Mathematical Modelling in Geography GEOG 2021 Environmental Remote
- Slides: 43
Mathematical Modelling in Geography GEOG 2021 Environmental Remote Sensing
Mathematical Modelling • What is a model? – an abstracted representation of reality • What is a mathematical model? – A model built with the ‘tools’ of mathematics • What is a mathematical model in Geography? – Use models to simulate effect of actual or hypothetical set of processes – to forecast one or more possible outcomes
Mathematical Modelling Functional model representation INPUTS I PROCESS f(I) O=f(I) OUTPUTS O
Type of Mathematical Model Main choice: – Statistical / empirical • ‘calibration model’ – Physically-based • model physics of interactions • in Geography, also used to include many empirical models, if it includes some aspect of physics – e. g. conservation of mass/energy - e. g. USLE • similar concepts: – Theoretical model – Mechanistic model
Type of Mathematical Model • May chose (or be limited to) combination in any particular situation • Definitions / use varies
Type of Mathematical Model Other options: – deterministic • relationship a=f(b) is always same – no matter when, where calculate it – stochastic • exists element of randomness in relationship – repeated calculation gives different results
Type of Mathematical Model Other options: – forward model • a=f(b) • measure b, use model to predict a – inverse model • b=f-1(a) • measure a, use model to predict b
Type of Mathematical Model E. g. : • forward model • inverse model
Type of Mathematical Model Practically, and especially in environmental modelling, always need to consider: • uncertainty – in measured inputs – in model – and so should have distribution of outputs • scale – different relationships over different scales • principally consider over time / space
Why Mathematical Modelling? 1. Improve process / system understanding – by attempting to describe important aspects of process/system mathematically e. g. – measure and model to planetary geology /geomorphology to apply understanding to Earth – build statistical model to understand main factors influencing system
Why Mathematical Modelling? 2. Derive / map information through surrogates e. g. : REQUIRE spatial distribution of biomass DATA spatial observations of microwave backscatter MODEL model relating backscatter to biomass Crop biomass map ? ? Soil moisture map ? ?
Why Mathematical Modelling? 3. Make past / future predictions from current observations (extrapolation) tend to use ‘physically-based’ models e. g. : short term: weather forecasting, economic models longer term: climate modelling
Why Mathematical Modelling? 4. Interpolation based on limited sample of observations - use statistical or physically-based models e. g. : – vegetation / soil surveys – political surveys
How useful are these models? • Model is based on a set of assumptions ‘As long as assumptions hold’, should be valid • When developing model – Important to define & understand assumptions and to state these explicitly • When using model – important to understand assumptions – make sure model is relevant
How do we know how ‘good’ a model is? • Ideally, ‘validate’ over wide range of conditions For environmental models, typically: – characterise / measure system – compare model predictions with measurements of ‘outputs’ • noting error & uncertainty ‘Validation’: essentially - how well does model predict outputs when driven by measurements?
How do we know how ‘good’ a model is? For environmental models, often difficult to achieve • can’t make (sufficient) measurements – highly variable environmental conditions – prohibitive timescale or spatial sampling required • systems generally ‘open’ – no control over all interactions with surrounding areas and atmosphere • use: – ‘partial validations’ – sensitivity analyses
How do we know how ‘good’ a model is? ‘partial validation’ • compare model with other models • analyses sub-components of system – e. g. with lab experiments sensitivity analyses • vary each model parameter to see how sensitive output is to variations in input – build understanding of: • model behaviour • response to key parameters • parameter coupling
Statistical / empirical models • Basis: simple theoretical analysis or empirical investigation gives evidence for relationship between variables – Basis is generally simplistic or unknown, but general trend seems predictable • Using this, a statistical relationship is proposed
Statistical / empirical models E. g. : • From observation & basic theory, we observe: – vegetation has high NIR reflectance & low red reflectance – different for non-vegetated FCC NDVI
Biomass vs NDVI: Sevilleta, NM, USA http: //sevilleta. unm. edu/research/local/plant/tmsvinpp/documents/sevsymp 2001_files/v 3_document. htm
Statistical / empirical models • Propose linear relationship between vegetation amount (biomass) and NDVI – Model fit ‘reasonable’, R-squared. 27 • Calibrate model coefficients (slope, intercept) • Biomass/ (g/m 2) = -136. 14 + 1494. 2*NDVI – biomass changes by 15 g/m 2 for each 0. 01 NDVI – X-intercept (biomass = 0) around 0. 10 • value typical for non-vegetated surface http: //sevilleta. unm. edu/research/local/plant/tmsvinpp/documents/sevsymp 2001_files/v 3_document. htm
Statistical / empirical models Dangers: • changing environmental conditions (or location) – lack of generality • surrogacy – apparent relationship with X through relationship of X with Y • Don’t have account for all important variables – tend to treat as ‘uncertainty’
Include season during which measurements made. . . • Biomass versus NDVI & Season • Examine inclusion of other factors: • Season: http: //sevilleta. unm. edu/research/local/plant/tmsvinpp/documents/sevsymp 2001_files/v 3_document. htm
Statistical / empirical models • Model fit improved, R-squared value increased to 38. 9% – Biomass = -200. 1 + 1683*NDVI + 25. 3*Season • biomass changes by 17 g/m 2 for each 0. 01 NDVI • X-intercept is 0. 104 for Spring and 0. 89 for summer http: //sevilleta. unm. edu/research/local/plant/tmsvinpp/documents/sevsymp 2001_files/v 3_document. htm
Estimated 116 Estimated. Live. Plant. Biomass: 1989 1990 Sep May http: //sevilleta. unm. edu/research/local/plant/tmsvinpp/documents/sevsymp 2001_files/v 3_document. htm
Estimated Live Plant Biomass: 1990 May 6 http: //sevilleta. unm. edu/research/local/plant/tmsvinpp/documents/sevsymp 2001_files/v 3_document. htm
Statistical / empirical models • Model ‘validation’ – should obtain biomass/NDVI measurements over wide range of conditions – R 2 quoted relates only to conditions under which model was developed • i. e. no information on NDVI values outside of range measured (0. 11 to 0. 18 in e. g. shown)
‘Physically-based’ models • e. g. Simple population growth • Require: – model of population Q over time t • Theory: – in a ‘closed’ community, population change given by: • increase due to births • decrease due to deaths – over some given time period t
‘Physically-based’ models • population change given by: • increase due to births • decrease due to deaths [1]
‘Physically-based’ models • Assume that for period t – rate of births per head of population is B – rate of deaths per head of population is D • We can write: [2] – Implicit assumption that B, D are constant over time t
‘Physically-based’ models • If model parameters, B, D, constant and ignoring age/sex distribution and environmental factors (food, disease etc) Then. . .
‘Physically-based’ models • As time period considered decreases, can write eqn [3] as a differential equation: • i. e. rate of change of population with time equal to birth-rate minus death-rate multiplied by current population • Solution is. . .
‘Physically-based’ models • Consider the following: – what does Q 0 mean? – does the model work if the population is small? – What happens if B>D (and vice-versa)? – How might you ‘calibrate’ the model parameters? [hint - think logarithmically]
‘Physically-based’ models Q B>D Q 0 B<D t
Other Distinctions • Another distinction – Analytical / Numerical
Other Distinctions • Analytical – resolution of statement of model as combination of mathematical variables and ‘analytical functions’ – i. e. “something we can actually write down” – e. g. biomass = a + b*NDVI – e. g.
Other Distinctions • Numerical – solution to model statement found e. g. by calculating various model components over discrete intervals • e. g. for integration / differentiation
Which type of model to use? • Statistical – advantages • simple to formulate • generally quick to calculate • require little / no knowledge of underlying (e. g. physical) principles • (often) easy to invert – as have simple analytical formulation
Which type of model to use? • Statistical – disadvantages • may only be appropriate to limited range of parameter • may only be applicable under limited observation conditions • validity in extrapolation difficult to justify • does not improve general understanding of process
Which type of model to use? • Physical/Theoretical/Mechanistic – advantages • if based on fundamental principles, applicable to wide range of conditions • use of numerical solutions (& fast computers) allow great flexibility in modelling complexity • may help to understand process – e. g. examine role of different assumptions
Which type of model to use? • Physical/Theoretical/Mechanistic – disadvantages • more complex models require more time to calculate – get a bigger computer! • Supposition of knowledge of all important processes and variables as well as mathematical formulation for process • often difficult to obtain analytical solution • often tricky to invert
Summary • Empirical (regression) vs theoretical (understanding) • uncertainties • validation – Computerised Environmetal Modelling: A Practical Introduction Using Excel, Jack Hardisty, D. M. Taylor, S. E. Metcalfe, 1993 (Wiley) – Computer Simulation in Physical Geography, M. J. Kirkby, P. S. Naden, T. P. Burt, D. P. Butcher, 1993 (Wiley)
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