UNIVERSITY COLLEGE LONDON
EXAMINATION FOR INTERNAL STUDENTS
MODULE CODE : GEOGG121
ASSESSMENT : GEOGG121A
PATTERN
MODULE NAME : Analytical and Numerical Methods
DATE : 9 January 2019
TIME : 10:00
TIME ALLOWED : 2 hours
This paper is suitable for candidates who attended classes for this
module in the following academic year(s):
Year
2017
EXAMINATION PAPER CANNOT BE REMOVED FROM THE EXAM HALL. PLACE EXAM
PAPER AND ALL COMPLETED SCRIPTS INSIDE THE EXAMINATION ENVELOPE
2017/18-Geography-Geography-EXAM-Geography
© 2018 University College London
Stationary Required Amount
Answer booklets 2
Calculators Allowed Standard
TURN OVER
GEOGG121 1 PLEASE TURN OVER
GEOGG121: Analytical and Numerical Methods
Time allowed: 2 hours
 Answer any 2 questions
 Answer each question in a separate answer book
 Include diagrams where they assist in providing an explanation
 All questions carry equal weighting. Where questions are split into sub-categories, the
marks for each part are shown in [ ]
 You are advised to spend an equal amount of time on each question
 Tables and / or charts will be provided
 Calculators are permitted
QUESTION ONE
i) Give a linear differential equation relating attenuation of incident radiation, ϕ Wm-2,
with distance z through a medium of extinction coefficient k m-1. [10 marks]
ii) Derive the general solution of this equation. Show your working. [20 marks]
iii) Solar radiation incident on the Earth’s surface is measured as follows for 3 solar zenith
angles θ (angle from the vertical):
Noting that the path length z through the atmosphere will be a function of θ, use these
values and the expression derived in part ii) to show that the extinction coefficient k, is
approximately 0.4 and that the solar constant (i.e. ϕ at z = 0 at the top of the atmosphere)
is ~1385 Wm-2. [70 marks]
QUESTION TWO
i) State Bayes Theorem, defining each of the terms. [10 marks]
ii) Describe ONE key advantage and ONE key disadvantage of the use of Bayes Theorem
to calculate probability, and outline why Bayes’ Theorem has been considered
controversial. [40 marks]
iii) Suppose a screening test to detect faulty loaves of bread in a bread factory is 99%
accurate in producing a true positive and 97% accurate in producing a true negative (i.e.
the test produces 3 times as many false negatives as false positives). Suppose also that
5% of the loaves are in fact faulty. Given these figures, is the screening test useful?
Explain your answer. [50 marks]
Solar zenith angle, θ ϕ(z) Wm-2
30° 871
45° 785
60° 620
GEOGG121 2 END OF PAPER
QUESTION THREE
i) Describe why model fitting and parameter inversion/estimation are important in
quantitative science, giving examples where possible, in particular distinguishing
between linear and non-linear models. [50 marks]
ii) Describe how we formulate the problem of inversion using a cost or error function and
outline ONE method for finding the minimum of such a function. [40 marks]
iii) Outline one metric we might use to judge the goodness of fit of a model to a set of
observations. [10 marks]
QUESTION FOUR
i) Outline how Monte Carlo (MC) sampling can be used for estimating the value of a
function. Describe the key advantages and disadvantages of the MC approach,
including how the error of a MC estimate varies. [80 marks]
ii) Give a specific example of MC sampling in a modelling application, making clear
why MC is used in this case. [20 marks]
QUESTION FIVE
Outline the advantages and disadvantages of using physically-based and empirical models in
the physical sciences. [100 marks]