geog 409: advanced spatial analysis & modelling © j.m. piwowar1modelling in action hardisty, et...

© J.M. Piwowar 1Geog 409: Advanced Spatial Analysis & Modelling Modelling in Action

Modelling in Action

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Modelling Examples

Cascading models. Process-response models. Stochastic models. Feedback models.


Cascading Models

Models where system components are linked by flows of mass and/or energy.

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A Simple Hydrologic Model

Object: To determine the channel discharge of a small drainage basin for a given rainfall.

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The “Catchwater” Drainage Basin


Rainfall

Precipitation less evaporation losses. Assume rainfall falls evenly over the

basin. R = rainfall * surface area

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Surface Storage

The amount of rainfall that is detained on the surface.

Assumptions: There is some pre-existing surface storageOne-half of rainfall is retained in surface

storage.

S = S + 0.5*R

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Surface-Channel Discharge

The proportion of water in Surface Storage that immediately runs-off.

QSC = surface storage * constant1

Initial Assumption: constant1 = 0.9

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Surface-Ground Discharge

The proportion of water in Surface Storage that percolates into the ground.

QSG = surface storage * constant2


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Groundwater Storage

The amount of water in the unsaturated soil layers and in the saturated zone below the water table.

Assumptions: There is some pre-existing

groundwater storage One-half of percolating

rainfall is retained in groundwater storage.

G = G + 0.5*QSG

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Ground-Channel Discharge

The proportion of water in Ground Storage that seeps back to the surface.

SGC = ground storage * constant3


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Channel Discharge

The sum of the water run-offs from the surface and from the ground.

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Time-Dependent or Evolutionary Models

One or more of the input parameters varies through time.

Example – a monthly extension of the drainage basin model.Assumptions:

The only time-dependent parameter is the monthly rainfall; and

The initial surface storage and the initial ground storage for a month is the mean surface and ground storage from the preceding month.


Spatially-Dependent Models

One or more of the input parameters varies by location.

Example – a hemispheric extension of the drainage basin model.Assumptions:

Rainfall decreases linearly from wet-tropical (Deep South) to dry-arid (Deep North); and

Each drainage basin is the same size and have initial surface and ground storages of 1000 cu.m.


Process-Response Models

If there is a change in process, the system will respond in order to develop a new form.

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An example:Modelling the boundary layer flow over

a desert dune field.Determine the variation in wind speed

with height above the sand surface.Deterministic; no feedbacks



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Model I Wind speed (u) is directly proportional

to the height above the sand (z) and the free wind speed at 100 cm above the surface (U).

u = c * U * z Where c is a constant (the coefficient

of proportionality).• The coefficient of proportionality is a constant used in

models between two quantities of different dimension.

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Model IIWind speed (u) is directly proportional

to the logarithm of the height above the sand (z) and the free wind speed at 100 cm above the surface (U).

u = c * U * LOG(z)Where c is a constant (the coefficient of

proportionality).

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Stochastic Models

Models that include a random element.Slightly different outputs are produced

for each run.


Stochastic Models

An example:Modelling the variations in minimum

daily temperatures.The minimum temperature (Tmin) is equal

to the absolute minimum (Tabs) plus a random element (R) related to the thermal capacity of air masses (C) and the number of hours of sunlight (S).

Tmin = Tabs + R * C* S


Stochastic Models


Feedback Models

Models where the model outputs affect the inputs and processes within the system.Feedbacks can be either positive or

negative.


Feedback Models

An Example:Modelling the evolution of a hillslope

profile through time.Erosion at each site is directly

proportional to the difference in height between that site and the next one further up the slope (i.e. it is proportional to the gradient of the hillslope).


Feedback Models

The hillslope is divided into 15 sites; the elevation of the first one is fixed at 1000m.

Height at a site at the present time (Z1,t) = previous height at the site (Z1,t-1) less the difference in height from the site above it to the site (Z2,t-1 – Z1,t-1)

Where C is the coefficient of proportionality, a constant

Z1,t = Z1,t-1 – (Z2,t-1 – Z1,t-1) * C


Feedback Models

geog 409: advanced spatial analysis & modelling © j.m. piwowar1modelling in action hardisty, et...

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