<?xml version="1.0" encoding="ISO-8859-1"?><article xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance">
<front>
<journal-meta>
<journal-id>0122-5383</journal-id>
<journal-title><![CDATA[CT&F - Ciencia, Tecnología y Futuro]]></journal-title>
<abbrev-journal-title><![CDATA[C.T.F Cienc. Tecnol. Futuro]]></abbrev-journal-title>
<issn>0122-5383</issn>
<publisher>
<publisher-name><![CDATA[Instituto Colombiano del Petróleo (ICP) - ECOPETROL S.A.]]></publisher-name>
</publisher>
</journal-meta>
<article-meta>
<article-id>S0122-53832018000200099</article-id>
<article-id pub-id-type="doi">10.29047/01225383.86</article-id>
<title-group>
<article-title xml:lang="en"><![CDATA[A GPU IMPLEMENTATION OF THE SECOND ORDER ADJOINT STATE THEORY TO QUANTIFY THE UNCERTAINTY ON FWI]]></article-title>
<article-title xml:lang="es"><![CDATA[RESULTSUNA IMPLEMENTACIÓN EN GPU DE LA TEORÍA DEL ESTADO ADJUNTO DE SEGUNDO ORDEN PARA LA CUANTIFICACIÓN DE LA INCERTIDUMBRE EN RESULTADOS DE FWI]]></article-title>
</title-group>
<contrib-group>
<contrib contrib-type="author">
<name>
<surname><![CDATA[Abreo]]></surname>
<given-names><![CDATA[Sergio-Alberto]]></given-names>
</name>
<xref ref-type="aff" rid="Aff"/>
</contrib>
<contrib contrib-type="author">
<name>
<surname><![CDATA[Ramírez]]></surname>
<given-names><![CDATA[Ana-B]]></given-names>
</name>
<xref ref-type="aff" rid="Aff"/>
</contrib>
<contrib contrib-type="author">
<name>
<surname><![CDATA[Reyes]]></surname>
<given-names><![CDATA[Oscar-Mauricio]]></given-names>
</name>
<xref ref-type="aff" rid="Aff"/>
</contrib>
</contrib-group>
<aff id="Af1">
<institution><![CDATA[,Universidad Industrial de Santander  ]]></institution>
<addr-line><![CDATA[Bucaramanga ]]></addr-line>
<country>Colombia</country>
</aff>
<pub-date pub-type="pub">
<day>00</day>
<month>12</month>
<year>2018</year>
</pub-date>
<pub-date pub-type="epub">
<day>00</day>
<month>12</month>
<year>2018</year>
</pub-date>
<volume>8</volume>
<numero>2</numero>
<fpage>99</fpage>
<lpage>111</lpage>
<copyright-statement/>
<copyright-year/>
<self-uri xlink:href="http://www.scielo.org.co/scielo.php?script=sci_arttext&amp;pid=S0122-53832018000200099&amp;lng=en&amp;nrm=iso"></self-uri><self-uri xlink:href="http://www.scielo.org.co/scielo.php?script=sci_abstract&amp;pid=S0122-53832018000200099&amp;lng=en&amp;nrm=iso"></self-uri><self-uri xlink:href="http://www.scielo.org.co/scielo.php?script=sci_pdf&amp;pid=S0122-53832018000200099&amp;lng=en&amp;nrm=iso"></self-uri><abstract abstract-type="short" xml:lang="en"><p><![CDATA[ABSTRACT The second order scattering information provided by the Hessian matrix and its inverse plays an important role in both, parametric inversion and uncertainty quantification. On the one hand, for parameter inversion, the Hessian guides the descent direction such that the cost function minimum is reached with less iterations. On the other hand, it provides a posteriori information of the probability distribution of the parameters obtained after full waveform inversion, as a function of the a priori probability distribution information. Nevertheless, the computational cost of the Hessian matrix represents the main obstacle in the state-of-the-art for practical use of this matrix from synthetic or real data. The second order adjoint state theory provides a strategy to compute the exact Hessian matrix, reducing its computational cost, because every column of the matrix can be obtained by performing two forward and two backward propagations. In this paper, we first describe an approach to compute the exact Hessian matrix for the acoustic wave equation with constant density. We then provide an analysis of the use of the Hessian matrix for uncertainty quantification of the full waveform inversion of the velocity model for a synthetic example, using the 2D acoustic and isotropic wave equation operator in time.]]></p></abstract>
<abstract abstract-type="short" xml:lang="es"><p><![CDATA[RESUMEN La información de dispersión de segundo orden proporcionada por la matriz Hessiana y su inversa juegan un papel importante en la inversión paramétrica y en la cuantificación de la incertidumbre. Para la inversión de parámetros, el Hessiano guía la dirección de descenso de manera que se alcanza el mínimo de la función de costo en un menor número de iteraciones. Por otro lado, proporciona información a posteriori de la distribución de probabilidad de los parámetros obtenidos luego de usar la inversión de onda completa, como una función de la distribución de probabilidad a priori. Sin embargo, el costo computacional de la matriz Hessiana representa el principal obstáculo de este método para su uso práctico sobre datos sintéticos o datos reales. La teoría del estado adjunto de segundo orden proporciona una estrategia para calcular la matriz Hessian exacta, reduciendo su costo computacional, ya que cada columna de la matriz se puede obtener realizando dos propagaciones hacia adelante y dos hacia atrás. En este artículo, primero mostramos una metodología para calcular la matriz Hessiana exacta usando la ecuación de onda acústica con densidad constante. Luego, proporcionamos un análisis del uso de la matriz Hessiana para la cuantificación de la incertidumbre de la inversión de onda completa en un ejemplo sintético, utilizando como operador la ecuación de onda acústica 2D, isotrópa con densidad constante en el dominio del tiempo.]]></p></abstract>
<kwd-group>
<kwd lng="en"><![CDATA[Inverse theory]]></kwd>
<kwd lng="en"><![CDATA[Waveform inversion]]></kwd>
<kwd lng="en"><![CDATA[Numerical modelling]]></kwd>
<kwd lng="es"><![CDATA[Teoría inversa]]></kwd>
<kwd lng="es"><![CDATA[Inversión de onda]]></kwd>
<kwd lng="es"><![CDATA[Modelado numérico]]></kwd>
</kwd-group>
</article-meta>
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