Geog 409: Advanced Spatial Analysis & Modelling © J.M. Piwowar1Modelling in Action Hardisty, et al., Computerised Environmental Modelling. Chichester: Wiley

Geog 409: Advanced Spatial Analysis & Modelling © J.M. Piwowar2Modelling in Action Modelling Examples  Cascading models.  Process-response models.  Stochastic models.  Feedback models.

Geog 409: Advanced Spatial Analysis & Modelling © J.M. Piwowar3Modelling in Action Cascading Models  Models where system components are linked by flows of mass and/or energy. Hardisty, et al., Computerised Environmental Modelling. Chichester: Wiley

Geog 409: Advanced Spatial Analysis & Modelling © J.M. Piwowar4Modelling in Action A Simple Hydrologic Model  Object: To determine the channel discharge of a small drainage basin for a given rainfall. Hardisty, et al Computerised Environmental Modelling: A Practical Introduction Using Excel. Chichester: Wiley.

Geog 409: Advanced Spatial Analysis & Modelling © J.M. Piwowar5Modelling in Action Hardisty, et al Computerised Environmental Modelling: A Practical Introduction Using Excel. Chichester: Wiley. The “Catchwater” Drainage Basin

Geog 409: Advanced Spatial Analysis & Modelling © J.M. Piwowar6Modelling in Action Rainfall  Precipitation less evaporation losses.  Assume rainfall falls evenly over the basin.  R = rainfall * surface area Hardisty, et al Computerised Environmental Modelling: A Practical Introduction Using Excel. Chichester: Wiley.

Geog 409: Advanced Spatial Analysis & Modelling © J.M. Piwowar7Modelling in Action Surface Storage  The amount of rainfall that is detained on the surface.  Assumptions:  There is some pre-existing surface storage  One-half of rainfall is retained in surface storage.  S = S + 0.5*R Hardisty, et al Computerised Environmental Modelling: A Practical Introduction Using Excel. Chichester: Wiley.

Geog 409: Advanced Spatial Analysis & Modelling © J.M. Piwowar8Modelling in Action Surface-Channel Discharge  The proportion of water in Surface Storage that immediately runs-off.  Q SC = surface storage * constant 1  Initial Assumption: constant 1 = 0.9 Hardisty, et al Computerised Environmental Modelling: A Practical Introduction Using Excel. Chichester: Wiley.

Geog 409: Advanced Spatial Analysis & Modelling © J.M. Piwowar9Modelling in Action Surface-Ground Discharge  The proportion of water in Surface Storage that percolates into the ground.  Q SG = surface storage * constant 2  Initial Assumption: constant 2 = 0.6 Hardisty, et al Computerised Environmental Modelling: A Practical Introduction Using Excel. Chichester: Wiley.

Geog 409: Advanced Spatial Analysis & Modelling © J.M. Piwowar10Modelling in Action 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*Q SG Hardisty, et al Computerised Environmental Modelling: A Practical Introduction Using Excel. Chichester: Wiley.

Geog 409: Advanced Spatial Analysis & Modelling © J.M. Piwowar11Modelling in Action Ground-Channel Discharge  The proportion of water in Ground Storage that seeps back to the surface.  S GC = ground storage * constant 3  Initial Assumption: constant 3 = 0.3 Hardisty, et al Computerised Environmental Modelling: A Practical Introduction Using Excel. Chichester: Wiley.

Geog 409: Advanced Spatial Analysis & Modelling © J.M. Piwowar12Modelling in Action Channel Discharge  The sum of the water run-offs from the surface and from the ground. Hardisty, et al Computerised Environmental Modelling: A Practical Introduction Using Excel. Chichester: Wiley.

Geog 409: Advanced Spatial Analysis & Modelling © J.M. Piwowar13Modelling in Action 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.

Geog 409: Advanced Spatial Analysis & Modelling © J.M. Piwowar14Modelling in Action 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.

Geog 409: Advanced Spatial Analysis & Modelling © J.M. Piwowar15Modelling in Action Process-Response Models  If there is a change in process, the system will respond in order to develop a new form. Hardisty, et al., Computerised Environmental Modelling. Chichester: Wiley

Geog 409: Advanced Spatial Analysis & Modelling © J.M. Piwowar16Modelling in Action Process-Response Models  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

Geog 409: Advanced Spatial Analysis & Modelling © J.M. Piwowar17Modelling in Action Process-Response Models Hardisty, et al Computerised Environmental Modelling: A Practical Introduction Using Excel. Chichester: Wiley.

Geog 409: Advanced Spatial Analysis & Modelling © J.M. Piwowar18Modelling in Action Process-Response Models  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. Hardisty, et al Computerised Environmental Modelling: A Practical Introduction Using Excel. Chichester: Wiley.

Geog 409: Advanced Spatial Analysis & Modelling © J.M. Piwowar19Modelling in Action Process-Response Models  Model II  Wind 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). Hardisty, et al Computerised Environmental Modelling: A Practical Introduction Using Excel. Chichester: Wiley.

Geog 409: Advanced Spatial Analysis & Modelling © J.M. Piwowar20Modelling in Action Process-Response Models

Geog 409: Advanced Spatial Analysis & Modelling © J.M. Piwowar21Modelling in Action Stochastic Models  Models that include a random element.  Slightly different outputs are produced for each run.

Geog 409: Advanced Spatial Analysis & Modelling © J.M. Piwowar22Modelling in Action Stochastic Models  An example:  Modelling the variations in minimum daily temperatures.  The minimum temperature (T min ) is equal to the absolute minimum (T abs ) plus a random element (R) related to the thermal capacity of air masses (C) and the number of hours of sunlight (S).  T min = T abs + R * C* S

Geog 409: Advanced Spatial Analysis & Modelling © J.M. Piwowar23Modelling in Action Stochastic Models

Geog 409: Advanced Spatial Analysis & Modelling © J.M. Piwowar24Modelling in Action Feedback Models  Models where the model outputs affect the inputs and processes within the system.  Feedbacks can be either positive or negative.

Geog 409: Advanced Spatial Analysis & Modelling © J.M. Piwowar25Modelling in Action 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).

Geog 409: Advanced Spatial Analysis & Modelling © J.M. Piwowar26Modelling in Action 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 (Z 1,t ) = previous height at the site (Z 1,t-1 ) less the difference in height from the site above it to the site (Z 2,t-1 – Z 1,t-1 )  Where C is the coefficient of proportionality, a constant  Z 1,t = Z 1,t-1 – (Z 2,t-1 – Z 1,t-1 ) * C

Geog 409: Advanced Spatial Analysis & Modelling © J.M. Piwowar27Modelling in Action Feedback Models