Model Resolution Exploration

Students create and investigate a physical model to explore how the resolution of a mathematical model impacts model results.

Learning Goal

  • Students will understand how resolution impacts what we can understand from a model.
  • Students will learn about the benefits and drawbacks of high and low-resolution models.

Materials

For each group of four:

  • Plastic container or cardboard box, about the size of a shoebox
  • Rice, sand, beans, or other material to fill the box
  • Legos
  • A thin rod such as a bamboo skewer
  • 2 copies of the low and high-resolution grid paper

Preparation

  • Make two copies of the low-resolution grid paper the two copies of the high-resolution grid paper for each group.
  • Prepare the plastic containers by building shapes out of legos, placing the shapes in the plastic containers, and covering them with sand or rice. Build each set-up in the location where students will investigate it so as not to disturb the legos. (Alternatively, have student groups create models for another group.)

Directions

  1. Without mentioning anything about modeling, pass out the containers and tell the students their goal is to create an accurate depiction of the object in the container without moving the sand or rice that covers it.
  2. Pass out two copies of the low-resolution grid paper. Instruct students to make holes at the intersection points on one of the grids and to use the other to draw the model on.
  3. Instruct students to place the grid paper with holes over the top of the container and use the skewer as a probe to determine the shape of the object, drawing the shape of the object on the second copy of the grid paper. With the low resolution, they will have to make a lot of guesses as to the exact shape, but that’s the idea.
  4. For a quantitative exploration, have students use a ruler with the skewer to accurately measure the height of the object in order to create a 3-dimensional model. The height could be represented on the 2-dimensional drawing by color or shading.
  5. Repeat steps 2, 3, and 4 with the higher resolution graph paper. Students should get a much more accurate picture of the shape of the object.
  6. Allow students to remove the object from the container and compare it to their drawings.
  7. Have student groups post their drawings from the low resolution and high-resolution scenarios side by side and discuss the benefits each. They should notice that the low-resolution scenario took less time and was less accurate, the high-resolution scenario took more time and was more accurate.
  8. Ask students what they could do to create a perfectly accurate representation of the object. (You would need to make grid paper with an infinite number of locations to test with the skewer.)
  9. Show students images of how Earth system models used to understand climate change have increased in resolution over time.

Background

Global climate models (GCMs) use mathematical equations to describe the behavior of factors of the Earth system that impact climate. These factors include dynamics of the atmosphere, oceans, land surface, living things, and ice, plus energy from the Sun. Sophisticated climate models are increasingly able to include details such as clouds, rainfall, evaporation, and sea ice. Thousands of climate researchers use global climate models to better understand the long-term effects of global changes such as increasing greenhouses gases or decreasing Arctic sea ice. The models are used to simulate conditions over hundreds of years so that we can predict how our planet's climate will likely change.

"Resolution" is an important concept in many types of modeling, including climate modeling. Spatial resolution specifies how large (in degrees of latitude and longitude or in km or miles) the grid cells in a model are.

Although we know that traits like temperature vary continuously over the surface of the Earth, calculating such properties for the entire globe is beyond the reach of even the fastest supercomputers. Instead, a climate model places "virtual weather stations" at intervals around the modeled Earth and reports the calculated properties at each station. Models use grids of "cells" to establish the locations of the "virtual weather stations." A typical climate model might have grid cells with a size of about 100 km (62 miles) on a side. The "virtual weather stations" are located at the corners of the grid cells.

Models can be generated with higher or lower resolutions. The grid cells could be reduced in size to 50 km. This would mean that more cells cover Earth's surface, increasing spatial resolution. Or the grid cells could be enlarged to 200 km. This would mean fewer grid cells and decreased spatial resolution. More, smaller cells increase the amount of computing time because there are more "virtual weather stations" at which atmospheric variables must be calculated. Higher-resolution models provide much more detailed information but take lots more computing time. As a general rule, increasing the resolution of a model by a factor of two means about ten times as much computing power will be needed (or that the model will take ten times as long to run on the same computer).

Model grids for atmospheric (including climate) models are three-dimensional, extending upward through our atmosphere. Early climate models typically had about 10 layers vertically; more recent ones often have about 30 layers. Because the atmosphere is so thin compared to the vast size of our planet, vertical layers are much closer together as compared to the horizontal dimensions of grid cells. Vertical layers might be spaced at 11 km intervals as compared to the 100 km intervals for horizontal spacing.