annual precipitation in the Lancang River basin decreased by 10–20% in 2003, which was a particularly dry year for the whole Mekong River basin (He et al. 2006; Li and He 2008). However, since the annual precipitation series is stationary, it can be concluded that the annual precipitation did not result for the jump in the water level series.
The cascade hydropower development in the Lancang River is considered next. On one hand, the operation of these water conservancy projects regulates runoff and affects water levels in the river, which make a reduction of the water level fluctuations for 2005–2014 as shown in Fig. 7; on the other hand, these projects also greatly change the river’s shape and alter water levels and navigation conditions. There are six large dams constructed in the mainstream of Lancang River (Fig. 1), and these dams have different influences on water processes due to their storage capacities and catchment areas. Considering the jump point is 2004, it can be concluded that the construction of Xiaowan, Jinghong and Nouzahdu contributed greatly to the water level variation, since these three dams began to be constructed in 2002, 2003 and 2004, respectively. Especially, the storage capacity of the Nouzhadu dam ranks the highest among these six dams as presented in Table 2. Although the Dachaoshan dam began to run in 2003, the reservoir does not have multiyear regulation performance. Thus, the annual water level is still mainly affected by the construction of water conservancy projects.
4.4 Impacts of water level variation on navigation safety
Comparison of statistical parameters of water level under different environments
Environmental conditions  Statistical parameters  

Mean  Variance coefficient  Skew coefficient  
Past  542.8050  0.0099  0.0028 
Current  542.1624  0.0099  0.0028 
For a ship with a certain cargo volume, a lower water level means a greater risk of grounding. Currently, the channel from Jinghong to the boundary (No. 243) between China and Myanmar meets the fifth navigation standards. The channel scale is 2.0 m × 40 m × 300 m (water depth × bottom width × bending radius), and it is navigable for 300ton ships. In general, the draft of a 300ton ship is about 1.3 m. As the results indicate in the Sect. 4.2, when the guaranteed rate is between 90 and 98%, the designed water level for the current environment considering nonstationarity is less than that without considering nonstationarity. The difference between these two is approximately − 0.60 m, which accounts for around 46% of the ship draft. Therefore, the water level variation is a very important factor that cannot be ignored for navigation safety.
In “LancangMekong River Commercial Vessel Passage Agreement” (2001), the LNWL at the Yunjinghong station is determined as 534.69 m for the guaranteed rate of 95%. According to Table 2 and Fig. 6, when the LNWL is 534.69 m, the guaranteed rate for the current environment will be less than 90%. Therefore, if the hydrological nonstationarity is ignored, the designed water level cannot meet the guaranteed rate requirement. This would make the shipping engineering design, water depth maintenance and other activities based on the LNWL become unreasonable and unreliable.
5 Conclusions
In this study, we proposed a nonstationary statistical approach for the design of LNWL in inland waterways based on hydrological component analysis. With this approach, the stationary series can be generated to design the LNWL for different environments. Compared to conventional methods, which directly handle the observed hydrological data, the proposed nonstationary statistical approach is able to provide more reasonable design parameters in a changing environment.
This nonstationary statistical approach was applied to design the LNWL in the downstream of the Lancang River. Results indicate that the LNWL is 534.90 m, 534.77 m and 534.65 m with guaranteed rates of 90, 95 and 98% in the past environment, respectively, while it is 534.27 m, 534.14 m, and 534.03 m in the current condition, respectively. The LNWL for the current environment with and without the assumption of nonstationarity were compared. Results show that in the current environment, the LNWL considering nonstationarity is about 0.60 m less than that without considering nonstationarity. This difference equals to approximately 46% of the draft of standard ships (300 tons) in the downstream of the Lancang River, which cannot be ignored when estimating the maximum ship capacity.
In summary, the approach proposed in this paper can be used in inland navigation to calculate the minimum navigable water levels in different places and environments. Further research should be performed to deal with the changing environment rather than only considering water level variations. This would provide a more reasonable reference for channel planning and design, and navigation safety assessment.
Notes
Acknowledgements
The authors gratefully acknowledged the valuable hydrological data and information provided by the Hydrology Bureau of Yunnan Province. This study was financially supported by the National Natural Science Foundation of China (Nos. 51579181, 91547205, 91647110, 51779176), and the Youth Innovation Promotion Association CAS (No. 2017074).
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