Nonlinear data assimilation for ocean applications

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1 Nonlinear data assimilation for ocean applications Peter Jan van Leeuwen, Manuel Pulido, Jacob Skauvold, Javier Amezcua, Polly Smith, Met Ades, Mengbin Zhu

2 Comparing observations and models Ensure they represent the same thing Comparison is useless unless uncertainties in observations and models are taken into account (representation errors) Data assimilation does that in a systematic way DA can be used for: - Forecasting - Study what real ocean has done over time period - Model parameter estimation - Systematic model improvement

3 What is data assimilation? Bayes theorem: The solution is a pdf!

4 What are representation errors? Extra uncertainty that arises in data assimilation because model and observations have different representation of reality.

5 How do they arise in DA? Start from Bayes Theorem: Representation differences between model and observations, so representation errors, arise in the likelihood, not in the prior!

6 What does the likelihood mean? Observation equation: However, the observations do appear as in Bayes Theorem, with each possible model state x as given, not the truth x t. Interpretation: is the likelihood of that observation when the truth would be given by x. t

7 The likelihood via observation space Assume high-resolution observations to be assimilated in a (relatively) low-resolution model. We have to calculate: so we need a relation between the two scales.

8 Relation between scales Introduce the high-resolution variable in observation space Now use: is the instrument error is the representation error is the convolution of the two

9 Gaussian errors For Gaussian instrument and representation errors we find: Using we can perform the integration:

10 How can we calculate C o? Use a whole set of observations: And calculate their statistics (e.g. mean and variance) Note that we need the statistics of y given the average over the model grid box:

11 Calculation of C 0 from a time series Only take those times into account when So at the times indicated by the green lines.

12 Warning: Hence one should not take the variance of all observations but the variance of those observations that have an average equal to H(x)!

13 Superobbing Can t we just assimilate the average of the observations so do superobbing? Yes, but the representation error is still there!!!

14 Naïve approach Assume each measurement is a realisation of the box average, in which case However, we will get the same equation if we would use a set of realisations of the full model grid box, so this cannot be correct!

15 Variance of the mean It is crucial to realise that we have to see each observation itself as a realisation of the model box average, so we have to condition this variance on the model state! Assume that each of the observations has variance, conditoned on the model state, of: And the representation error is correlated with correlation coefficient between any two observations.

16 Variance from superobbing Hence we find: So superobbing is NOT free of representation errors. Note that if M increases the representation error remains and becomes the dominant term! (Indeed assimilating the mean or all observations gives the same result as assimilation each separately.)

17 Nonlinear DA: Standard Particle filter Ensemble size has to grow with exp(n y ) to avoid degeneracy.

18 Transportation particle filters Particle flow Before observations Prior After observations Posterior

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sha1_base64="z02amt//0wo0nwuhpxiggldfhok=">aaab7xicbvbnswmxej34wetx1aoxybhqpeykomeif48v7ae0s8mm2ty2myxjvixl/4mxd4p49f9489+ytnvq1gcdj/dmmjkxjoib63nfagv1bx1js7bv3n7z3dsvhrw2juo1zq2qhnltkbgmugqny61g7uqzeoectclrzdrvptjtujl3dpywicydysnoixvsm+qzytnzr1t2qt4mejn4osldjnqv9nxtk5rgtfoqided30tskbftorvsuuymhiwejsiadryvjgymygbxtvcpu/o4utqvthim/p7isgzmoa5dz0zs0cx6u/e/r5pa6criuexsyysdl4psga3c09dxn2tgrrg7qqjm7lzmh0qtal1arrecv/jymmmev32v6t9dlgvxerwfoiytqiapl1cdw6hdayg8wdo8whts6aw9o4956wrkz47gd9dnd/gtjrc=</latexit> Particle flows The particles are propagated in artificial time s via dx ds = f s(x) The question is: How to choose the vector flow field f s (x)? The corresponding pdf of x evolves via the Liouville s = r x (p s (x)f s (x)) with p 0 (x) =p(x) and lim p s(x) =p(x y) s!1 p(x y) Flow field should reduce distance between p s (x) and.

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sha1_base64="z02amt//0wo0nwuhpxiggldfhok=">aaab7xicbvbnswmxej34wetx1aoxybhqpeykomeif48v7ae0s8mm2ty2myxjvixl/4mxd4p49f9489+ytnvq1gcdj/dmmjkxjoib63nfagv1bx1js7bv3n7z3dsvhrw2juo1zq2qhnltkbgmugqny61g7uqzeoectclrzdrvptjtujl3dpywicydysnoixvsm+qzytnzr1t2qt4mejn4osldjnqv9nxtk5rgtfoqided30tskbftorvsuuymhiwejsiadryvjgymygbxtvcpu/o4utqvthim/p7isgzmoa5dz0zs0cx6u/e/r5pa6criuexsyysdl4psga3c09dxn2tgrrg7qqjm7lzmh0qtal1arrecv/jymmmev32v6t9dlgvxerwfoiytqiapl1cdw6hdayg8wdo8whts6aw9o4956wrkz47gd9dnd/gtjrc=</latexit> A distance measure: KL divergence The KL divergence between the present pdf and the posterior is given by: Z ps (x) KL(p s )= p s (x) log dx p(x y) The change in KL due to the flow field f s (x) is given by KL(p s+ ) KL(p s ) dkl(f s )= lim!1 Z h i = p s (x) (r x log p(x y)) T f s (x)+r x f s (x) dx We need to find the flow field that maximizes this change: Complex optimization problem for the flow field

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sha1_base64="i1ijibp2nefqamkf9dbbll5f/iy=">aaab6nicdvdlssnafl2pr1pfvzdubovgkislteuu6mzlrfuanptjdnionuzczeqooz/gxouibv0id/6n04egogcuhm65l3vvcrlolhacdyu3tr6xuzxfluzs7u0ffa+p2ipojaetevnydgoskgectjttnhytsxeucnojjldzv3nppwkxunpthporhgkwmok1kw7dgrous47tlsuev0eoxatwxofcemervrwqcm1ngrks0bwu3/vdmkqrfzpwrftpdrltz1hqrjidffqpogkmezyipumfjqjys8wpm3rmlceky2lkalrqv09kofjqggwmm8j6rh57c/evr5fqso5ntcsppoisf4uprzpg87/rkelknj8agolk5lzexlhiok06brpc16fof9iu267hn5vs43ivrx5o4btowyuanoaamtacain4gcd4trj1al1yr8vwnlwaoyyfsn4+acoujh0=</latexit> <latexit sha1_base64="i1ijibp2nefqamkf9dbbll5f/iy=">aaab6nicdvdlssnafl2pr1pfvzdubovgkislteuu6mzlrfuanptjdnionuzczeqooz/gxouibv0id/6n04egogcuhm65l3vvcrlolhacdyu3tr6xuzxfluzs7u0ffa+p2ipojaetevnydgoskgectjttnhytsxeucnojjldzv3nppwkxunpthporhgkwmok1kw7dgrous47tlsuev0eoxatwxofcemervrwqcm1ngrks0bwu3/vdmkqrfzpwrftpdrltz1hqrjidffqpogkmezyipumfjqjys8wpm3rmlceky2lkalrqv09kofjqggwmm8j6rh57c/evr5fqso5ntcsppoisf4uprzpg87/rkelknj8agolk5lzexlhiok06brpc16fof9iu267hn5vs43ivrx5o4btowyuanoaamtacain4gcd4trj1al1yr8vwnlwaoyyfsn4+acoujh0=</latexit> The change in KL due to the flow field dkl f s rkl dkl = hrkl,f s i

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sha1_base64="za4jnyibesdv/jnrkrvuco8ppio=">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</latexit> <latexit sha1_base64="za4jnyibesdv/jnrkrvuco8ppio=">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</latexit> Kernel embedding Recall that the change in KL is given by: Z h i dkl(x) = p s (x) (r x log p(x y)) T f s (x)+r x f s (x) dx And we have dkl = hrkl,f s i. <latexit sha1_base64="llzpgijeqnvpgeunv13+dl+wby0=">aaacg3icbvbns8naen3u7/ov9ehlsqgepcrf0itq9cloqcgq0jqy2u7apztn2n0ijfr/epgvepggicfbg//gbzudwh/m8hhvht15ysq4np735zrmzufmfxaxyssrq2vr7sbmju4yxbdbepgouxa0ci6xybgrejcqhdguebv2t0f+7t0qzrn5bqyptmlosh5xbszkbbfwob+gxzqqgbnbqxyf0kbckicex+ztqk1pohi3z101cdtuxat6y9bp4hekqgpctt2pojowlezpmactm76xmlyoynamcfgomo0psd50swmphbh1kx/fnqs7vunqkfg2pkfj9edgdrhwgzi0kzgynv7rjct/vgzmoqnwzmwagzrs8lcucwosogqkdrhczstaemck279s1gmfzng4yzye/+/j0+smvvw9qn91ukmffheskm2yq/aitw5jnzyrs9igjdyqj/jcxp1h59l5c94noywn2nkiv+b8fgpyfp9x</latexit> Instead of solving the optimization problem directly we embed the flow in a Reproducing Kernel Hilbert Space, so f s (x) =hk(x, ),f s ( )i F Using this kernel embedding we find: rkl = Z p s (x)[k(x 0,x)r x log p(x 0 y)+r x K(x 0,x)] dx 0

23 <latexit sha1_base64="tceabf9bk3hpcpagahpg4qxniey=">aaacfhicbvc7sgnbfj31bxxflw0gg6aiyvcelsycnkjsrdcjka1hdni3gzydxwbuimhnr9j4kzywitha2pk3th6frwmdh3po5c49qskfqdf9dkamz2bn5hcwc0vlk6tr+fwnuoltzahgyxnrq4azkejbdqvkueo0sciq0aiuz4z+4wa0ebg6xh4cryh1lqgfz2ildn7flxdihfuvcysj5qrdvd2jvhbdht6dub8si6qnzuxkoj0vuev3bpqxebnsibnu2/kpvxpznakfxdjjmp6byctjggwxmmj5qyge8wvwhaalikvgwtnoqahdsuqhhrg2tyedqd8nmhyz048cm4wy9sxvbyj+5zvtdi9bmvbjiqd4efgysooxhtzeo0idr9m3hhet7f8p7zhnonoec7ye7/fjf0n9ooi5re/isfa6ndsxqlbintklhjkijxjoqqrgolknj+szvdgpzppz6ryno1pozgat/idz/gu1s52s</latexit> <latexit sha1_base64="tceabf9bk3hpcpagahpg4qxniey=">aaacfhicbvc7sgnbfj31bxxflw0gg6aiyvcelsycnkjsrdcjka1hdni3gzydxwbuimhnr9j4kzywitha2pk3th6frwmdh3po5c49qskfqdf9dkamz2bn5hcwc0vlk6tr+fwnuoltzahgyxnrq4azkejbdqvkueo0sciq0aiuz4z+4wa0ebg6xh4cryh1lqgfz2ildn7flxdihfuvcysj5qrdvd2jvhbdht6dub8si6qnzuxkoj0vuev3bpqxebnsibnu2/kpvxpznakfxdjjmp6byctjggwxmmj5qyge8wvwhaalikvgwtnoqahdsuqhhrg2tyedqd8nmhyz048cm4wy9sxvbyj+5zvtdi9bmvbjiqd4efgysooxhtzeo0idr9m3hhet7f8p7zhnonoec7ye7/fjf0n9ooi5re/isfa6ndsxqlbintklhjkijxjoqqrgolknj+szvdgpzppz6ryno1pozgat/idz/gu1s52s</latexit> <latexit sha1_base64="tceabf9bk3hpcpagahpg4qxniey=">aaacfhicbvc7sgnbfj31bxxflw0gg6aiyvcelsycnkjsrdcjka1hdni3gzydxwbuimhnr9j4kzywitha2pk3th6frwmdh3po5c49qskfqdf9dkamz2bn5hcwc0vlk6tr+fwnuoltzahgyxnrq4azkejbdqvkueo0sciq0aiuz4z+4wa0ebg6xh4cryh1lqgfz2ildn7flxdihfuvcysj5qrdvd2jvhbdht6dub8si6qnzuxkoj0vuev3bpqxebnsibnu2/kpvxpznakfxdjjmp6byctjggwxmmj5qyge8wvwhaalikvgwtnoqahdsuqhhrg2tyedqd8nmhyz048cm4wy9sxvbyj+5zvtdi9bmvbjiqd4efgysooxhtzeo0idr9m3hhet7f8p7zhnonoec7ye7/fjf0n9ooi5re/isfa6ndsxqlbintklhjkijxjoqqrgolknj+szvdgpzppz6ryno1pozgat/idz/gu1s52s</latexit> <latexit sha1_base64="tceabf9bk3hpcpagahpg4qxniey=">aaacfhicbvc7sgnbfj31bxxflw0gg6aiyvcelsycnkjsrdcjka1hdni3gzydxwbuimhnr9j4kzywitha2pk3th6frwmdh3po5c49qskfqdf9dkamz2bn5hcwc0vlk6tr+fwnuoltzahgyxnrq4azkejbdqvkueo0sciq0aiuz4z+4wa0ebg6xh4cryh1lqgfz2ildn7flxdihfuvcysj5qrdvd2jvhbdht6dub8si6qnzuxkoj0vuev3bpqxebnsibnu2/kpvxpznakfxdjjmp6byctjggwxmmj5qyge8wvwhaalikvgwtnoqahdsuqhhrg2tyedqd8nmhyz048cm4wy9sxvbyj+5zvtdi9bmvbjiqd4efgysooxhtzeo0idr9m3hhet7f8p7zhnonoec7ye7/fjf0n9ooi5re/isfa6ndsxqlbintklhjkijxjoqqrgolknj+szvdgpzppz6ryno1pozgat/idz/gu1s52s</latexit> <latexit sha1_base64="ob0h5k/ce6sl83s/jkvf07aufxa=">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</latexit> <latexit sha1_base64="ob0h5k/ce6sl83s/jkvf07aufxa=">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</latexit> <latexit sha1_base64="ob0h5k/ce6sl83s/jkvf07aufxa=">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</latexit> <latexit sha1_base64="ob0h5k/ce6sl83s/jkvf07aufxa=">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</latexit> Particle implementation The evolution equation for each particle is now dx i ds = f s(x i )= rkl(x i ) This leads to an iterative scheme x (j) (j 1) i = x i rkl x (j 1) i until rkl(x) < KL

24 <latexit sha1_base64="4cto8wfpv3dtwgwasygp3qeyxxw=">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</latexit> <latexit sha1_base64="4cto8wfpv3dtwgwasygp3qeyxxw=">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</latexit> <latexit sha1_base64="4cto8wfpv3dtwgwasygp3qeyxxw=">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</latexit> <latexit sha1_base64="4cto8wfpv3dtwgwasygp3qeyxxw=">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</latexit> Convergence LU, Lu and Nolan (2018) proof that: 1. The large N limit converges t p s = r [p s (rhk, p s i hk, p s r log p(x y)i)] 2. This PDE is well-posed and has unique solution 3. This solution converges to posterior pdf for large s limit.

25 Illustration 1-D case

26 Particle Flow in 1-D case

27 Lorenz96 model 1000-dimensional N=20 All variables observed every 6 hour.

28 Conclusions Great care must be taken in comparing observations and models. Data assimilation merges information from observations and models in most consistent way The ocean data-assimilation problem can be highly nonlinear. Particle Flows transport samples from the prior to samples from the posterior The iterative mapping PF is robust in low- to intermediatedimensional systems. We are working on the real case! Cost equivalent to 3DVar on each particle. Model error covariance Q is essential.