# Snow

I live in the tropics for most of my life, so to me, precipitation equals rainfall. Until yesterday, when I read a paper which said in the summer, “precipitation is mostly in the form of liquid”, did I realize that precipitation = rainfall + snowfall. Well, I’ve experienced only one snowy winter so far. Maybe in the future I should find some work dealing with snow, it should be fun.

# Some reservoir concepts

Draft ratio is the ratio between mean annual withdrawal and mean annual inflow.

$D = \dfrac{\mathbb{E}(Q_{out})}{\mathbb{E}(Q_{in})}$

Storage ratio is the ratio between the total reservoir volume and mean annual withdrawal.

$S = \dfrac{V}{\mathbb{E}(Q_{in})}$

Some reliability concepts:

• Annual frequency reliability ($R_a$) is the percentage of number of failure years over the total number of years in record.
• Time reliability ($R_t$) is the percentage of the total failure duration over the total record time.
• Volume reliability ($R_v$) is the percentage of total shortage volume over the total required volume in record.

Some uncertainty concepts for reservoir modelling:

• Model uncertainty: the uncertainty due to the model structure.
• Parameter uncertainty: the uncertainty due to parameter estimation method.
• Intrinsic uncertainty (of the system): uncertainty that arises when assuming that the system will behave in the same way in the future as when it is modelled.

Model uncertainty and parameter uncertainty can be reduced with longer record lengths, but intrinsic uncertainty may not.

While the hydrologic aspects of a reservoir can be modelled universally, the socioeconomic aspects are very site specific and is up to the practitioner. Klemeš et al (1981) reported that long-memory and short-memory models are not statistically different (statistical significance), but the small statistical differences may lead to significantly different socioeconomic outcomes (socioeconomic significance). This idea reminds me of a thought my advisor told me at the beginning of our project: a model’s value may be more important than its skill. I’ll keep that in mind.

### References

Klemeš, V., Srikanthan, R., & McMahon, T. A. (1981). Long-memory flow models in reservoir analysis: What is their practical value? Water Resources Research, 17(3), 737–751. http://doi.org/10.1029/WR017i003p00737

# A little dendrochronology

The science of dendrochronology derives its name from the Greek words δένδρον (dendron, meaning tree limb) and χρόνος (khronos, meaning time). In a nutshell, the name means tree dating (Wikipedia). Many species of trees are known to develop annual growth rings, such as those in Figure 1. The outermost ring corresponds to the most recent year (the current year for a living tree, or the year when the tree died), and the innermost ring corresponds to the tree’s first year. Thus, by counting the number of rings, scientists can determine the age of a tree. Furthermore, the width of these rings are correlated to moisture, as the growth of a tree increases with higher rainfall. Therefore, by calibrating the ring widths with some environmental measurements, scientists can derive important information about the past climate, such as precipitation and drought.

Figure 1: An example of annual growth rings in a tree. Source: Wikipedia

Ring width measurements can be done without cutting open a tree. The equipment involved is called an increment borer, which helps scientists bore a small hole in the tree trunk and draw out a small core that contains a segment for each ring (Figure 2).  For a tutorial on how to use the borer, check out this video, and many others on YouTube. To increase accuracy, scientists usually need to take two cores from a tree, and sample from many trees at one site. After the cores are collected, the rings on each of them are measured, often automatically with a computer image processor. The result is a collection of ring width time series. Many factors affect the growth of a tree, some are due to the tree itself, and some are due to the environment. As a result, each ring width time series contains both endogenous and exogenous growth and thus needs to be standardized before use. Standardization involves three main steps (Cook and Kairiukstis, 1990, p. 104): (i) fitting a growth curve that represents the endogenous growth rate, (ii) for each year, dividing the observed ring width by the fitted value to get a dimensionless annual index, which is the ratio between the total growth and the endogenous growth, and (iii) using a statistical procedure to obtain a mean index time series across all trees at the site. The final result is called a tree ring chronology, which can then be used to derive information about past climate.

Figure 2: An increment borer and two cored tubes, with a ruler for size reference. Source: Wikipedia.

Tree ring data are available in the public domain on the International Tree Ring Data Bank (ITRD). An interactive user interface and a collection of standardized chronology can be found on the dendrobox project. Interested readers can find more details in Cook and Kairiukstis (1990). It is a classic textbook on dendrochronology, and is available in a the public domain here.

### References

Cook, E. R. and Kairiukstis, L. A. (1990). Methods of dendrochronology. Applications in the
Environmental Sciences. Kluwer Academic Publishers.

Zang, C. (2015). Dendrobox – An interactive exploration tool for the International Tree Ring Data bank. Dendrochronologia, 33:31-33.

### Photo credits

Figure 1: (Wikipedia) No machine-readable author provided, Arpingstone assumed (based on copyright claims). No machine-readable source provided, own work assumed (based on copyright claims). Public Domain, https://commons.wikimedia.org/w/index.php?curid=447580

Figure 2: (Wikipedia) By Hannes Grobe/AWI – Own work, CC BY-SA 2.5, https://commons.wikimedia.org/w/index.php?curid=1135047

# Deterministic versus stochastic models

I had an interesting talk with Sean today over lunch. We discussed whether rainfall is a stochastic or deterministic process. At one point, I asked how we should consider a flip of a coin then. Sean brought up the chaoticity theory which says that instead of using a stochastic model, you can have a deterministic one and only change the initial conditions in order to have different outputs. There are some papers on that on Journal of Hydrology.

This is definitely something to check out, though I certainly have to focus on what I am doing.

# Variance of sample mean in an autocorrelated stochastic process

Let $X$ be a stochastic process with mean $\mathbb{E}(X) = \mu$, and variance $\mathbb{V}(X) = \sigma^2$. Let $X_1,..., X_n$ be an observed time series of $X$.

A good estimator for $\mu$ is $\overline{X} = \sum_{i = 1}^{n} X_i$ . We know that if the observations are IID, $\mathbb{V}(\overline{X}) = \frac{\sigma^2}{n}$. However, if the observations are not IID, the variance will be larger – this is what I learned recently in Loucks (2005). The authors skipped some details of the book, so I worked them out. Below is a detailed proof.

\begin{aligned} \mathbb{V}(\overline{X}) &= \mathbb{E}\left((\overline{X}-\mu)^2\right) \\ &= \mathbb{E}(\overline{X}^2 - 2\mu\overline{X} + \mu^2)\\ &= \frac{1}{n^2} \mathbb{E} \left( \left(\sum_{i=1}^{n} X_i\right)^2 - 2n\mu\sum_{i=1}^{n}X_i + n^2\mu^2\right)\\ &= \frac{1}{n^2} \mathbb{E} \left( \left(\sum_{i=1}^{n} X_i\right)\left(\sum_{j=1}^{n} X_j\right) - 2\left(\sum_{i=1}^{n}X_i\right)\left(\sum_{j=1}^{n}\mu\right) + \sum_{i=1}^{n}\sum_{j=1}^{n}\mu^2\right)\\ &= \frac{1}{n^2}\mathbb{E}\left(\sum_{i=1}^{n}\sum_{j=1}^{n}X_iX_j - \mu X_i - \mu X_j + \mu^2 \right)\\ &= \frac{1}{n^2}\mathbb{E}\left(\sum_{i=1}^{n}\sum_{j=1}^{n}(X_i - \mu)(X_j - \mu)\right)\\ \end{aligned}

Now, on one hand, the summands where $i = j$ can be grouped; on the other hand, note that when $j \neq j$, there is one summand for $j > i$ and one summand for $j < i$. Therefore,

$\displaystyle \mathbb{V}(\overline{X}) = \frac{1}{n^2}\mathbb{E}\left(n\sum_{i=1}^{n}(X_i - \mu)^2 + 2\sum_{i=1}^{n}\sum_{j=i+1}^{n}(X_i - \mu)(X_j - \mu)\right)$           (1)

Let $k = j - i$, in other words, $k$ denotes the lag between the $j^{\text{th}}$ and $i^{\text{th}}$ timesteps. (1) becomes

\begin{aligned} \mathbb{V}(\overline{X}) &= \frac{1}{n}\mathbb{E}\left(\sum_{i=1}^{n}(X_i - \mu)^2\right) + \frac{2}{n^2}\mathbb{E}\left(\sum_{i=1}^{n}\sum_{k=1}^{n-1}(X_i - \mu)(X_{i+k} - \mu \right) \\ &= \frac{\mathbb{V}(X)}{n} + \frac{2}{n^2}\sum_{k=1}^{n-1}\sum_{i=1}^{n-k}\text{Cov}(X_i,X_{i+k})\\ &= \frac{\sigma^2}{n} + \frac{2}{n^2}\sum_{k=1}^{n-1}(n-k)\rho(k)\sigma^2\\ &= \frac{\sigma^2}{n}\left(1 + 2\sum_{k=1}^{n-1}\left(1 - \frac{k}{n}\right)\rho(k)\right) \end{aligned}

where $\rho(k)$ is the lag-$k$ autocorrelation and is defined as

$\displaystyle \rho(k) = \frac{\text{Cov}(X_i, X_{i+k})}{\sigma^2}$

Observe that compared to the IID case, the variance of the sample mean estimator is inflated by a factor bigger than 1. Furthermore, it can be checked that this factor does not decrease as $n$ increases. We conclude that the sample mean of an autocorrelated time series always has a bigger standard error than that of an IID time series with the same variance.

References

Loucks, D.P et al. (2005). Water Resources Systems Planning and Management (Chapter 7, p. 198-201). UNESCO Publication. (The book is publicly available on the UNESCO website)

# Danube

I remember a poem we learned in primary school

Man tells the Danube
I shall block you with steel and concrete
So that from heights
You fall
Swiftly

So that trains
Go faster
So that machines
Never stop

So that from now on
The flow of your water
Will no longer be wasted
But it will bring
Bread, electricity, and coal
For my people to use to their hearts' content

So that the fields
Rumble with tractors and engines

So that lights lit up all nights
In homes
And on the streets
###### (Translated from Vietnamese to English to the best of my ability. The Vietnamese version, which was in turn translated from Russian, can be found here)

Such was the thinking of the past. Following this paradigm, man has built more than 40 dams and created over 500 reservoirs in the Danube’s catchment. The river is heavily regulated, causing substantial reduction in its floodplains – from 26,000 km² to merely 6000 km² over the last 5 decades, and adverse disturbance in its sediment and nutrient transport. These effects, together with heavy industrial, agricultural and urban activities of 85 million people in the catchment, have raised serious ecological and environmental concerns.

Nowadays, we no longer think that river flow is a waste. The environmental impacts of hydropower production have been recognized. The 13 riparian countries that share the Danube have agreed to manage the river together.

Along the Mekong, however, dams are still sprouting up. Will the Mekong countries come to a peaceful agreement the way the Danubians did?

### References

Loucks, D.P et al. (2005). Water Resources Systems Planning and Management. UNESCO Publication. (The book is publicly available on the UNESCO website).

# A river changing course (and should it?)

When I was little, I was fascinated by a storybook about a family living by a river. One day, the grandmother told her grandson that the river god had been angry in the past, throwing earth, trees, houses and people across the banks. The boy asked his father later that day about what grandma told, and his father explained to him that was a natural phenomenon when a river changed its course.

I wondered how it would look in real life.

Last week, while I was reading a textbook, the phrase “river changing course” came up, which reminded me about the childhood story. Curious, I looked the phrase up and found this gif from reddit

Now isn’t that breathtaking? This whole thing only takes place in less than 30 years. This is what a river does without human intervention.

From the hint “Reserva comunal El Sira” in the top left corner, I was able to locate this place – the Ucayali river in Peru. You see, the river is still meandering and doing what a river does to this day.

Now, what happens if people live along the river banks? In the old days, grandma’s story would be true. But mankind has learned to regulate rivers with concrete banks, dikes, dams and barrages. Rivers’ courses become fixated. The floodplains (flat land areas around the rivers) are better protected from floods, and human populations along major river banks increase rapidly over time. As a result, economic value of the land behind dykes mounts up, and a flood event would cause ever more severe consequences. For centuries, dykes have been built higher and higher to protect the increasingly costly hinterland. But is that a sustainable way to live by the water?

The Dutch, at least, have realized that may not be the way. After two major floods in 1993 and 1995, they embarked on a huge project called “Room for the river”, in which they made modifications to flood protection structures at over 30 locations across the country. These modifications include setting back or lowering dykes, building dykes at different locations, and dredging new channels. The overall objective is to give more rooms for the water, and to let the floodplains do what they are meant to do – storing floods. Consequently, mean water levels are reduced by about 10 – 30 cm, and rather surprisingly, they actually managed to increase waterfront activities with a new island, new dwelling mounds, floating piers and viewing platforms.

This project is scheduled to complete this year. I learned about it in 2010 while doing my Master’s in Delft. It has been featured in the news on various occasions. Time will tell whether this is a successful endeavour, but I believe that in order to live in harmony with water, especially in a seemingly warming world, it is better that mankind adapt to nature than try to brutally regulate it.

### References

1. Loucks, D.P et al. (2005). Water Resources Systems Planning and Management. UNESCO Publication. (The book is publicly available on the UNESCO website).
2. Room for the river project website
3. Room for the river on Scientific American