Math Excursion:

FILLING IN THE GAPS — Acquiring and Analyzing Satellite and Ground-Based Data

Adapted from "A high performance Interactive Image Spread Sheet " (A.F. Hasler, K. Palaniappan, M. Manyin, and J. Dodge; Computers in Physics, Vol. 8, No. 3, May/Jun 1994; pp. 325-342) and "How When Affects What: Part 3–Local Temperature, Global Changes" (Ground Truth Studies Teacher Handbook).

Objective
Students will monitor daily rainfall to compute average daily rainfall and the estimate of total rainfall for the month. They will compare how their readings and computations are analogous to the methods used by researchers in computing similar data from satellite observations.
Background
The SSM/I satellite instrument receives passive microwave emissions from Earth to measure moisture levels worldwide. The instrument obtains data from ice, rainfall, oceans, clouds, and other sources of moisture in and below the atmosphere. On any given day, the geometrical position of the satellite instrument relative to Earth permits it to record data from most–but not all –of the planet's surface. This is because the satellite instruments cannot "see" all points simultaneously. Consequently, the moisture image for both hemispheres reveals patterns of diamond-shaped blackout patches from which no data has been recorded. Throughout a month-long observational window, however, all surface points eventually come into view, so that moisture readings can be captured for the entire Earth. Therefore, moisture data for a given point may be recorded for 25 out of 31 days, but for another point of data may be recorded for all 31 days. Inestimating total rainfall for the month for all points on Earth, the computation may be based on a limited set of daily readings. When a daily reading is not made for a given point, this "gap" in the data must be accounted for mathematically in the computation. To explore how scientists "fill in the gaps" for missing data from instruments such as SSM/I, students will monitor and record daily rainfall measurements for most–but not all–days in a one month observational window.
Materials

1 rain gauge per student
pencil
paper
calculator

Procedure

Have each student set up a rain gauge at home. Students should record daily rainfall for a one month period. For a given gauge, each measurement must be made at the same time of day, although it is NOT imperative that a reading be made every day of the month. Empty the gauge after each reading, and empty the gauge even on days for which NO reading is made.
Each student should compute his or her average daily rainfall measurements by adding together the daily readings and dividing by the number of measurements taken.
Compare the average daily rainfall computed by each student for the entire class.
Each student should multiply his or her average daily rainfall by the number of days in the month to compute the estimate of total rainfall for the month. The execution of this step is essentially "filling in the gaps" for days in which no data was available.

Questions

How might the method used for measuring rain in the gauge introduce error into the readings? How might varying the time each day the reading is taken introduce error?
How might failing to measure rainfall on a very rainy day affect the computation of the average daily rainfall?
Assuming that most students live in relatively close proximity to one another, and have essentially "measured the same rain," propose reasons why average daily rainfall computations may differ from student to student.
Why is it acceptable to "fill in the gaps" when computing the estimate of total rainfall for the month when a measure is not made each and every day? How can you decide on the level of acceptability?
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Last modified prior to September, 2000 by the Windows Team

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