- Rain and Snow
2: Streamflow Prediction
This module is the second
of three which are designed to lead to the prediction of streamflow
for a river in a particular region. Temperature obviously affects the
type of precipitation we get, and the amount of precipitation affects
the amount of water in the river. A model for temperature was developed
in the first module in this series. In this module, we will construct
a model for average precipitation in the region.
Observations of precipitation trends, visualization of a model
- Write down average monthly
precipitation amounts for the area in which you live. You can estimate
these or consult a website for accurate data. If it snows in your
area, then record this as water from snow melt, the ratio is about
10:1, snow to water.
- During what period of
the year is precipitation primarily in the form of snow?
- What units could be used
to measure precipitation?
- Draw a coordinate system
with months listed on the horizontal axis and precipitation on the
Then plot your average precipitation on this system.
- What kind of function
would best fit the points on your graph?
- Would the function be
different for different parts of the world?
- Why do you think it might
be important to know about the average precipitation amounts?
- How does temperature affect
- Sketch a graph of the
function that might describe average monthly precipitation in your
area for one year.
- Do you think that average
precipitation is cyclic, i.e., does it repeat its pattern year after
- How would mathematics
be useful in the analysis of precipitation?
The mathematical topics required for this study are listed in the menu
to the left. Click on the topics if you need to learn more or refresh
We will measure precipitation in inches of water, even if the precipitation
is snow. (Note: We specify "inches of water" to distinguish
between inches of snow; 10 inches of snow only produce approximately
1 inch of water.) The function that describes precipitation will be
of the variable t, 0 < t < 12, where t represents monthly time
of year. For example, t = 5 corresponds to the end of May, t = 4.5 corresponds
to mid-May, and t = 9.3 corresponds to the point in time which is .3
into the month of October. Also, since the function is, in general,
cyclic (repeats year after year), t = 0 and t = 12 correspond to exactly
the same time, the very beginning of January and the very end of December
(midnight on New Year's Eve).
In order to approximate the
average precipitation for any month, we use the mid-month value for
t. For example, for average precipitation for the month of May, we use
t = 4.5. The table below shows the t values to use to estimate averages
for each month of the year.
We will fit data provided with a sine function, denoted by P(t). Although
the data provided is for average monthly precipitation, we will assume
that P(t) gives the rate of precipitation at time t, 0 < t < 12.
Thus, even though it is unrealistic, we are assuming that it is always
raining in this part of the country; this assumption allows us to easily
define a precipitation function.
Average monthly precipitation
is, on a short term basis, a cyclic phenomenon repeating year after
year. If a region gets 3 inches of rain in August as an average over
many years, then a good guess is that the same region will get close
to 3 inches in August of next year and the year after also. The function
you derive in this section will be defined for one year but copies of
its graph could be placed end-to-end to extend the domain for as many
years as desired. However, the domain of this function will be only
one year, i.e., from t = 0 to t = 12, or 0 < t < 12, where t denotes
the time of year.
Finally, use the midpoint
of any month to approximate the average precipitation for that month.
- The average maximum precipitation
of 3 inches occurs at the end of May (t = 5).
- The average minimum precipitation
of 1 inch occurs at the end of November (t = 11).
- Average monthly precipitation
is cyclic with period 12 months.
Apply mathematical knowledge and Tool Chest applets to analyze precipitation.
- Make points from the
data provided: the first coordinate will be time t and the second
coordinate will be average precipitation for that time. Plot these
points using the applet.
- Let P(t) denote the
average monthly precipitation in month t. Use the information provided
in the Assumptions to find the sine function that fits the data
- Graph the function P(t)
on the applet screen with the data points showing one year, January
through December. Check to be sure the graph goes through the points.
- Use the function to
determine the rate of precipitation at each of the times. A) t =
1.4 B) t = 11
- Use the function to
determine the average precipitation in: A) April; B) August; C)
- When will the rate of
precipitation be 2 inches per month?
Examine the model and its implications
- Do you think that this
model would work for long periods of time, i.e., decades or centuries?
Why or why not?
- What do you see as advantages
to the use of this function? Disadvantages?
- What are some properties
of the sine function that make it particularly appropriate to use
in this situation?
- Do you know other functions
that might also be appropriate for use in this study?