Inverse Problems, Information Content and Stratospheric Aerosols

Transcription

1 Inverse Problems, Information Content and Stratospheric Aerosols May 22, 214

2 Table of contents 1 Inverse Problems 2 Aerosols 3 OSIRIS 4 Information Content 5 Non-Linear Techniques 6 Results

3 Inverse Problems - Overview Given the output of a system, determine the system s state Often more difficult than the forward problem First described mathematically by Victor Ambartsumian in 1929 but didn t take off until the 5 s Most people, if you describe a train of events to them, will tell you what you the result would be. They can put those events together in their minds, and argue from them that something will come to pass. There are few people, however, who, if you told them the result would be able to evolve from their own inner consciousness what the steps were which led up to that result. This power is what I mean when I talk of reasoning backwards, or analytically. -Sherlock Holmes 1887

4 Inverse Problems - Overview Show up all the time in geophysics, but pretty much every field has them Often ill-posed or underdetermined, although can be over-determined as well. I ll talk about some methods I ve used for investigating and solving the ill-posed/underdetermined variety First, a look at the inverse problem I ve been investigating

5 Aerosols Small droplets (occasionally crystals) of sulphuric acid and water in the stratosphere Reflect solar radiation, decreasing warming (.5 C surface cooling after Pinatubo eruption) Highly variable due to volcanic events, stratospheric circulation patterns IPCC points to aerosols as one of the biggest unknowns in climate forecasting

6 Aerosols To fully describe aerosol we would need to know the particle size distribution and total amount of aerosol. This is pretty much impossible with OSIRIS so we settle for a few simpler parameters Integral properties of the distribution such as total number density, volume, or extinction Assume a distribution shape and try to retrieve parameters of that shape Lognormal is common and well supported by other data ( (ln r ln r g ) 2 ) dn(r) dr = n aer r ln(σ g ) 2π exp 2 ln(σ g ) 2 Leaves only 3 aerosol parameters - mode radius, mode width and number density

7 Aerosols A few typical lognormal distributions

8 OSIRIS OSIRIS is a Canadian instrument launched in 21 Measures radiance along a single line of sight Satellite nods up and down to produce vertical profiles Used to retrieve vertical profiles of ozone, NO 2, BrO and aerosols Solar Illumination Line of Sight Satellite Earth Atmosphere

9 OSIRIS I meas = s1 s J(s) = k sc(s) k(s) k(s)j(s)e τ(s,s 1) ds 4π I diff (s, Ω)p(s, Θ)dΩ Solar Illumination Line of Sight Attenuate J(s) back to satellite Satellite Integrate I diff and rescatter to get J(s) Earth Atmosphere

10 OSIRIS An example of some OSIRIS measurements Ozone visible around 3 and 6 nm NO 2 around 44 nm Broadband aerosol signature >5 nm Spectral Radiance (photons/m 2 /s/nm/st) Wavelength (nm)

11 OSIRIS Goal is to use this broadband aerosol signature to determine size and quantity of aerosol particles A few problems that make this tricky: Finite measurements of continuous distributions (underdetermined system) Aerosol signal is broadband, limiting useful wavelengths (non-orthogonal measurements) Measurements at each height contain information from all heights (non-orthogonal measurements) This altitude coupling also makes the problem non-linear To get an idea of what s possible first take a look at the information content of the measurements

12 Information Content Twomey explored a very nice way of determining information content of linear systems by looking at the eigenfunctions/vectors of the measurements. Say we have some linear system: y = K(s )x(s )ds y = Kx s Then the Jacobian (or kernel), K, is the sensitivity of the measurement, y, to changes in the state, x. K i j = dk i dx j

13 Information Content If we can predict a measurement before its taken, then that measurement provides no new information. This is possible if K is rank deficient, or, K i = a 1 i a j K j (s). Since if this is true we can also reconstruct the measurement y i = a 1 i a j K j x(s )ds = a 1 k a l y l j i s j i l k

14 Information Content For noisy measurements this doesn t need to be strictly true, and if we can reconstruct a measurement within error, then it is redundant. The covariance matrix C ij = K i (s )K j (s )ds s tells us the orthogonality of the measurements. The number of measurements that contain unique information is then given by number of eigenvalues of C that exceed the relative error Λ i Λ 1 > ɛ 2

15 Linearizing the OSIRIS Measurements To apply this to OSIRIS we first have to make the problem linear. Assume changes in aerosol don t change the diffuse field I diff Assume most of the information is coming from the tangent point (lowest altitude we re looking through) Attenuation back to the satellite is negligible This lets us write the radiance in terms of some aerosol properties, and the measurement y as an integral over a particle size distribution. I aer n aer σ aer p aer (Θ) y i = K i (r) dv (r) dr dr

16 Linearizing the OSIRIS Measurements Take 4 measurements, at 2 different wavelengths and 2 different satellite geometries - how much info? K4 K3 K2 K λ = 75 nm Θ = 6 λ = 75 nm Θ = 118 λ = 1.53 µm Θ = 6 λ = 1.53 µm Θ = Radius (µm) Multiple Scatter Single Scatter 4

17 Linearizing the OSIRIS Measurements This gives normalized eigenvalues of 1.,.8,.4,.1 Λi Λ 1 = 1.,.28,.21,.12 Meaning only 1% of the 4 th measurement is new information, resulting in a 1 error magnification. Implying that it is probably not a good idea to extract 4 pieces of information from these 4 measurements. (Unless the measurements were very accurate)

18 Using the Information What can we do with these 3ish pieces of information? It might be possible to define some particle size distribution with 2 parameters and retrieve that, however, just because we have three pieces of information doesn t mean we can retrieve any 3 pieces. The work by Twomey (Also see Backus and Gilbert) provides a method of determining which quantities we re most sensitive to, and a maximum possible error.

19 An Orthogonal Framework Dealing with non-orthogonal functions is hard, so first we make our measurements orthogonal using Principal Component Analysis K T φ T = K T QΛ 1/2 Where Q and Λ are the eigenvectors and values of C = KK T respectively. Arbitrary Units st Component 2 nd Component 3 rd Component 4 th Component Radius (µm) 4

20 An Orthogonal Framework With an orthogonal set it s easy to reconstruct an arbitrary function (fourier decomposition, spherical harmonics, etc...), so we can test our ability to reconstruct various parameters. So, lets say we want to retrieve a property, P, that depends on our desired quantity dv (r)/dr P = Ψ(r) dv (r) dr dr

21 An Orthogonal Framework If we can write Ψ(r) in terms of our kernels, then it can be obtained directly from measurements P = Ψ(r) dv (r) dr dr dv (r) a i K i (r) dr = dr i i We can decompose P using our orthogonal functions P = dv (r) (Ψ φ i )φ i (r) dr dr i a i y i The accuracy of our retrieved quantity P, is then dictated by how well we can reconstruct Ψ(r) using our kernels

22 An Orthogonal Framework Applying this to the OSIRIS data we can test how well we can retrieve a couple aerosol properties, total number of particles, n aer, and total amount of light scattered by aerosols, k aer. Ψ naer = 3 4πr 3 Ψ kaer = 3 4πr 3 σ aer (r)

23 An Orthogonal Framework Number density is very poorly reconstructed, however extinction is actually quite good, implying extinction is a much more robust quantity for retrieval. However, errors may still be large if the majority of particles are small. Ψnaer(r) (1/µm 3 ) 4 2 Ψnaer(r) = 3 4πr 3 i (Ψ φ i)φi(r) Ψkaer (r) = 3 4πr3 σaer(λ, r) i (Ψ φ i)φi(r) 2 1 Ψkaer (r) (cm2 /µm 3 ).1 1 Radius (µm) 4

24 A Linear Inversion We can also use these orthogonal function to perform an inversion. First, write the retrieved quantity as a linear combination of our kernels (plus whatever we can t reconstruct) dv (r) dr = i a i φ i (r) + O N Then our measurements would be y i = K i (r) dv (r) dr = dr K i (a i φ i )dr which yields dv (r) = K T QΛ 1 Q T y dr Note the Λ 1 term, that may cause large error magnification for small eigenvalues i

25 A Linear Inversion As an example, I attempted to retrieve a few different particle size distributions using OSIRIS measurements dv (r)/dr (1 5 µm 1 ) rg=.8µm V =-1.8e-7 σg=1.3 rg=.15µm V =1.7e-6 σg=1.3 rg=.35µm V =5.4e-6 σg= rg=.8µm V =1.3e-6 σg=1.6 rg=.15µm V =3.9e-6 σg=1.6 rg=.35µm V =5.2e-6 σg= Radius (µm) rg=.8µm V =3.4e-6 σg=1.9 rg=.15µm V =4.7e-6 σg=1.9 rg=.35µm V =4.1e-6 σg=

26 A Non-Linear Inversion Linear analysis is nice for getting a grasp on the problem, but too inaccurate for actual retrievals - need a non-linear technique. System Output, y meas A priori System State, x System Model y mod = F(x) Retrieved System State Update State x (n) x (n+1)

27 A Non-Linear Inversion Provided you have a good forward model, the key here is updating the state vector. Lots of ways to do this Jacobian based methods: Newton, Gradient descent, Levenberg-Marquardt Quick to converge, slow to iterate Multiplicative methods: Chahine relaxation, Algebraic reconstruction Quick to iterate slow to converge Statistical methods: Optimal Estimation Requires detailed statistical knowledge of the system More complicated methods Genetic algorithms, simulated annealing Difficult implementation

28 A Non-Linear Inversion Most OSIRIS products use the Multiplicative Algebraic Reconstruction Technique (MART) since the measurements are not too non-linear and Jacobians are expensive to calculate. x (n+1) i = x (n) i j i y measij y modij (x (n) ) W kij However, aerosol measurements are more non-linear as a function of particle size and have multiple dimensions (number density and mode radius), so this method provides very slow convergence.

29 A Non-Linear Inversion Instead the aerosol retrieval uses the Levenberg-Marquardt algorithm (K T K + γdiag(k T K))δ = K T (y meas y mod ) x (n+1) = x (n) + δ This is basically a combination of Gauss-Newton s method and gradient descent, but generally provides faster convergence and improved robustness Downside is that it requires computation of the Jacobian, K, which takes time.

30 A Non-Linear Inversion The final retrieval algorithm 1 Assume aerosol consists of a lognormal distribution with variable mode radius and number density 2 Use two OSIRIS measurements at each altitude, one at 75 nm and one at 153 nm 3 Simulate measurements using forward model and update radius and density using LM 4 Compare simulation results with measurements and if they don t agree repeat step 2

31 Results