# Group Presentations

## Quantitative Problems with GPS

### Connect Field and Lab

**M. Jordan, B. Kerbert, V. Schielack**

Mapping:

- Calculations of length, scale, area resolution—geometry
- Surveying: trig problems = surveying
- Average characteristics of area/volume, spatial statistics

Earth Process Dynamics

- modeling plate movements
- erosion and other geomorph
- hydrologic and biogeochemical processes

3 topics:

- vectors
- rates
- dynamic system modeling

Geodesy: size and shape of earth

## Total Volume of Water from a Storm

### Math Approach

Flood height: H = f(t) - a function of time

Discharge: D = g(H) - a function of flood height

So, D = g(f(t)), is also a function of time.

The volume of water discharged over time

or

Can be approximated using Riemann sum:

V=S g(f(t))

*See photocopied graphs*

### Geoscience Approach

[nothing written in the notes]

## Find the Circumference of the Earth Using a Shadow Stick

Ratio to determine the sun's angle at two different locations (Eratosthenes)

*Diagram on photocopies*

## Estimating Hoizontal Wind from Dopplar Radar

[Note: handwritten copy makes it difficult to distinguish between v, u, u or other similar variables, so there are undoubtedly errors in the transcription.]

Cartesian system: x, y, z; v1, v

Spherical system: r, j, q, ur, uj

Derive a relation between ur and v1:

r^{2} = x^{2} + y^{2}

Take D/Dt

2r (Dr/Dt) = 2x (Dx/Dt) + 2y (Dy/Dt) => divide by 2r

(Dr/Dt)= ur ; (Dx/Dt) = v ; (Dy/Dt) = u

ur = (x/r) u + (y/r)v

Approximate v and u as constant V and U

What are they?

Try to find V and U, then minimize equation error

Total error in a horizontal cross-section:

[Do we need long process of working out error equation—includes some lovely integrals, greek letters, derivatives and other such excitement]

Result: aU + bV = c where a, b and c are known constants determined by
ur, x, y, r

Similarly, dJ / dV = 0 and dV + eV = f

2 linear algebra equations for V and U, solve them as:

U = answer

V = answer

## g Periodic Phenomena—Harmonic analyses (too hard?)

### e.g. tides, sea-levels change

Summation of sinusoids

[greek letter that looks like a four that isn't on character map] log-normal
distribution and sediment grain-size analysis

|A|, w, f

simply: A sin (+)

## Radiometric Decay

Geologic context: how we date rocks

Mathematical context: exponential function and logarithmic

Depending on how presented, could address all 5 of the "over-arching" quantitative skills

## Crystallization/Melting

Crystallization/melting of solid solutions mineral (olivine) as single model for magma melting/crystasllization
*See graph on photocopy*

## Predicting Catastrophic Events

**Dobler, Hutchings, Ormand**

Geological contexts:

- Earthquakes
- Floods
- Storms

Mathematical Contexts:

- Modeling
- Prediction
- Error Analysis
- Probability

## Erosion and Deposition Rates

Geological Topic: Erosion and deposition rates

Examples: Grand Canyon (Colorado River), Lake Mead (Hoover Dam)

Math Skills:

- Estimation of Volume
- Rates and rates of change of rates, rates of change of volume
- Modeling

## Groundwater Containment Modeling

**Chris Gellasch, Janet Andersen, Albert Hsui**

- Install monitoring wells
- Sampling plan
- Collect samples & well volumes (averaging)
- Sample analyses (QA/QC)
- Plot data on survey map
- Sketch contours for water table and concentration of containment
- Determine gradients 3C and 4h

## Flow Problem (Heat or Water)

**Cathy Summa, Linda Eroh, Moe Muldoon, Steve Leonhardi**

Examples of geologic context:

- Groundwater flux to well, river, under dam
- Continental rifting, 3 sediment transport and mixing

Mathematical context: Differential equations

## Geologic hyperspectral Remote Sensing

Deriving viv wind from a radar display (dr/dt)

Mean annual temps on a given j to get a mean G

Atmospheric stability correlates with [??]

Measuring pressure gradients across middle latitude and tropical cyclones

Effects of wind on structure (Bernoulli)

# Large Group Work

## Predicting how high people will bounce on a trampoline on different planets

Geoscience Approach

Find gravity on each planetMath Approach

Mass of the person who jumps does not change

Weight divided by gravity

The higher the weight, the lower the jump

F= m * a

## Tophographic Maps

- Contour lines
- Vertical exaggeration—profiles
- Orientation

The math:

Slope

Rates

Gradient

Max/min

Directional derivatives

Scales (rep. fraction)

Interpretation

2D and 3D Functions

Applications/Projects:

Pipelines/canals/aquaducts

Orienteering

Optimization (lots)

Meterology

Petrology

## True Dip vs. Apparent Dip

- Visualization
- Difficulties/obstacles
- Trigonometry
- Approximation
- 2D and 3D
- Error approximation

*Diagram on photocopy*