An Exponentially Damped Water-Level Trend with Limited Data

If the water level is now 200 cm and rose by 10 cm over the past hour, will it reach 250 cm in five hours? That follows if the rate stays constant. Recent readings alone do not establish that the trend will last.

In a monitoring system at one station in Surabaya, the readings received in real time do not always form a pattern as tidy as that example. Some stay very stable, some move up and down, and others change only briefly. A few recent numbers alone are not enough to establish the direction of the water level.

The forecast inputs are also limited to water-level history from the same sensor. Rainfall, discharge, and upstream conditions are not included. The approach therefore starts with the patterns available in that history.

The readings are therefore checked and summarized before estimating a trend. The recent direction is then carried forward while gradually reducing its influence. This note explains that approach and the steps leading to its equation.

Approach and main assumptions

Trend damping is an established forecasting idea, including in damped Holt forecasting.¹² The approach here starts from the latest water level, estimates a rate using linear regression, and applies exponential damping to that rate.

Sensor distance readings are grouped into five-minute intervals. Each interval needs at least five distinct reading times, with all readings passing the data checks. Missing values, zero or negative distances, distances above the reference, and conflicting values at the same time are rejected.

Each interval is represented by its median, the middle value after sorting. This reduces the influence of an individual distant reading. For example, 149, 150, 150, 151, and 200 cm have a median of 150 cm but a mean of 160 cm. Water level is the sensor reference height minus the median distance.

The system requires six complete hours of history to check continuity. A separate rule flags a potentially stuck reading when exactly the same number keeps arriving for six hours. Both durations are initial settings rather than requirements of the equation. Further experiments are needed to choose suitable durations for each sensor.

The trend uses the last 13 points, spanning one hour at five-minute spacing. There are 12 gaps and 13 points when both endpoints are included. Linear regression finds the straight line that best fits these points. Its slope becomes the rate $b$, measured in cm/hour.

Deriving the equation

Let $H_t$ be the starting water level and $u$ the elapsed time since the forecast begins. Assume the rising or falling rate fades according to:

$$v(u)=b{e}^{-u/\tau }$$

At $u=0$, the rate is still $b$. The exponential factor then becomes smaller. The parameter $\tau$ controls the fading timescale. With $\tau =1$ hour, about 37% of the initial rate remains after one hour. This is an assumption of the model here, rather than a rate equation quoted directly from the Holt paper.

To obtain the accumulated change over $h$ hours, integrate the rate:

$$\Delta H(h)={\int }_0^{h}b{e}^{-u/\tau }du$$

The indefinite integral is $-b\tau e^{-u/\tau }+C$. Substitute the ending time $h$ and subtract the result at the starting time 0:

$$\begin{array}{rl}\Delta H(h)& ={\left[-b\tau e^{-u/\tau }\right]}_0^h\\ & =-b\tau e^{-h/\tau }+b\tau \\ & =b\tau (1-e^{-h/\tau }).\end{array}$$

Add this change to the starting water level:

$${\tilde{H}}_{t+h}=H_t+b\tau (1-e^{-h/\tau }).$$

This estimate already includes damping. A separate step clips it between 0 and the sensor reference height $R$:

$${\hat{H}}_{t+h}=\min \left(R,\max \left(0,{\tilde{H}}_{t+h}\right)\right).$$

Damping reduces the trend's influence. Clipping only changes estimates outside the measurement range. For example, a forecast of 310 cm becomes 300 cm if $R=300$ cm.

Example and insights

With $H_t=200$ cm, $b=10$ cm/hour, and $\tau =1$ hour, the one-hour forecast is:

$$200+10(1-e^{-1})=206.32\text{ cm}.$$

The rate at the end of that hour is 3.68 cm/hour, while the total rise over the hour is 6.32 cm. They differ because the assumed rise is faster at the start and gradually slows.

Forecasts from +1 to +5 hours are 206.32, 208.65, 209.50, 209.82, and 209.93 cm. These illustrative values approach 210 cm because of the damping assumption, not because the river has a physical ceiling of 210 cm.

The current calculation uses a one-hour tau. A separate experiment compared 0.5, 1, and 2 hours, representing faster and slower fading. Selection used mean absolute error on validation data. Hang Tuah selected one hour, while Pucang and Menur selected half an hour. Those results apply to the tested candidates and observations, rather than establishing the waterways' physical response times.

The main lesson is that a short equation still contains choices that need testing. The median determines how readings are summarized. The six-hour history determines when a forecast is allowed. The regression window and tau determine how the trend continues. The next step is to compare these settings per sensor on separate test data, using persistence and a linear trend as reference forecasts.

Footnotes

1. E. S. Gardner, Jr. and E. McKenzie, “Forecasting Trends in Time Series,” Management Science, 31(10), 1237–1246, 1985. DOI: 10.1287/mnsc.31.10.1237. ↩

2. R. J. Hyndman and G. Athanasopoulos, Forecasting: Principles and Practice, 3rd edition, OTexts, 2021. Section 8.2, “Damped trend methods”. ↩