Inverse methods in glaciology

Transcription

1 Inverse methods in glaciology Martin Truer University of Alaska Fairbanks McCarthy, Summer School, / 20

2 General Problem Setting Examples of inverse problems Solution methods 2 / 20

3 Outline General Problem Setting Examples of inverse problems Solution methods General Problem Setting 3 / 20

4 General Problem Common situation in geophysics: You have observables (data d) General Problem Setting 4 / 20

5 General Problem Common situation in geophysics: You have observables (data d) You have a certain understanding of the world that is expressed in a set of equations (forward model G ) General Problem Setting 4 / 20

6 General Problem Common situation in geophysics: You have observables (data d) You have a certain understanding of the world that is expressed in a set of equations (forward model G ) You would like to derive a set of parameters (model m) General Problem Setting 4 / 20

7 Forward model The forward model consists of an equation or a set of equations that can calculate observables from model parameters: G : m d General Problem Setting 5 / 20

8 Forward model The forward model consists of an equation or a set of equations that can calculate observables from model parameters: G : m d We would like to go the other way, but G might not have a well-dened inverse General Problem Setting 5 / 20

9 Forward model The forward model consists of an equation or a set of equations that can calculate observables from model parameters: G : m d We would like to go the other way, but G might not have a well-dened inverse Finding m from d is often an ill-posed problem General Problem Setting 5 / 20

10 Ill-posed problems The problem might not have a solution General Problem Setting 6 / 20

11 Ill-posed problems The problem might not have a solution The problem might have many solutions General Problem Setting 6 / 20

12 Ill-posed problems The problem might not have a solution The problem might have many solutions The solution might be badly dened, i.e. small changes in input lead to large changes in output General Problem Setting 6 / 20

13 Ill-posed problems The problem might not have a solution The problem might have many solutions The solution might be badly dened, i.e. small changes in input lead to large changes in output Honest mathematicians keep their hands o such problems General Problem Setting 6 / 20

14 Linear inverse problems Linear means that the map G is linear General Problem Setting 7 / 20

15 Linear inverse problems Linear means that the map G is linear If G is linear, the data d can be written as a scalar product in an appropriate space: d = (g, m) General Problem Setting 7 / 20

16 Linear inverse problems Linear means that the map G is linear If G is linear, the data d can be written as a scalar product in an appropriate space: d = (g, m) In discretized form this is d = Gm, where G is a matrix of dimension N M General Problem Setting 7 / 20

17 Linear inverse problems Linear means that the map G is linear If G is linear, the data d can be written as a scalar product in an appropriate space: d = (g, m) In discretized form this is d = Gm, where G is a matrix of dimension N M For linear inverse problems useful theorems can be derived (such as existence of solutions, etc General Problem Setting 7 / 20

18 Linear inverse problems Linear means that the map G is linear If G is linear, the data d can be written as a scalar product in an appropriate space: d = (g, m) In discretized form this is d = Gm, where G is a matrix of dimension N M For linear inverse problems useful theorems can be derived (such as existence of solutions, etc Non-linear problems are much more dicult. Often the methods involve linearization and iteration. General Problem Setting 7 / 20

19 Outline General Problem Setting Examples of inverse problems Solution methods Examples of inverse problems 8 / 20

20 Examples of inverse problems Finding the seismic velocity structure of the Earth from measurements of seismic arrival times Examples of inverse roblems 9 / 20

21 Examples of inverse problems Finding the seismic velocity structure of the Earth from measurements of seismic arrival times Finding oil by active seismics Examples of inverse roblems 9 / 20

22 Examples of inverse problems Finding the seismic velocity structure of the Earth from measurements of seismic arrival times Finding oil by active seismics Finding a brain tumor with a CAT scan Examples of inverse roblems 9 / 20

23 Examples of inverse problems in glaciology Finding basal conditions (e.g. slipperiness) from surface velocity observations Examples of inverse roblems 10 / 20

24 Examples of inverse problems in glaciology Finding basal conditions (e.g. slipperiness) from surface velocity observations Finding past accumulation from radar layers Examples of inverse roblems 10 / 20

25 Examples of inverse problems in glaciology Finding basal conditions (e.g. slipperiness) from surface velocity observations Finding past accumulation from radar layers Finding ice thickness from gravity anomalies Examples of inverse roblems 10 / 20

26 Examples of inverse problems in glaciology Finding basal conditions (e.g. slipperiness) from surface velocity observations Finding past accumulation from radar layers Finding ice thickness from gravity anomalies Finding initial conditions for ice sheet models given all available observations Examples of inverse roblems 10 / 20

27 Outline General Problem Setting Examples of inverse problems Solution methods Solution methods 11 / 20

28 Norm minimization Example: Finding velocities at the base of a glacier from surface observations Solution methods 12 / 20

29 Norm minimization Example: Finding velocities at the base of a glacier from surface observations The discretized problem has many solutions. How do you choose one? Solution methods 12 / 20

30 Norm minimization Example: Finding velocities at the base of a glacier from surface observations The discretized problem has many solutions. How do you choose one? Minimize a property of the solution that can be expressed as a norm Solution methods 12 / 20

31 Norm minimization Example: Finding velocities at the base of a glacier from surface observations The discretized problem has many solutions. How do you choose one? Minimize a property of the solution that can be expressed as a norm The choice of norm determines the solution (user input or a prior) Solution methods 12 / 20

32 Generating data to be used in an example Solution methods 13 / 20

33 Finding the smallest solution Solution methods 14 / 20

34 Finding the smoothest solution Solution methods 15 / 20

35 Accounting for errors Observations always have errors Solution methods 16 / 20

36 Accounting for errors Observations always have errors Forward models are always imperfect Solution methods 16 / 20

37 Accounting for errors Observations always have errors Forward models are always imperfect A perfect t to data is neither expected nor desirable Solution methods 16 / 20

38 Accounting for errors Observations always have errors Forward models are always imperfect A perfect t to data is neither expected nor desirable We nd models that t data within a certain tolerance: Gm d T Solution methods 16 / 20

39 Finding a solution within a tolerance Solution methods 17 / 20

40 Bayesian methods Information is thought of as probability distribution Solution methods 18 / 20

41 Bayesian methods Information is thought of as probability distribution Ex: Observations with random errors ρ(d) exp (d/σ) 2 Solution methods 18 / 20

42 Bayesian methods Information is thought of as probability distribution Ex: Observations with random errors ρ(d) exp (d/σ) 2 One makes a prior assumption about the model parameters ρ(m) Solution methods 18 / 20

43 Bayesian methods Information is thought of as probability distribution Ex: Observations with random errors ρ(d) exp (d/σ) 2 One makes a prior assumption about the model parameters ρ(m) Apply Bayes' Theorem: ρ(m d) = rho(m)ρ(d m) ρ(d) Solution methods 18 / 20

44 Bayesian methods Information is thought of as probability distribution Ex: Observations with random errors ρ(d) exp (d/σ) 2 One makes a prior assumption about the model parameters ρ(m) Apply Bayes' Theorem: ρ(m d) = rho(m)ρ(d m) ρ(d) The probability of m given d is equal to the prior assumption times the probability of d given m (the forward model) divided by the probability distribution of the data Solution methods 18 / 20

45 Iterative methods Make an assumption about the model parameters Solution methods 19 / 20

46 Iterative methods Make an assumption about the model parameters Calculate the mist to observations Solution methods 19 / 20

47 Iterative methods Make an assumption about the model parameters Calculate the mist to observations Calculate a correction to lower the mist Solution methods 19 / 20

48 Iterative methods Make an assumption about the model parameters Calculate the mist to observations Calculate a correction to lower the mist Stop once the data are t well enough Solution methods 19 / 20

49 Commonalities of inverse methods Many methods turn out to be identical on a fundamental level Solution methods 20 / 20

50 Commonalities of inverse methods Many methods turn out to be identical on a fundamental level Each method nds a solution and not the solution Solution methods 20 / 20

51 Commonalities of inverse methods Many methods turn out to be identical on a fundamental level Each method nds a solution and not the solution A solution of the inverse problem is a set of model parameters that is consistent with the forward model and the data within errors Solution methods 20 / 20

52 Commonalities of inverse methods Many methods turn out to be identical on a fundamental level Each method nds a solution and not the solution A solution of the inverse problem is a set of model parameters that is consistent with the forward model and the data within errors Each method involves a number of assumptions Solution methods 20 / 20