[CIG-SEISMO] Mineos problem

[Li Zhao]

The way MINEOS calculates the normal modes is to find the roots (eigenfrequencies) of a determinant function derived from the free boundary condition on the surface of the Earth. So we could expect that if the determinant function of a mode exponentially decays towards the surface (as it happens to Stoneley modes) then its eigenfrequency cannot be accurately solved and a small error in eigenfrequency will lead to exponential growth in the eigenfunctions. This problem can only be solved by a complete reformulation of the eigenvalue problem, e.g. by converting the PDE system to a weak form or changing the way of finding the eigenfrequencies for the problematic modes. 

By the same token, any mode with substantial exponential behavior in its eigenfunctions is potentially problematic because any error in its eigenfrequency value or in the integration of its radial ODE system may be lethal. A simple way to increase the accuracy is to reduce the grid size in the integration. But there are always some modes that MINEOS can never calculate accurately. The good news is that those modes do not have meaningful contribution to the seismograms on the surface. 

Cheers, 

Li Zhao
Institute of Earth Sciences
Academia Sinica
Taipei 11529 
Taiwan


※ 引述《"Guust Nolet" <nolet at geoazur.unice.fr>》之郵件內容： 
>I am not a Mineos expert, but I think one should consider the physical reality of degenerate modes and not remain focused on the numerics. If two modes are truly degenerate, *any* combination of the regular mode and the Stoneley mode is an acceptable eigenfunction. I assume your Mineos eigenfunctions do satisfy the equations and boundary conditions and are not perse wrong, you just get something you do not want.
>
>One could counter that the modes are only degenerate to numerical precision, but that ignores the fact that in the real, attenuating Earth, the modes have a finite width, so that frequencies probably overlap even with exact calculus. Of course one could formulate the eigenproblem for an attenuating medium, but that is not what Mineos does and I do not think anyone has considered it worthwhile to program such a problem.
>
>But I suggest that if you wish to separate the two into modes with Stoneley vs. regular character, you should just optimize some quality (e.g. energy near the CMB) and get the linear combination of the two modes you want for the Stoneley mode, then find the second mode by orthogonalization.
>
>Best wishes,
>
>Guust Nolet
>https://www.geoazur.fr/GLOBALSEIS/nolet/
>
>
>※ 引述《"Dimitri Komatitsch" <komatitsch at lma.cnrs-mrs.fr>》之郵件內容： 
>
>Hi all,
>
>I am not a Mineos expert myself but I know Li Zhao (cc'ed) worked on an 
>optimized version back in 2007 (which he called Mineos2007), thus let me 
>cc him to see if he knows how to fix that.
>
>Best wishes,
>Dimitri.
>
>On 03/03/2016 17:03, Jeroen Tromp wrote:
>> I think this is a known problem with the version of mineos available via
>> CIG. In fact, I believe the main author of mineos, Guy Masters, advises
>> against using it...
>>
>> My recommendation is reaching out to Guy.
>>
>> Best regards,
>>
>> Jeroen
>>
>> On 3/3/16 10:00 AM, Matthew Knepley wrote:
>>> We have a problem with the eigenfunctions calculated by Mineos at near
>>> degenerate eigenvalues.
>>>
>>> Mineos works fine to calculate the eigenfrequencies of normal modes,
>>> but there is a problem for the eigenfunctions. For those different
>>> modes with very close eigenfrequencies in the mathematical point of
>>> view(see Figure f-l), Mineos regards them as degeneracy in the
>>> numerical point of view, and therefore it gets a linear combination of
>>> pure R mode's and pure Stoneley mode's eigenfunctions, i.e. the
>>> eigenfunctions are not orthogonalized. For example, based on Okal's
>>> classification [1], 2S25  and 3S25 should be Stoneley mode and R mode,
>>> respectively. But the eigenfunctions from Mineos have both surface
>>> oscillations (the feature of R modes) and exponentially decaying along
>>> CMB (the feature of Stoneley modes).(see Figure 2S25 and 3S25)
>>>
>>> Since the eigenfunction is not calculated properly, the group velocity
>>> is also not reliable. According to Okal's paper [1], for each branch
>>> of R modes or Stoneley modes, the group velocity should be continuous.
>>> And we notice that there are some "bad points" in the group velocity
>>> picture, which are exactly the modes we mention before. (see Figure
>>> groupV_R2)
>>>
>>> What eigensolver is being used inside Mineos? The standard symmetric
>>> eigensolvers in LAPACK handle this problem I believe.
>>>
>>>   Thanks,
>>>
>>>     Jingchen and Matt
>>>
>>> References:
>>>
>>> [1] Okal, Emile A. "A physical classification of the earth's
>>> spheroidal modes." Journal of Physics of the Earth 26.1 (1978): 75-103.
>>>
>>>
>>>
>>> _______________________________________________
>>> CIG-SEISMO mailing list
>>> CIG-SEISMO at geodynamics.org
>>> http://lists.geodynamics.org/cgi-bin/mailman/listinfo/cig-seismo
>>
>>
>>
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>
>-- 
>Dimitri Komatitsch
>CNRS Research Director (DR CNRS), Laboratory of Mechanics and Acoustics,
>UPR 7051, Marseille, France    http://komatitsch.free.fr
>