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

  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 should be used to measure the rate of precipitation?
  6. What kind of graph would be used to predict the rate of precipitation?
  7. Do you think that a graph describing the 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.

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 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

 

Further, the value 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.

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 end-to-end 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

  1. The average precipitation is 3.2 inches in March.
  2. The average precipitation is .75 inches in November.
  3. Precipitation at the end of December, t = 12 (and the beginning of January, t = 0), is 1.67 inches per month.
  4. 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 linear regression applet.
  2. Print the plot screen from the applet and connect adjacent data points with straight lines; this will be the precipitation graph.
  3. 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.
  4. Use the graph to determine the rate of precipitation at each of the times. (A) t = 1.4 (B t = 11
  5. Use the graph to determine the average precipitation in April, in August and in December
  6. When will the rate of precipitation be 2 inches per month?
  7. 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 mid-October 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(mid-point). For half months, take half of the average precipitation.

  8. Find the total precipitation in inches (water from snow and rain) for each month of the year.
  9. Determine the total amount of water from snow for the entire year.

Reflection

Examine solutions

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