- Rain and Snow
Part 2: Streamflow Prediction
In this module
we will construct a model which gives the rate of precipitation
in a certain region of the country.
Understanding the model, visualizing the model
- Which month
would you predict has the greatest amount of precipitation in
the area where you live? Which month has the least amount?
- Why would
it be important to be able to predict precipitation?
- Which type
of precipitation, rain or snow, allows for more available water
over long periods of time? Why?
- What affects
the type of precipitation received?
- What units
do you think could be used to measure precipitation? The rate
- What kind
of function could be used to describe the rate of precipitation?
- Do you
think that a function that describes the average rate of precipitation
is cyclic, i.e., does it repeat its pattern year after year?
a graph that might describe precipitation in your region over
a period of one year.
- How do
you think mathematics might be used for the prediction of future
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 your memory.
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 piecewise function, denoted by
R(t). Although the data provided is for average temperatures,
we will assume that R(t) gives the rate of precipitation at time
t, 0 < t < 12, R(t) is measured in inches per month. Thus,
even though it is unrealistic, we are assuming that it is always
raining in this part of the country; this assumption allows us
easily define a precipitation function.
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.
The average precipitation is 3.2 inches in March.
The average precipitation is .75 inches in November.
Precipitation at the end of December, t = 12 (or the beginning
of January, t = 0), is 1.67 inches per month.
The rate of precipitation is linear between data points.
Apply mathematical knowledge and Tool Chest applets to analyze
points from the data provided. For Assumptions 1 and 2, the
first coordinate will be mid-month t value and the second
coordinate will be average precipitation for that month; for
Assumption 3, use the information provided. Plot these points
using the applet.
the plot screen from the applet and connect adjacent data
points with straight lines; this will be the graph of the
precipitation function R(t).
- Use regression
to find the equations for each of the three lines in your
graph. Round coefficients to two decimal places.
three equations define the piecewise function R(t) that describes
the rate of precipitation in inches of water per month at
time t. Write this piecewise function in its proper form.
- Use the
function to determine the rate of precipitation at each of
(A) t = 1.4 (B) t = 6 (C) t = 11
- Use the
function to determine the average precipitation in: A) April;
B) August;C) December
will the rate of precipitation be 2 inches per month?
As a general rule, temperature determines whether it rains
or snows, and when it is above freezing, how fast the existing
snow will melt. These are our main concerns in the development
of this model. We assume that when the average monthly temperature
is above 32 degrees F, precipitation means rain and whatever
snow there is will melt; otherwise precipitation is in the
form of snow and it sticks. In this region, the average temperature
goes above freezing at the end of April and then goes below
freezing again in mid-October (See Part 1, Temperature, of
Streamflow Prediction.). So, from the first of the year through
April and from end of October through the end of the year,
precipitation is in the form of snow; otherwise precipitation
is rain.In order to answer questions 8 - 9, assume that the
total precipitation for any given month is the average precipitation
for that month, i.e., R(mid-point). For half months, take
half of the average precipitation.
the total precipitation in inches (water from snow and rain)
for each month of the year.
the total amount of water from snow for the entire year.
- Do you
think the piecewise function was a good model in this case?
are the advantages to using a piecewise function for this
- Can this
model be used to predict rain on a given day? Why or why not?
- Do you
think that some of the assumptions are unreasonable? Why?
- Can this
model be used to predict precipitation over longer periods
of times, say years or decades? Why or why not?
this model work in all areas of the world? If not, are there
parts of the world that would use some other type of function?