Analysis of the covariance matrix in FWI through density of covariance maps

  • Anyeres Neider Jimenez Universidad de Antioquia, Colombia
  • Juan Carlos Muñoz Cuartas Universidad de Antioquia, FACoM, Colombia
  • Sheryl Avendaño Universidad de Antioquia, Colombia
  • Leonardo Gómez Bernal Instituto de Geociȇncias da Universidade Federal da Bahia
Keywords: Full Waveform Inversion, Resolution, Correlation Matrix


Full waveform inversion (FWI) is a tool for the inversion of seismic data. There are several sources of uncertainty in the results provided by FWI. The quantification of such uncertainties has been studied through the resolution matrix (Res), which rests on a quadratic approximation that interprets the Hessian matrix as the posterior covariance matrix. Despite efforts in the use of Res, there is no published analysis of the uncertainties contained in the full correlation matrix, (R). Our approach leads to build the full R matrix, which, at the end of the day, is the final quantity that includes all the information associated with uncertainties.
We focused on uncertainties related to variation in the starting models of the FWI, and thus propose a method to study the full R matrix, which is-called the Density of Correlation Map, D. By using the D map, we found that the highest uncertainty zones in the FWI inverted model are near the sources, the model boundaries, and the interfaces. We argue that D can be a complement for the study and estimation of uncertainties in FWI.

Author Biography

Anyeres Neider Jimenez, Universidad de Antioquia, Colombia

Academics and employments

  1.  2007. Physicist, University of Antioquia. Monography: "Discos rotantes en relatividad General", Advisor: M.Sc. Alonso Sepúlveda Soto.
  2.  2011. M.Sc., University of Antioquia, Thesis: "Doubly special relativity theory by conformal transformations", Advisor: PhD Daniel Jaramillo.
  3.  2012. Temporary lecturer at University of Antioquia and Metropolitan Technological Institute. From 2007 to 2012.
  4.  2014. Academic Coordinator (Profesor Ocasional) of the Ocean Engineering Program, University of antioquia. Engineering Faculty. From 2012 to 2014.
  5.  2017. Internship: Universidade federal da Bahia, Bahia-Brasil. Centro de Pesquisa em Geofísica e Geologia da UFBA. Advisor: PhD Reynam Pestana. 
  6.  2018. PhD Thesis accepted, at University of Antioquia, “Full waveform inversion and uncertainties quantification”. 

International Events:

a) International School in physics education, Rionegro Antioquia, 2007.

b) Workshop on Physics and Technology at CERN, Bogotá Colombia. 2010-10-20, poster: "Conformal Transformations for Modified Lorentz symmetry of DSRMS Model”

c) Joint ICTP-Twas School on coherent state transforms, time-frequency and time-scale analysis, applications. June 2nd-20th 2014. ICTP, Trieste, Italy.

d) Educational tour, 2014, Paris. Professor: Tariq Alkhalifah.

Applied Courses

a) Geophysical inverse problems, professor: Mrinal Kanti Sen, april 2015.

b) Métodos de migración RTM y WWE 3D, January 2015, professor: Reynam da Cruz Pestana.


a) 2012. Non Linear Lorentz Transformation and Doubly Special Relativity,

b) PNUD, “Employment Observatory, 2013”, Urabá: Región invencible. “Statistical study of employment in Antioquia Colombia”. Red de Observatorios Regionales del Mercado de Trabajo - Red ORMET Programa de las Naciones Unidas para el Desarrollo – PNUD. ISBN: 978-958-8815-09-1.

c) Creation of educational content for virtual education, “Guía de estudio para Aula Semilla, Ude@”, faculty of engineering, university of Antioquia. Mechanical physics.

d) Anyeres A. Jiménez, J. C. Muñoz-Cuartas, Sheryl Avendaño. “Integral modelling of propagation of incident waves in a laterally varying medium: An exploration in the frequency domain”,


Biswas R, Sen M (2017) 2d full-waveform inversion and uncertainty estimation using the reversible jump Hamiltonian Montecarlo. SEG Technical Program Expanded Abstracts, DOI

Lailly P (1983) The seismic inverse problem as a sequence of before stack migrations. In: Bednar JB, Redner R, Robinson E, Weglein A (eds) Conference on inverse scattering: theory and application, SIAM, pp 206-220

Tarantola A (1984) Inversion of seismic reflection data in the acoustic approximation. Geophysics, 49(8):1259-1266

Brossier R, Operto S, Virieux J (2009) Seismic imaging of complex structures by 2d elastic frequency-domain full-waveform inversion. Geophysics, DOI

Brenders A, Albertin U, Mika J (2012) Comparison of 3d time- and frequency-domain waveform inversion: Benefits and insights of a broadband, discrete-frequency strategy. SEG Technical Program Expanded Abstracts, DOI https: //

Vigh D, Starr B, Kapoor J, Li H (2010) 3d full waveform inversion on a gulf of Mexico waz data set. SEG Technical Program Expanded Abstracts, DOI

Fitchner A, Trampert J (2011) Resolution analysis in full waveform inversion. Geophys J Int, DOI

Nocedal J (1980) Updating quasi-Newton matrices with limited storage. Mathematics of Computation, 35(151):773-782

Liu D, Nocedal J (1989) On the limited memory BFGS method for large scale optimization: Mathematical Programming. Springer, DOI

Brossier R, Operto S, Virieux J (2010) Which data residual norm for robust elastic frequency-domain full waveform inversion? Geophysics, DOI

Virieux J, Operto S (2009) An overview of full-waveform inversion in exploration geophysics. Geophysics, DOI

Boonyasiriwat C, Valasek P, Routh P, Cao W, Schuster G, Macy B (2009) An efficient multiscale method for time- domain waveform tomography. Geophysics, DOI

Kennett B, Nolet G (1978) Resolution analysis for discrete systems. Geophys J R astr Soc, 53:413-425

Williamson P (1991) A guide to the limits of resolution imposed by scattering in ray tomography. Geophysics, DOI

Meles G, Greenhalgh S, Maurer H, Green A, van der Kruk J (2012) Gpr full-waveform sensitivity and resolution analysis using an fdtd adjoint method. IEEE Transactions on Geoscience and Remote Sensing, DOI 10.1109/IWAGPR.2011.5963863

Tao Y, Sen M (2013) Frequency-domain full wave inversion with scattering-intregral approach and its sensivity analysis. Journal of Geophysics and Engineering, DOI 10.1088/1742-2132/10/6/065008

Abreo, S. A., Ramírez- Silva, A. B., & Reyes- Torres, O. M. (2018). A GPU implementation of the second order adjoint state theory to quantify the uncertainty on FWI results. CT&F - Ciencia, Tecnología Y Futuro, 8(2).

Tarantola A (2005) Inversion Problem Theory and methods for model parameter estimation. SIAM

dos Santos A, Pestana R (2015) Time-domain multiscale fullwaveform inversion using the rapid expansion method and efficient step-length estimation. Geophysics, DOI https: //

Plessix RE (2006) A review of the adjoint-state method for computing the gradient of a functional with geophysical applications. Geophysical Journal International, DOI

Nocedal J, Wright SJ (2006) Numerical optimization, 2edn. Springer

Ma Y, Hale D (2012) Quasi-newton full-waveform inversion with a projected hessian matrix. Geophysics, DOI https: //

Leeuwen TV, Mulder WA (2010) A correlation-based misfit criterion for wave-equation traveltime tomography. Geophysical Journal International, DOI

Madagascar Development Team (2012) Madagascar Software, Version~1.4.

How to Cite
Jimenez, A. N., Muñoz Cuartas, J. C., Avendaño, S. ., & Gómez Bernal, L. (2020). Analysis of the covariance matrix in FWI through density of covariance maps. CT&F - Ciencia, Tecnología Y Futuro, 10(1), 93–105.


Download data is not yet available.
Scientific and Technological Research Articles


Crossref Cited-by logo
QR Code

Some similar items: