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## Precipitation - 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.

Comprehension

Understanding the model, visualizing the model

Questions

1. Which month would you predict has the greatest amount of precipitation in the area where you live? Which month has the least amount?
2. Why would it be important to be able to predict precipitation?
3. Which type of precipitation, rain or snow, allows for more available water over long periods of time? Why?
4. What affects the type of precipitation received?
5. What units do you think could be used to measure precipitation? The rate of precipitation?
6. What kind of function could be used to describe the rate of precipitation?
7. 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?
8. Sketch a graph that might describe precipitation in your region over a period of one year.
9. How do you think mathematics might be used for the prediction of future precipitation?
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 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.

 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

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.

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.

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 (or 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
1. Make 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.
2. Print 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).
3. Use regression to find the equations for each of the three lines in your graph. Round coefficients to two decimal places.
4. These 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.
5. Use the function to determine the rate of precipitation at each of the times.
(A) t = 1.4 (B) t = 6 (C) t = 11
6. Use the function to determine the average precipitation in: A) April; B) August;C) December
7. 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 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.
8. Approximate the total precipitation in inches (water from snow and rain) for each month of the year.
9. Approximate the total amount of water from snow for the entire year.

Reflection

Examine solutions

1. Do you think the piecewise function was a good model in this case?
2. What are the advantages to using a piecewise function for this model? Disadvantages?
3. Can this model be used to predict rain on a given day? Why or why not?
4. Do you think that some of the assumptions are unreasonable? Why?
5. Can this model be used to predict precipitation over longer periods of times, say years or decades? Why or why not?
6. 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 function?  