Lecture #15 Lecture Notes

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Clifton Parks
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1 Lecture #15 Lecture Notes The ocean water column s very much a 3-D spatal entt and we need to represent that structure n an economcal way to deal wth t n calculatons. We wll dscuss one way to do so, emprcal orthogonal functons, n what follows. A.1. EOF s Let us assume a representaton of the 3D soundspeed feld (though t can actually be ANY ocean feld) by c c( c( x, As we have been dong wth our other lnear nverse problems, let s thnk of ths n matrx form. In partcular, let s consder a real world case where we have done sx CTD profles to get the ocean sourdspeed profle at sx dfferent (x, ponts. Let s also consder 4 vertcal ponts of each profle as gvng a far samplng of the vertcal structure. (We could always use more f the vertcal profle has a lot of structure we ust pck four here as an example.) Then the sx CTD s and four levels let us form a 4x6 matrx, as you can see n my beautful handwrtng below: 1

2 Subtractng the mean soundspeed at each depth level (seen n equaton above) from each measurement, we get the 4x6 soundspeed varance matrx seen above,.e. the c A matrx. Ths s OK, but we really want to work wth square matrces. No problem we ust multply the A matrx by ts transpose, whch s a 6x4 matrx,.e. 2

3 Ths s nce, because the dmenson of the matrx s ust the number of vertcal levels we want to use, generally of order ~10. Often we have many CTD casts, or vertcal profles from towed CTD s lke Scanfsh, to deal wth, so that M>>N n general. The bg gan here s that T AA s a standard egenvalue problem! The (column vectors) belongng to each (the egenvalues whch represent the energy n each mode) are called the Emprcal Orthogonal Functons and wll be seen to be very useful. Let s dscuss ther propertes for a second. Frst, they are the modes of the covarance matrx they express the varablty of c(x, about the mean profle n a modal representaton. Second, they are emprcal because they are data based no theory or model nvolved. And they are orthogonal T because the relaton holds. There are also two other propertes of the EOF s that turn out to be very useful. Frst, they put the most energy possble nto the lowest modes,.e. they produce the reddest modal spectrum possble. Ths s useful because the ocean dynamcal modes also have a red spectrum, so that the EOF s provde a good match to the ocean dynamcs. And second, due to ths red propert one can often truncate the modal sum wth lttle or no harm to the results a nce computatonal savng. Let s look closer at the EOF modal representaton, whch s: c( x, c( N 1 a ( Ths expresson s nce, but have we done anythng better than wrte c(x, n a dfferent way? Um,yeah. We had c(x, only at our measurement stes, and now wth ths representaton, we can fnd c(x, everywhere n a way that s consstent wth the ocean dynamcs. To see ths, let s frst get the a at the measurement stes. c( x, N 1 a ( 3

4 Followng standard procedure, we use the orthogonalty of the EOF s to proect out the ( on the RHS,.e. we multply the equaton above by ( and ntegrate over z. So we have or ( c( x, dz a N 1 a c( x, ( dz ( ( dz We now have the EOF coeffcents at the (x, coordnates of our measurements. Ths s what we need, as we can nterpolate the a n the x-y plane on a mode by mode bass, as seen below! In the lttle cartoon above, we want the 3D oceanography along the source to recever acoustc lne for some acoustcs purpose or at the want spot. But we ust have the a at the measurement ponts n the regon. But, f we nterpolate those coeffcents n 2D (an easy enough exercse, as we wll see) on a mode by mode bass, we can get the feld anywhere we want, e.g. the want spot on the pcture, or along the S/R track. And not only can we do ths ths s perhaps the best, most consstent nterpolaton we can do, as we wll see n the next secton. A.2. Gauss Markov nterpolaton - theory We can nterpolate the EOF coeffcents n (x, va canned routne 2D nterpolators to get a 3D ocean feld, and there s no law to stop us. But there are better and worse ways to do the nterpolaton. In sayng ths, we re not talkng about effcenc but rather respectng the dynamc scales of the medum so that we don t nterpolate entrely uncorrelated ponts! We wll look here at the so-called Gauss Markov 4

5 nterpolator, whch s both effcent, and also takes nto account the scales of the ocean medum. Assume that a measurement (e.g. the a that consttute our measurement at each r=(x, poston) conssts of a true value plus some nose (whch can be computed from the error/nose n our c(x, measurements). So If the error s spatally uncorrelated, and also not correlated to the true value,.e. [ ] where s the error varance [ ] Then (wthout showng the proof) the least squares optmal estmator of gven by at any pont t s [ ] In the above, n s the number of measurements (data ponts) one uses n framng the estmate and the A,B matrces are gven by: [ ] ( ) ( ) where C s the spatal correlaton functon. We see that ths formalsm explctly nects the spatal scale(s) of the ocean medum nto the problem, so that the estmate for the soundspeed (or temperature or whatever) feld has both reasonable modal structure (the EOF modes and ocean modes are smlar) and also the correct correlaton scales for the ocean modes. Moreover, n dong the sums above, one only has to consder the n data ponts that are wthn a correlaton length of the pont we are tryng to nterpolate to. Ths makes the scheme more numercally effcent as well. Often, for ocean work, we can use a smple radal form for the correlaton functon, e.g. a Gaussan, where the wdth of the Gaussan s the radal spatal correlaton length. In consderng ocean eddes (as an example), we can take ths correlaton length to be the Rossby radus of deformaton, whch descrbes how the Corols effect moves the water n a crcular path gven the lattude and speed of the feature. Specfcall ( ) 5

6 phase velocty of the feature of nterest As an example, we can use ~ 50 km at 45 degrees for deep ocean eddes, and set the correlaton length to ths number. Another way to get the correlaton functon, f one has lots of data, s the data-data correlaton functon, whch we defne va ( ) ( ) ( ) ( ) ( ) We wll generally get a more rregularly shaped correlaton functon from the real data, but t wll be a correct one. We also not n passng that for the EOF s (and any other such modal representaton), each mode has ts own correlaton length,.e.. A.3. Gauss Markov nterpolaton some examples Let me post some old calculated examples of usng the Gauss Markov nterpolator n varous crcumstances. The frst example s a Harvard Open Ocean model secton of the Gulf Stream. We look at the soundspeed contours at some depth (whch doesn t so much matter for now), and get the sold lne result from the model. Now, f we remove all the grd ponts of the model except for the ones wth astersks (a drastc decmaton of the data), we want to retreve an nterpolated verson of the Gulf Stream SSP usng a Gauss Markov nterpolaton. Usng the data-data correlaton functon, we were able to get the dashed lne result a very good match to the orgnal 3D feld! 6

7 Another useful example to show s how one can nterpolate through bathymetrc change usng the Gauss Markov nterpolator. By lookng at the average soundspeed at each depth level, whch can have a dfferent number of ponts, and assgnng the water soundpeed n the sedment to be the average of the water column ponts at that level, we get reasonable answers, e.g. 7

8 An example of ths s nterpolatng n the vcnty of a seamount. Ths s a crude old MATLAB result, but shows qute well that the nterpolaton wth and wthout bathymetry s consstent. The only dfference n the results s nsde the seamount, where the result s meanngless anyway! 8

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10 MIT OpenCourseWare Acoustcal Oceanography Sprng 2012 For nformaton about ctng these materals or our Terms of Use, vst:

2x x l. Module 3: Element Properties Lecture 4: Lagrange and Serendipity Elements

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