## MA4DA-Theory and Techniques of Data Assimilation

Module Provider: Mathematics and Statistics
Number of credits: 10 [5 ECTS credits]
Level:7
Terms in which taught: Autumn term module
Pre-requisites: MA1LIN Linear Algebra or MA1LA Linear Algebra and MT24C Numerical Methods for Environmental Science or MA2NA1 Numerical Analysis I
Non-modular pre-requisites:
Co-requisites:
Modules excluded: MA3DA Theory and Techniques of Data Assimilation
Module version for: 2016/7

Module Convenor: Dr Amos Lawless

Summary module description:
This module introduces the theory of data assimilation and its applications.

Aims:
Mathematical models for simulating physical, biological and economic systems are now often more accurate than the data that is available to drive them. In particular, complete information describing the initial state of an evolutionary system is seldom known. In this case it is desirable to use the measured output data that is available from the system over an interval of time, in combination with the numerical model equations, to derive accurate estimates of the expected system behaviour. The problem of constructing a state-estimator, or observer, is the dual of the feedback control design problem. For very large nonlinear systems arising in numerical weather prediction and in ocean circulation modelling, traditional control system design techniques are not practicable, and 'data assimilation' schemes are used instead to generate accurate state-estimates. The aim of these schemes is to incorporate observed data into computational simulations in order to improve the accuracy of the numerical forecasts.

Assessable learning outcomes:
At the end of the course, students will (i) appreciate the principles of data assimilation theory; (ii) be able to apply techniques of data assimilation to simple problems and (iii) be able to discuss and apply data assimilation theory beyond that covered in lectures.

Outline content:
• Purpose of data assimilation. Observations and their availability. Different applications.
• Data assimilation for a general linear system. Sequential and 4D variational.
• Observability matrix.
• Minimum variance analysis for simple scalar problem. Equivalence to least squares for Gaussian errors.
• Bayes' theorem. Maximum a posteriori equivalent to minimum variance for Gaussian errors.
• Least squares for 3D problem from Bayes theorem.
• Derivation of Best Linear Unbiased Estimate (BLUE) as solution of 3D problem.
• Analysis error covariance matrix.
• Background error covariance matrix and how it affects analysis increments in BLUE.
• 3D-Var. Idea of numerical minimization. Gradient of cost function with nonlinear observation operator.
• Control variable transform to represent background error covariance matrix.
• 4D-Var theory - Calculation of gradient using Lagrange multipliers.
• Assumptions of 4D-Var.
• Estimating the quality of the analysis from the Hessian.
• Kalman filter.
• Extended Kalman filter.
• Relationship between Kalman filter and 4D-Var.

Brief description of teaching and learning methods:
Lectures supported by problem sheets and guided reading.

Contact hours:
 Autumn Spring Summer Lectures 20 Guided independent study 80 Total hours by term 100.00 Total hours for module 100.00

Summative Assessment Methods:
 Method Percentage Written exam 80 Set exercise 20

Other information on summative assessment:
Two problem sheets and one examination paper.

Formative assessment methods:
Problem sheets.

Penalties for late submission:
The Module Convenor will apply the following penalties for work submitted late, in accordance with the University policy.

• where the piece of work is submitted up to one calendar week after the original deadline (or any formally agreed extension to the deadline): 10% of the total marks available for the piece of work will be deducted from the mark for each working day (or part thereof) following the deadline up to a total of five working days;
• where the piece of work is submitted more than five working days after the original deadline (or any formally agreed extension to the deadline): a mark of zero will be recorded.

• The University policy statement on penalties for late submission can be found at: http://www.reading.ac.uk/web/FILES/qualitysupport/penaltiesforlatesubmission.pdf
You are strongly advised to ensure that coursework is submitted by the relevant deadline. You should note that it is advisable to submit work in an unfinished state rather than to fail to submit any work.

Length of examination:
2 hours.

Requirements for a pass:
A mark of 50% overall.

Reassessment arrangements:
One examination paper of 2 hours duration in August/September - the resit module mark will be the higher of the exam mark (100% exam) and the exam mark plus previous coursework marks (80% exam, 20% coursework).