Study area and dataset collection

HLR is separated from Tien River in Tan Phu Commune, Chau Thanh District, BTP, creating a natural border between Bao and Minh islet. It has 72 km long, from 12 to 15 m in-depth, and from 1,200 to 1,500 m (over 3,000 m at estuary) in width. During the rainy season, average river flows are approximately 3,300–3,400 m/s, while around 800–850 m/s in the dry season16.

There are six saltwater monitoring stations (from estuary to upstream) situated in An Thuan-AT (Tiem Tom harbor, Ba Tri District), Son Doc-SD (Hung Le Commune, Giong Trom District), Phu Khanh-PK (Phu Khanh Commune, Thanh Phu District), My Hoa-MH (Ben Tre city), An Hiep-AH (An Hiep Commune, Chau Thanh District), and Vam Mon-VM (Phu Son Commune, Cho Lach District) (Figure 1). In each station, the saltwater monitoring data were collected one time per week for a period of 23 weeks (from January to June that is the dry season in Mekong Delta). The river saltwater monitoring data from 2012 to 2019 were provided by BTHMFC (available at http://www.bentre.gov.vn/Lists/ThongTinCanBiet/TongQuat.aspx). The present study forecast the SI in HLR from Jan 1-Jan 8 (week 1) to Jun 4-Jun 11 (week 23) of 2020 based on saltwater monitoring data from 2012 to 2019 .

ARIMA models description and application

ARIMA was first formed by Box and Jenkin in 19769. The general equation of successive differences at the th difference of X is briefly expressed as follows:

, where is the different order, and B is the backshift operator

The successive difference at one-time lag equals to:

In this situation, the general non-seasonal ARIMA () is as follows:

, where is an auto-regressive operator of order , is a moving average operator of order , and

A general nonseasonal/seasonal ARIMA ()x()s model with nonseasonal parameters , seasonal parameters , and seasonality s that consists of several terms: A nonseasonal autoregressive term of order , a onseasonal differencing of order , a nonseasonal moving average term of order , a seasonal autoregressive term of order , a seasonal differencing of order , a seasonal moving average term of order . ARIMA(0,1,1)x(0,1,1)s–seasonal and nonseasonal MA terms of order 1 which was a common nonseasonal/seasonal ARIMA model. For a more detailed description of the terminology, see Box and Jenkins (1976)9, Bowerman and O’Connell (1987)17, and Pankraz (1991)18.

ARIMA modeling was developed using Statgraphics Centurion ver. 18 software. Model performance was evaluated using the root mean squared error (RMSE), the mean absolute error (MAE), the mean absolute percentage error (MAPE), the mean error (ME), the mean percentage error (MPE)19.

Map visualizations

An Inverse Distance Weighting (IDW) method in ArcGIS 10.3 was used to interpolate forecast point data to create continuous surface maps20:

where was the property at location i; was the property at location was the distance from to was the number of sampled locations, and was the inverse-distance weighting power.

Long-term saline intrusion data in Ham Luong River from 2012 to 2019

The saline concentration data in HLR for eight years that is obtained from the BTHMFC and Figure 2 presented the basic trends of the collected data. Overall, the saltwater concentration in HLR increased from February to April. The maximum saltwater occurred at the end of March or the beginning of April in which was the driest months in the year. Subsequently, the saltwater concentration decreased slightly in late May and fell rapidly in early June because of the seasonal change with rainfall in May. In early June, it is the beginning of the rainy season with much rainfall than those in May; therefore, the saline concentration decreased rapidly in the whole river. Notably, the highest saltwater concentration in HLR was observed in 2016 because of a severe El Niño, BTP experienced serious SI. The maximum saltwater concentration was 31.50 ‰ (05/02/2016), 26.01‰ (03/12/2016), 14.50‰ (03/12/2016), 12.40 (03/05/2016), 9.90‰ (03/12/2016), and 6.7% (03/12/2016) observed in AT, SD, PK, MH, AH, and VM, respectively. Saltwater (approximately 10‰) expanded through HLR by up to 50-60 km, considered to be the most extensive SI in the last 90 years.

The ARIMA model for the forecast of saline intrusion in Ham Luong River

In AT station, the highest saline concentration of 25.34 ‰ is observed in week 6, followed by 21.25‰ (week 10) and 21.16‰ (week 9). Furthermore, week 12 was expressed as the highest saltwater concentration (13.24‰), week 5 (8.95‰), week 12 (4.67‰), week 4 (1.68‰), week 11 (0.72‰). By contrast, the lowest saltwater concentration of 12.46 ‰ is observed in week 23. The saltwater concentration measured from 5.09 (week 22) to 13.24 (week 12), 4.31 (week 22)-9.40 (week 12), 1.61 (week 22) to 4.67 (week 12), 0.00 (week 22)-1.49 (week 12), and 0.00 (week 22)-0.72 (week 11) in SD, PL, MY, AH, and VM, respectively. Clearly, at the beginning of the rainy season (from May 28 to Jun 11) observed with the lowest saltwater concentration. In turn, saline intrusion began in mid-March, saltwater entered deep to inland (). Table 1 showed an overview of the monthly average of the forecasted saltwater concentration for all stations in HLR from January to June 2020. Generally, the saltwater concentration increased from January to March and then decreased from April to June. The maximum saltwater occurred in February and March while the lowest saltwater was observed in early June. Figure 3 showed the historical data, the forecasts, and the forecast limits (95% P.I.)

Testing forecast models

A normal probability plot of the residuals can be displayed in Figure 4. If the residuals come from a normal distribution, they should fall close to the line. In fact, the residual plot in AT, SD, PK, AH showed some curvature away from the line while MH and VM did not.

There are five tests have been run to determine whether or not the residuals form a random sequence of numbers. If a p-value for each test is greater than or equal to 0.05, we can not reject the hypothesis that the series is random at the 95.0% or higher confidence level. ARIMA forecasting model in AT, SD, PK, AH passed five tests while MH and VM did not (Table 2).

The perspective view of the saline intrusion in Ham Luong River in 2020 is predicted by the ARIMA model

At the beginning of the dry season (January), the saltwater levels of 10‰ will have occurred in a location where between Mo Cay Nam and Thanh Phu District, over 50 km away from Ham Luong estuary. Also, the saltwater levels from 5-10‰ will cover almost all of Giong Trom and half of Mo Cay Nam District. These districts in upstream such as Chau Thanh and Cho Lach District will be covered by under 2‰ (Figure 5A). Subsequently, at the driest month (February and March), saltwater will be intruded into an area within 60-70 km from the mouth of HLR; therefore all of Giong Trom and Mo Cay Nam District will be affected with the saltwater rate 10‰. Ben Tre City and a small part of Chau Thanh District will be covered by under 5‰ (Figure 5B, C). Finally, at the beginning of the rainy season (early June), saltwater will be pushed away from the inland. The saltwater levels of 10‰ will be observed in Ba Tri District, approximately 10km away from the estuary (Figure 5F).

Based on the forecasting results of the ARIMA model, saltwater with 5‰ will be entered up to 60-70 km deep inland that means Ben Tre city (areas with the highest population) and Chau Thanh District (areas with large-scale fruit production) seems to be affected by SI. Outcomes of this study are useful for reducing damages caused by the saline intrusion in the Mekong Delta, also BTP in saline season 2020.

The ARIMA model: advantages and disadvantages

Forecast is an activity to calculate or predict future events or situations, usually as a result of rational study or analysis of suitable data21. The accurate information for saline forecast will become more and more difficult to predict due to climate change and extreme weather22. In recent years, there are several quantitative forecast techniques available such as ARIMA models, Random walk models, Trend models, or Exponential Smoothing. Generally, ARIMA models are considered as statistical theory and mathematically complex techniques while the others are defined as simple prediction techniques. Therefore, the ARIMA model has been regarded as the most efficient prediction technique in hydrology12. In the empirical research, many advantages of the ARIMA model were found and support it as a proper way in especially short-term time series forecasting23. The ARIMA model requires fewer the prior data inputs to generalize the forecast., only needs endogenous variables and does not need to use other exogenous variables. Basically, this model is relatively more robust and efficient than other complex structural models in relation to short-run predictions24. However, the main limitation of ARIMA is the lack of a deterministic cause25. In addition, many traditional techniques for time series forecast, such as ARIMA, which assume that the series is generated from linear processes and as a result might be inappropriate for most real-world problems that are nonlinear27, 26. This problem has now been circumvented through large numbers of past data inputs, stochastic events, and the accuracy of past data inputs that must be enhanced.