Lake Sonoma

Task

California is in the fifth year of a drought. Towns have been asked to decrease water use across the state. The graph below shows the water storage in Lake Sonoma, a reservoir in Northern California from 1988, shortly after it was first filled, until February 1, 2015.
 

1. What questions do you have looking at the graph?
2. Describe some of the features of the graph.
3. What might explain these features?  (Consider data collection, weather events and reservoir management decisions.)

4. Based on the given information, should the region that depends on Lake Sonoma for water use implement additional water conservation measures?

 

IM Commentary

The graph shows the water storage in Lake Sonoma, a reservoir in Northern California. The data for Lake Sonoma is freely available at http://cdec.water.ca.gov/cgi-progs/stationInfo?station_id=WRS.

(At http://cdec.water.ca.gov/, one can access the sensor for many more locations.) More information is available at http://www.scwa.ca.gov/current-water-supply-levels/ comparing water storage levels in recent years.

The graph of the water storage in Lake Sonoma has many features that are interesting to interpret. The features are connected to weather events or water agency management of the reservoir. Some of the features are easier to explain than others. Students might recognize that during the summer months the water level goes down and in the winter it goes up. They might be confused by the sudden changes. The extremely abrupt changes are caused by sensor errors, but the sudden peaks above 250,000 AF are explained by water management rules that require a buffer in the water storage for larger storm events. The instructor can fill in some of this information after students have a chance to identify the features of the graph.

The units on the vertical axis of the graph describe the water storage (volume) in the reservoir using acre-feet (AF). This is an unfamiliar volume measure that is nevertheless widely used in water management. One acre-foot is the volume of water it takes to cover one acre of land with one foot of water.

This task asks students to describe features of a graph. It provides an opportunity to introduce (or use) mathematical terminology that makes communication easier and more precise, such as: periodic behavior, maxima, minima, outliers, increasing, decreasing, slope.

This task shows many features of mathematical modeling. It presents a mathematical representation of a real world situation that students are asked to interpret. The data are not “cleaned up”, so students have to decide which features matter and which features do not matter.

This is a critical step in the modeling cycle—does your interpretation make sense in the context of the problem? Students have likely not seen acre-fee before and will have to learn how to interpret them.

To use this task, an instructor should be prepared to guide students to resources about background information or to fill in some of the information after the students ask about the different features of the graph.

An instructor can modify the task and turn it into a more open investigation by not giving a specific graph but rather directing the students to the website where they can generate their own graphs and tables of data for different time periods. Weather data is available for many regions and depending on the severity of recent weather events, it might make a compelling context for students to investigate how bad the weather during a recent season was by comparison with other years or regions of the country.

 

 

Solution

a) Questions

• What are the units on the vertical axis?

• What is the meaning of the blips in the graph? How can there be -187 AF stored? How can the storage go from 225000 AF to 387817 AF and back in a very short time?

• Why are the minima at the beginning and at the end of the time period shown lower than the ones in the middle?

b) Features:

• There is a lot of regularity in the graph.

• Between 1999 and 2006 the graph is almost periodic with sudden increases followed by a maximum, followed by less steep decreases. The maxima have strange sudden decreases that are followed by then less steep decreases.

• A lot of the maxima are around 250,000 AF, a lot of the minima are around 200,000 AF.

• The 2015 minimum of about 135,000 AF is the lowest minimum since 1991.

c) Possible Explanations:

• AF stands for acre-feet, the common unit to report water storage in reservoirs. An acre-foot is the amount of water that it takes to cover one acre of area with one foot high of water. An acre of area is 4840 square yards or 43,560 square feet.

• The extreme blips are data collection errors. The loggers are somehow malfunctioning and record data that is not really there.

• The shorter/less extreme blips can be explained by the water storage management of Lake Sonoma. When the water level goes above 245,000 AF, water is released to bring the level below 245,000 AF so that there is enough storage capacity left for extreme inflow amounts caused by large rain events.

• The graphs show a certain periodicity with yearly period. This is caused by the normal rain pattern in Northern California: Rain in the winter and spring, followed by dry weather in the summer and fall. For the reservoir this means sudden increases in water level in the beginning of the year followed by slower outflow the rest of the year.

• During a “normal” rain year, the water level peaks at or above the maximum storage capacity of 245,000 AF and decreases to 200,000 AF throughout the summer. Partly this is caused by the seasons and partly by the reservoir management that requires a buffer for large rain events and an outflow to support fish lifecycles in the Russian River and local water demand.

• The data show that there was less than usual rainfall in the last few years. The rainfall in 2012 just barely caused the water storage to approach 245,000 AF and in 2014 the storage just got back to 185,000 AF.

• No matter what the rain year was like, the outflow throughout the summer and fall seems to be the same. This caused a much lower minimum before the rain started in 2014, since the yearly maximum was very low at 185,000 AF.

d) As discussed in part c), the water storage in the last two years has been lower than at any time since the reservoir was first completely filled. February 1 is still early for the Northern California rainy season, which usually last through April. A positive sign is that the February 2015 water level is already above the season maximum from the previous year. It might be too early to make a definite decision. (A natural next step is to look up the current data to continue the discussion.)