Streamflow
Precipitation
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.
Comprehension
Understanding
the model, visualizing the model
Questions
 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
should be used to measure the rate of precipitation?
 What kind
of graph would be used to predict the rate of precipitation?
 Do you
think that a graph describing the rate of precipitation is cyclic,
i.e., does it repeat its pattern year after year?
 Sketch
a graph that might describe precipitation in your region over
a period of one year.
Acquisition
Mathematical Topics
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.
Information
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 equation that describes precipitation will independent
variable t, 0 < t < 12, where t represents monthly
time of year. Also, since the graph 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 midmonth 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.
Jan.

Feb.

Mar.

Apr.

May

Jun.

Jul.

Aug.

Sep.

Oct.

Nov.

Dec.

.5

1.5

2.5

3.5

4.5

5.5

6.5

7.5

8.5

9.5

10.5

11.5

Further, the
value t = 5 corresponds to the end of May, t = 4.5 corresponds
to midMay, and t = 9.3 corresponds to the point in time which
is .3 into the month of October.
We will fit
data provided with a piecewise graph. There will be three pieces,
each piece will be a linear graph. Although the data provided
is for average precipitation, we will assume that the graph indicates
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 find a precipitation graph.
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 graph you derive in this section will be
defined for one year but copies could be placed endtoend to
extend the domain for as many years as desired. However, the period
for 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.
Assumptions
 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 (and the beginning of January,
t = 0), is 1.67 inches per month.
 The rate
of precipitation is linear between data points.
Application
Apply mathematical
knowledge and Tool Chest applets to analyze precipitation.
Questions
 Make points
from the data provided. For Assumptions 1 and 2, the first coordinate
will be midmonth 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 linear
regression applet.
 Print the
plot screen from the applet and connect adjacent data points
with straight lines; this will be the precipitation graph.
 Use regression
to find the three equations for each of the three lines in your
graph; denote these by A = mt + b, B = mt + b, and C = mt +
b, respectively. Round coefficients to two decimal places. These
three equations define the piecewise graph that describes the
rate of precipitation in inches of water per month at time t.
Note that the first equation goes from t = 0 to t = 2.5, the
second from t = 2.5 to t = 10.5, and the third from t = 10.5
to t = 12.
 Use the
graph to determine the rate of precipitation at each of the
times. (A) t = 1.4 (B t = 11
 Use the
graph to determine the average precipitation in April, in August
and in December
 When 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 midOctober (See Part 1, Temperature, of Streamflow
Prediction.). So, from the first of the year through April and
from midOctober through the end of the year, precipitation
is in the form of snow; otherwise precipitation is rain.
In order
to answer questions 7  8, assume that the total precipitation
for any given month is the average precipitation for that month,
i.e., R(midpoint). For half months, take half of the average
precipitation.
 Find the
total precipitation in inches (water from snow and rain) for
each month of the year.
 Determine
the total amount of water from snow for the entire year.
Reflection
Examine solutions
 Do you
think the piecewise graph is a good model in this case?
 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?
 Would this
model work in all areas of the world?If not, are there parts
of the world that would use some other type of graph