Seismic Reservoir Modeling. Dario Grana

Читать онлайн книгу.

Seismic Reservoir Modeling - Dario Grana


Скачать книгу
with associated matrix F; and (iii) the measurement errors ε are Gaussian bold-italic epsilon tilde script upper N left-parenthesis bold-italic epsilon semicolon bold 0 comma bold sigma-summation Underscript epsilon Endscripts right-parenthesis, with 0 mean and covariance matrix ε, and they are independent of m; then, the posterior distribution md is also Gaussian bold-italic m bar bold-italic d tilde script upper N left-parenthesis bold-italic m semicolon bold-italic mu Subscript m bar d Baseline comma bold sigma-summation Underscript m bar d Endscripts right-parenthesis with conditional mean μm∣d:

      (1.53)bold-italic mu Subscript m bar d Baseline equals bold-italic mu Subscript m Baseline plus bold sigma-summation Underscript m Endscripts bold upper F Superscript upper T Baseline left-parenthesis bold upper F bold sigma-summation Underscript m Endscripts bold upper F Superscript upper T Baseline plus bold sigma-summation Underscript epsilon Endscripts right-parenthesis Superscript negative 1 Baseline left-parenthesis bold-italic d minus bold upper F bold-italic mu Subscript m Baseline right-parenthesis

      and conditional covariance matrix m∣d:

      (1.54)bold sigma-summation Underscript m bar d Endscripts equals bold sigma-summation Underscript m Endscripts minus bold sigma-summation Underscript m Endscripts bold upper F Superscript upper T Baseline left-parenthesis bold upper F bold sigma-summation Underscript m Endscripts bold upper F Superscript upper T Baseline plus bold sigma-summation Underscript epsilon Endscripts right-parenthesis Superscript negative 1 Baseline bold upper F bold sigma-summation Underscript m Endscripts period

      For the proof, we refer the reader to Tarantola (2005). This result is extensively used in Chapter 5 for seismic inversion problems.

      Example 1.3

      We illustrate the Bayesian approach for linear inverse problems in a geophysical application. We assume that the model variable of interest is S‐wave velocity VS and that a measurement of P‐wave velocity VP is available. The goal of this exercise is to predict the conditional probability of S‐wave velocity given P‐wave velocity.

      We assume that S‐wave velocity is distributed according to a Gaussian distribution script upper N left-parenthesis upper V Subscript upper S Baseline semicolon mu Subscript upper S Baseline comma sigma Subscript upper S Superscript 2 Baseline right-parenthesis with prior mean μS = 2 km/s and prior standard deviation σS = 0.25 km/s (sigma Subscript upper S Superscript 2 Baseline equals 0.0625). We assume that the forward operator linking P‐wave and S‐wave velocity is a linear model of the form:

upper V Subscript upper P Baseline equals 2 upper V Subscript s Baseline plus epsilon period

      If the available measurement of P‐wave velocity is VP = 3.5 km/s, then the posterior distribution of S‐wave velocity given the P‐wave velocity measurement is Gaussian distributed script upper N left-parenthesis upper V Subscript upper S Baseline semicolon mu Subscript upper S bar upper P Baseline comma sigma Subscript upper S bar upper P Superscript 2 Baseline right-parenthesis with mean μS∣P:

mu Subscript upper S bar upper P Baseline equals 2 plus 0.495 times left-parenthesis 3.5 minus 2 times 2 right-parenthesis equals 1.75 k m slash normal s

      and standard deviation σS∣P:

sigma Subscript upper S bar upper P Baseline equals StartRoot 0.0625 minus 0.495 times 2 times 0.0625 EndRoot equals 0.025 k m slash normal s period

      If the available measurement of P‐wave velocity is VP = 4.5 km/s, then the mean μS∣P of the posterior distribution is:

mu Subscript upper S bar upper P Baseline equals 2 plus 0.495 times left-parenthesis 4.5 minus 2 times 2 right-parenthesis equals 2.25 k m slash normal s

      and the standard deviation is σS∣P = 0.025 km/s.

      The posterior standard deviation does not depend on the measurement but only on the prior standard deviation of the model variable and the standard deviation of the error.

      Конец ознакомительного фрагмента.

      Текст предоставлен ООО «ЛитРес».

      Прочитайте эту книгу целиком, купив полную легальную версию на ЛитРес.

      Безопасно оплатить книгу можно банковской картой Visa, MasterCard, Maestro, со счета мобильного телефона, с платежного терминала, в салоне МТС или Связной, через PayPal, WebMoney, Яндекс.Деньги, QIWI Кошелек, бонусными картами или другим удобным Вам способом.

/9j/4AAQSkZJRgABAQEBLAEsAAD/7SFqUGhvdG9zaG9wIDMuMAA4QklNBAQAAAAAAAccAgAAAgAA ADhCSU0EJQAAAAAAEOjxXPMvwRihontnrcVk1bo4QklNBDoAAAAAAOUAAAAQAAAAAQAAAAAAC3By aW50T3V0cHV0AAAABQAAAABQc3RTYm9vbAEAAAAASW50ZWVudW0AAAAASW50ZQAAAABDbHJtAAAA D3ByaW50U2l4dGVlbkJpdGJvb2wAAAAAC3ByaW50ZXJOYW1lVEVYVAAAAAEAAAAAAA9wcmludFBy b29mU2V0dXBPYmpjAAAADABQAHIAbwBvAGYAIABTAGUAdAB1AHAAAAAAAApwcm9vZlNldHVwAAAA AQAAAABCbHRuZW51bQAAAAxidWlsdGluUHJvb2YAAAAJcHJvb2ZDTVlLADhCSU0EOwAAAAACLQAA ABAAAAABAAAAAAAScHJpbnRPdXRwdXRPcHRpb25zAAAAFwAAAABDcHRuYm9vbAAAAAAAQ2xicmJv b2wAAAAAAFJnc01ib29sAAAAAABDcm5DYm9vbAAAAAAAQ250Q2Jvb2wAAAAAAExibHNib29sAAAA AABOZ3R2Ym9vbAAAAAAARW1sRGJvb2wAAAAAAEludHJib29sAAAAAABCY2tnT2JqYwAAAAEAAAAA AABSR0JDAAAAAwAAAABSZCAgZG91YkBv4AAAAAAAAAAAAEdybiBkb3ViQG/gAAAAAAAAAAAAQmwg IGRvdWJAb+AAAAAAAAAAAABCcmRUVW50RiNSbHQAAAAAAAAAAAAAAABCbGQgVW50RiNSbHQAAAAA AAAAAAAAAABSc2x0VW50RiNQeGxAcsAAAAAAAAAAAAp2ZWN0b3JEYXRhYm9vbAEAAAAAUGdQc2Vu dW0AAAAAUGdQcwAAAABQZ1BDAAAAAExlZnRVbnRGI1JsdAAAAAAAAAAAAAAAAFRvcCBVbnRGI1Js dAAAAAAAAAAAAAAAAFNjbCBVbnRGI1ByY0BZAAAAAAAAAAAAEGNyb3BXaGVuUHJpbnRpbmdib29s AAAAAA5jcm9wUmVjdEJvdHRvbWxvbmcAAAAAAAAADGNyb3BSZWN0TGVmdGxvbmcAAAAAAAAADWNy b3BSZWN0UmlnaHRsb25nAAAAAAA

Скачать книгу