“TRIANGULAR DISLOCATION: AN ANALYTICAL, ARTIFACT-FREE SOLUTION” (by Mehdi Nikkhoo and Thomas R. Walter) COMMENTED BY OLAF ZIELKE
“Paper of the Month” is a new blog series launched by the ECS-reps team. The aim is to absorb and share the reading experience of mature seismologists with regards to peer-review articles either recently published (less than 12 months) or older, if of particular note.
As an opener, we asked Olaf Zielke, a research scientist working at King Abdullah University of Science and Technology (KAUST), to choose a paper of particular interest for him and to tell us a little about the topic of the selected study and why he finds this particularly striking.
Before joining KAUST, Olaf worked as post-doc for two years at GFZ Potsdam. He obtained his PhD from Arizona State University in 2009. Olaf’s research focuses on the physics of earthquakes and faulting and its relation to crustal deformation and plate tectonics. He conducts data-based studies (e.g., using paleoseismic and tectono-geomorphic techniques) as well as simulation-based studies (e.g., performing multi-cycle numerical earthquake simulations and kinematic source inversion models), aiming to combine the respective outcomes into comprehensive conceptual models.
The paper chosen by Olaf for our blog is “Triangular dislocation: an analytical, artifact-free solution” by Mehdi Nikkhoo and Thomas R. Walter, published in Geophysical Journal International in March 2015. And here is his post…
So, what follows is going to be the first of hopefully many “Paper of the Month” reviews here on this blog. The goal is to highlight important, more or less recent work that bears potential (or has already proven) to have a lasting impact on the seismology community. My own research focuses on the physics of earthquake rupture so that the paper that I am going to present here is addressing this field of work. The actual choice was a bit arbitrary – after all, there are many very high-quality publications, as the one I’m going to talk about, that deserve being highlighted in this form (or another).
For this first review, I picked a paper that I believe is going to become a new standard in its domain. This paper by Nikkhoo and Walter (2015) is addressing elastic dislocation theory and presents a novel approach to calculate the displacement- and strain/stress-fields that are associated with for example earthquakes, volcanoes, and landslides. First, I want to briefly mention the relevance of calculating those fields, then write about the currently and commonly adopted approaches to do so (and their problems), and lastly how the new approach that is provided by Nikkhoo and Walter (2015) is dealing and solving those problems.
Elastic dislocation theory – the calculation the displacement and deformation at a given observation point in half- or full-space due to slip along a fault or fault patch – has a wide range of applications that may be divided into two groups: a) forward modeling applications and b) inversion applications. The prior includes for example (multi-cycle) numerical rupture simulations that aim to determine the recurrence characteristics of earthquakes. This setup requires to identify how much stress a certain amount of slip at one part of the fault is inducing at another part of the fault. Another forward modeling application is the computation of Coulomb failure stress changes due to the occurrence of an earthquake. An inversion application that is based on elastic dislocation theory is for instance the generation of a kinematic rupture model that quantifies the amount of coseismic slip along an earthquake rupture surface using GPS or InSAR data that constrain the displacement field at the surface. All of these applications require for fast and numerically stable formulations, preferably for finite-size dislocation sources (as opposed to point sources).
A range of analytical solutions for finite dislocation sources have been published in the past, for example by Okada (1985, 1992), Yeyakumaran et al. (1992), and Meade (2007). While these solutions (i.e., respective publications) have proven very useful in the past and some of them have reached an almost iconic status, they do face some challenges –related to numerical stability and to the capability to represent geometrically complex faults and fault systems. That is, representing the dislocation source with rectangular fault patches (e.g., Okada, 1985, 1992) prohibits the generation of closed fault surfaces for geometries other than planes or cylinders. This issue disappears once triangular source elements are used (e.g., Jeyakumaran et al., 1992; Meade, 2007). However, the present analytical formulations for rectangular and triangular fault elements both exhibit numerical instabilities, for example along the extensions of the dislocation source edges. While those singularities may not be of relevance if only a few fault patches and simple fault geometries are considered, they may start to become a severe nuisance and cause problems once the fault geometries become more complex and more highly discretized.
This is where Nikkhoo and Walter (2015) come into play. The new analytical solution that they provide is free of such artifacts. Because it is considering triangular dislocation elements, it is also possible to generate complex fault geometries. Aside from the actual analytical formulations, this publication also provides code implementations in MATLAB and C. This publication is therefore very beneficial to a wide range of investigations and I want to encourage acquiring the code (available via GitHub and supplementary information on the Geophys. J. Int. website).
But how did Nikkhoo and Walter achieve to remove the numerical artifacts (leaving only the intrinsic singularities along the edge of a dislocation)? The “trick” that was used is rooted in the setup of those triangular dislocations –each defined by three angular dislocations (see figure 3 from Nikkhoo and Walter, 2015). If an observation point P intersects one of the dislocation lines (that make the angular dislocations) then the numerical formulation is not stable and we observe a numerical artifact for P. However, the alternative representation of the same triangular dislocation puts P at the maximum possible distance to such lines. Thus, with the appropriate configuration–with respect to each observation point individually–any artifact singularity or numerical instability can be avoided.
Concluding, I want to recommend this paper and the tools provided with it to any reader that employs elastic dislocation theory in their research. I am convinced that “Nikkhoo and Walter (2015)” will become the new standard method for this kind of work, replacing the iconic representations by e.g., Okada (1992). Note that I here only gave a very brief summary of the paper’s content. Please have a look at the actual paper for more detail–it is certainly worth it!
Olaf Zielke — email contact: olaf.zielke at kaust.edu.sa — website: https://ces.kaust.edu.sa/Pages/Olaf-Zielke.aspx
Have you already read Nikkhoo and Walter (2015) and you want to share your opinion on this paper? Do you want to comment Olaf’s post? If so, please have your say in the space below!
If you are an experienced seismologist and want to be the next author of our “Paper of the Month” series, get in touch with the ECS-reps (Early Career Scientist representatives) of the Seismology Division (ecs-sm at egu.eu)!
Jeyakumaran, M., Rudnicki J. W. and Keer, L. M., 1992. Modeling slip zones with triangular dislocation elements. Bull. Seism. Soc. Am. 82, 2153–2169.
Meade, B.J., 2007. Power-law distribution of fault slip-rates in southern California. J. Geophys. Res., 34 (23).
Nikkhoo, M. & Walter, T. R., 2015. Triangular dislocation: an analytical, artifact-free solution. Geophysical Journal International, 201, 1117-1139. Doi: 10.1093/gji/ggv035.
Okada, Y., 1985. Surface deformation due to shear and tensile faults in a half-space. Bull. Seism. Soc. Am., 75, 1435-1154.
Okada, Y., 1992. Internal deformation due to shear and tensile faults in a half-space. Bull. Seism. Soc. Am., 82, 1018-1040.