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The potential for Full Waveform Fracture Inversion

Dr Mark Hildyard (SEE), Prof Andy Hooper (SEE)

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In the past few years there has been a massive explosion in the use of full-waveform inversion and indeed it has become a buzzword in the petroleum industry. A simple web search highlights that all large seismic processing companies now promote their own full waveform inversion techniques as improved methods leading to more accurate velocity models, important for higher resolution and complex geologies. While the techniques and computing power have only recently reached the point where they can be applied more routinely, research has spanned many decades. (See for example Tarantola (1984), or Bunks et al (1995). For a comprehensive review see Vireaux and Operto (2009), and for current capabilities see Warner et al (2013)).

In reality we do not want to simply produce a seismic velocity model but rather seek to know the detailed underlying structures which cause these velocities. A more complex problem then is to invert for the underlying structures causing these velocities. In particular, of great interest to a number of industries is to determine the structure and the mechanical properties of the fracture zones which correspond to some of the anisotropic regions determined in velocity models. This includes the petroleum industry for identifying fractured regions such as ‘sweet spots’, but this is equally important to locating underground repositories (nuclear industry), stability in mining, and to determine fractures in concrete for civil and structural engineering.

For decades, we have also been developing more accurate models of seismic wave interactions with fracture zones (Hildyard et al., 1995; Hildyard, 2001; Hildyard and Young, 2002; Hildyard, 2007). The time is ripe to investigate whether there is potential for applying inversion techniques to the inversion of fractures and fracture zones, in particularly to inversion of detailed mechanical properties. Fracture-related models are most accurately reproduced in fully elastic FD and FE models which are significantly more time and processing consuming than methods widely employed in waveform inversion (e.g. frequency domain methods). Even excluding practical questions on limitations due to computer power requirements and frequency bandwidths required in data, such a thought raises many questions. Is the problem well-posed? Can inversions be unique? Is there enough information in the wave-field? Surely the number of parameters is too large? Initially we should rather pose the question as under what conditions and restrictions can we achieve a stable and accurate inversion, proceeding from simplified cases to more complex.

As an example consider the model in Figure 1 comparing the wavefield for uniform stiffness fractures (LHS) and stress-dependent stiffness fractures (RHS). The figure is taken from an attempt to reproduce laboratory fracture experiments from Pyrak-Nolte et al (1990), and is published in Hildyard (2007). What was found was that previously assumed uniform fracture stiffness (of any value) could not account consistently for all waveforms in the experiment. Once a non-uniform stress state was postulated, and a (3 parameter) law linking fracture stiffness to stress was assumed, an exploration of parameter space produced a fit which explained the waveforms. The fit was indeed possible only for a narrow set of stress conditions and stiffness parameters. This latter point does not come through strongly in the paper. The results were obtained iteratively, not through inversion. However, it does beg the question as to under what conditions could an inversion be possible. Is it possible with known fractures and stress-field? If so, do we need to constrain fracture numbers and positions? How accurately do we need to first constrain the stress-field? Such fundamental questions need to be investigated to establish if there is potential for full-waveform fracture inversion.

Figure 1: Models attempting to match fracture experiments from Pyrak-Nolte et al (1990), modified from Hildyard (2007). Left shows the wavefield at different timesteps using uniform fracture stiffness, while stress-dependence assumed in the model on the right gives non-uniform fracture stiffness and a very different wavefield.


In this project, you will work with leading scientists at Leeds.

1.    You will gain an understanding of the physics and modelling of seismic wave interaction with fractures.

2.    You will investigate waveform inversion methods.

3.    You will experiment with and implement inversion algorithms to attempt to invert for fractures in very simple seismic models.

4.    You will iteratively increase the complexity of the fracture models and the inversion process.

5.    You will apply the methods to simplified experimental data.

6.    You will be the first to establish the limitations, requirements and potential for full waveform fracture inversion.

Potential for high impact outcome

This is novel work and potentially world-leading. The current interest in full waveform inversion makes it very topical. Useful results will receive long-term academic and industry interest.


The student will work under the supervision of Dr Mark Hildyard and Prof Andrew Hooper. The project provides a high level of specialist scientific training in: (i) Numerical modelling, particularly seismic wave modelling and interaction with fractures; (ii) Inversion techniques; (iii) Use of cutting-edge supercomputers. Supervision will involve regular supervisor meetings and interaction with researchers on related projects. The successful applicant will be actively encouraged to attend and present work at conferences and to publish papers. The successful applicant will have access to a broad spectrum of training workshops put on by the Faculty from training in numerical modelling, through to managing your degree or preparing for your viva (


  • Bunks, C., F. M. Saleck, S. Zaleski and G. Chavent (1995). "MULTISCALE SEISMIC WAVE-FORM INVERSION." Geophysics 60(5): 1457-1473.

  • Hildyard, M.W., (2007). Manuel Rocha Medal recipient: Wave interaction with underground openings in fractured rock, Rock Mechanics and Rock Engineering, Vol. 40, pp 531-561.

  • Hildyard, M.W. and Young, R.P. (2002), Modelling wave propagation around underground openings in fractured rock, Special issue on induced seismicity, ed. Trifu, Pure and Applied Geophysics, Vol. 159, pp. 247-276.

  • Hildyard, M.W., 2001. “Wave interaction with underground openings in fractured rock”, PhD Thesis, University of Liverpool, 2001.

  • Hildyard, M.W., Daehnke, A., and Cundall, P.A. (1995), WAVE: A computer program for investigating elastodynamic issues in mining. Proc. 35th U.S. Symp. on Rock Mech., June 1995, Balkema, pp. 519-524.

  • Tarantola, A. (1984). "INVERSION OF SEISMIC-REFLECTION DATA IN THE ACOUSTIC APPROXIMATION." Geophysics 49(8): 1259-1266.

  • Virieux, J. and S. Operto (2009). "An overview of full-waveform inversion in exploration geophysics." Geophysics 74(6): WCC1-WCC26.

  • Warner, M., A. Ratcliffe, T. Nangoo, J. Morgan, A. Umpleby, N. Shah, V. Vinje, I. Stekl, L. Guasch, C. Win, G. Conroy and A. Bertrand (2013). "Anisotropic 3D full-waveform inversion." Geophysics 78(2): R59-R80.

Related undergraduate subjects:

  • Applied mathematics
  • Computing
  • Engineering
  • Geophysics
  • Mathematics
  • Physics