A Problem in the Application of Inverse Methods to Tracer Data [and Discussion]
Attempts to apply inverse methods to the interpretation of tracer data usually seek some least squares solution of the flux divergence equations AC = S, where A is a matrix of transport coefficients, C a vector of concentrations and S a vector of sources/sinks. However, what is often really required...
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| Veröffentlicht in: | Philosophical transactions of the Royal Society of London. Series A: Mathematical and physical sciences 1988-05, Vol.325 (1583), p.85-91 |
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| container_title | Philosophical transactions of the Royal Society of London. Series A: Mathematical and physical sciences |
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| creator | Shepherd, J. G. Jenkins, W. J. Wunsch, C. J.-F. Minster |
| description | Attempts to apply inverse methods to the interpretation of tracer data usually seek some least squares solution of the flux
divergence equations AC = S, where A is a matrix of transport coefficients, C a vector of concentrations and S a vector of
sources/sinks. However, what is often really required is a set of values for the elements of A, which will give a satisfactory
prediction of the concentrations. This corresponds to finding a least squares solution of C = A$^{-1}$ S. The two problems
are not equivalent. The latter corresponds to an extensively reweighted version of the former, where the weights depend on
the solution (the elements of A). The former is linear in the elements of A: the latter is highly nonlinear. In addition,
the matrix A is invariably sparse, and required to be so. However, A$^{-1}$ is not, nor is it guaranteed that its inverse
will be if its elements are determined freely. It is not clear whether the standard methods of generalized inverse theory
are applicable to the more difficult `real' problem nor, if they are not, what other methods might be used. It is, however,
possible that solutions of the `real' problem, if they can be found, would be more informative. |
| doi_str_mv | 10.1098/rsta.1988.0044 |
| format | Article |
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divergence equations AC = S, where A is a matrix of transport coefficients, C a vector of concentrations and S a vector of
sources/sinks. However, what is often really required is a set of values for the elements of A, which will give a satisfactory
prediction of the concentrations. This corresponds to finding a least squares solution of C = A$^{-1}$ S. The two problems
are not equivalent. The latter corresponds to an extensively reweighted version of the former, where the weights depend on
the solution (the elements of A). The former is linear in the elements of A: the latter is highly nonlinear. In addition,
the matrix A is invariably sparse, and required to be so. However, A$^{-1}$ is not, nor is it guaranteed that its inverse
will be if its elements are determined freely. It is not clear whether the standard methods of generalized inverse theory
are applicable to the more difficult `real' problem nor, if they are not, what other methods might be used. It is, however,
possible that solutions of the `real' problem, if they can be found, would be more informative.</description><identifier>ISSN: 1364-503X</identifier><identifier>ISSN: 0080-4614</identifier><identifier>EISSN: 1471-2962</identifier><identifier>EISSN: 2054-0272</identifier><identifier>DOI: 10.1098/rsta.1988.0044</identifier><identifier>CODEN: PTRMAD</identifier><language>eng</language><publisher>London: The Royal Society</publisher><subject>Earth, ocean, space ; Error rates ; Exact sciences and technology ; External geophysics ; Inverse problems ; Least squares ; Linear equations ; Linear regression ; Matrices ; Nonlinearity ; Oceans ; Physics of the oceans ; Thermohaline structure and circulation. Turbulence and diffusion ; Transport phenomena ; Water conservation</subject><ispartof>Philosophical transactions of the Royal Society of London. Series A: Mathematical and physical sciences, 1988-05, Vol.325 (1583), p.85-91</ispartof><rights>Copyright 1988 The Royal Society</rights><rights>Scanned images copyright © 2017, Royal Society</rights><rights>1988 INIST-CNRS</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><cites>FETCH-LOGICAL-c419t-bb376a1bc565c0d4675b4d2c677283cb048d9808696605fc16d56dc5681049e03</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://www.jstor.org/stable/pdf/38102$$EPDF$$P50$$Gjstor$$H</linktopdf><linktohtml>$$Uhttps://www.jstor.org/stable/38102$$EHTML$$P50$$Gjstor$$H</linktohtml><link.rule.ids>309,310,314,776,780,785,786,799,828,23910,23911,25119,27903,27904,57995,57999,58228,58232</link.rule.ids><backlink>$$Uhttp://pascal-francis.inist.fr/vibad/index.php?action=getRecordDetail&idt=7805014$$DView record in Pascal Francis$$Hfree_for_read</backlink></links><search><creatorcontrib>Shepherd, J. G.</creatorcontrib><creatorcontrib>Jenkins, W. J.</creatorcontrib><creatorcontrib>Wunsch, C.</creatorcontrib><creatorcontrib>J.-F. Minster</creatorcontrib><title>A Problem in the Application of Inverse Methods to Tracer Data [and Discussion]</title><title>Philosophical transactions of the Royal Society of London. Series A: Mathematical and physical sciences</title><addtitle>Phil. Trans. R. Soc. Lond. A</addtitle><description>Attempts to apply inverse methods to the interpretation of tracer data usually seek some least squares solution of the flux
divergence equations AC = S, where A is a matrix of transport coefficients, C a vector of concentrations and S a vector of
sources/sinks. However, what is often really required is a set of values for the elements of A, which will give a satisfactory
prediction of the concentrations. This corresponds to finding a least squares solution of C = A$^{-1}$ S. The two problems
are not equivalent. The latter corresponds to an extensively reweighted version of the former, where the weights depend on
the solution (the elements of A). The former is linear in the elements of A: the latter is highly nonlinear. In addition,
the matrix A is invariably sparse, and required to be so. However, A$^{-1}$ is not, nor is it guaranteed that its inverse
will be if its elements are determined freely. It is not clear whether the standard methods of generalized inverse theory
are applicable to the more difficult `real' problem nor, if they are not, what other methods might be used. It is, however,
possible that solutions of the `real' problem, if they can be found, would be more informative.</description><subject>Earth, ocean, space</subject><subject>Error rates</subject><subject>Exact sciences and technology</subject><subject>External geophysics</subject><subject>Inverse problems</subject><subject>Least squares</subject><subject>Linear equations</subject><subject>Linear regression</subject><subject>Matrices</subject><subject>Nonlinearity</subject><subject>Oceans</subject><subject>Physics of the oceans</subject><subject>Thermohaline structure and circulation. Turbulence and diffusion</subject><subject>Transport phenomena</subject><subject>Water conservation</subject><issn>1364-503X</issn><issn>0080-4614</issn><issn>1471-2962</issn><issn>2054-0272</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>1988</creationdate><recordtype>article</recordtype><recordid>eNp9kEtr3DAUhU1oIWnabRZZadGtJ1fWw_KmMOTRBFJSkikUShGyLHc0OJaRNAnTXx_ZLqGhNCtJ3HO-c3Wy7AjDAkMlTnyIaoErIRYAlO5lB5iWOC8qXrxJd8JpzoB838_ehbABwJiz4iC7WaKv3tWduUe2R3Ft0HIYOqtVtK5HrkVX_YPxwaAvJq5dE1B0aOWVNh6dqajQD9U36MwGvQ0hOX6-z962qgvmw5_zMPt2cb46vcyvbz5fnS6vc01xFfO6JiVXuNaMMw0N5SWraVNoXpaFILoGKppKgOAV58BajXnDeJPUAgOtDJDDbDFztXcheNPKwdt75XcSgxzrkGMdcqxDjnUkw8fZMKigVdd61Wsbnl2lAAZ4lJFZ5t0u7e-0NXEnN27r-_T8Pzy85rq9Wy1xRegDKZjFTBAJgmAoMcdC_rbDhBsFMgmkDWFr5CR7GfNv6vGcugnR-eevkNRRkYaf5uHa_lo_Wm_ki90mlHZ9NH2cUqc8wWS77To5NG0CnLwKcLshIf6ykicPBcXr</recordid><startdate>19880525</startdate><enddate>19880525</enddate><creator>Shepherd, J. G.</creator><creator>Jenkins, W. J.</creator><creator>Wunsch, C.</creator><creator>J.-F. Minster</creator><general>The Royal Society</general><general>Royal Society of London</general><scope>IQODW</scope><scope>AAYXX</scope><scope>CITATION</scope></search><sort><creationdate>19880525</creationdate><title>A Problem in the Application of Inverse Methods to Tracer Data [and Discussion]</title><author>Shepherd, J. G. ; Jenkins, W. J. ; Wunsch, C. ; J.-F. Minster</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c419t-bb376a1bc565c0d4675b4d2c677283cb048d9808696605fc16d56dc5681049e03</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>1988</creationdate><topic>Earth, ocean, space</topic><topic>Error rates</topic><topic>Exact sciences and technology</topic><topic>External geophysics</topic><topic>Inverse problems</topic><topic>Least squares</topic><topic>Linear equations</topic><topic>Linear regression</topic><topic>Matrices</topic><topic>Nonlinearity</topic><topic>Oceans</topic><topic>Physics of the oceans</topic><topic>Thermohaline structure and circulation. Turbulence and diffusion</topic><topic>Transport phenomena</topic><topic>Water conservation</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Shepherd, J. G.</creatorcontrib><creatorcontrib>Jenkins, W. J.</creatorcontrib><creatorcontrib>Wunsch, C.</creatorcontrib><creatorcontrib>J.-F. Minster</creatorcontrib><collection>Pascal-Francis</collection><collection>CrossRef</collection><jtitle>Philosophical transactions of the Royal Society of London. Series A: Mathematical and physical sciences</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Shepherd, J. G.</au><au>Jenkins, W. J.</au><au>Wunsch, C.</au><au>J.-F. Minster</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>A Problem in the Application of Inverse Methods to Tracer Data [and Discussion]</atitle><jtitle>Philosophical transactions of the Royal Society of London. Series A: Mathematical and physical sciences</jtitle><stitle>Phil. Trans. R. Soc. Lond. A</stitle><date>1988-05-25</date><risdate>1988</risdate><volume>325</volume><issue>1583</issue><spage>85</spage><epage>91</epage><pages>85-91</pages><issn>1364-503X</issn><issn>0080-4614</issn><eissn>1471-2962</eissn><eissn>2054-0272</eissn><coden>PTRMAD</coden><abstract>Attempts to apply inverse methods to the interpretation of tracer data usually seek some least squares solution of the flux
divergence equations AC = S, where A is a matrix of transport coefficients, C a vector of concentrations and S a vector of
sources/sinks. However, what is often really required is a set of values for the elements of A, which will give a satisfactory
prediction of the concentrations. This corresponds to finding a least squares solution of C = A$^{-1}$ S. The two problems
are not equivalent. The latter corresponds to an extensively reweighted version of the former, where the weights depend on
the solution (the elements of A). The former is linear in the elements of A: the latter is highly nonlinear. In addition,
the matrix A is invariably sparse, and required to be so. However, A$^{-1}$ is not, nor is it guaranteed that its inverse
will be if its elements are determined freely. It is not clear whether the standard methods of generalized inverse theory
are applicable to the more difficult `real' problem nor, if they are not, what other methods might be used. It is, however,
possible that solutions of the `real' problem, if they can be found, would be more informative.</abstract><cop>London</cop><pub>The Royal Society</pub><doi>10.1098/rsta.1988.0044</doi><tpages>7</tpages></addata></record> |
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| source | JSTOR Mathematics & Statistics; Jstor Complete Legacy |
| subjects | Earth, ocean, space Error rates Exact sciences and technology External geophysics Inverse problems Least squares Linear equations Linear regression Matrices Nonlinearity Oceans Physics of the oceans Thermohaline structure and circulation. Turbulence and diffusion Transport phenomena Water conservation |
| title | A Problem in the Application of Inverse Methods to Tracer Data [and Discussion] |
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