[CIG-MC] computing spherical harmonic functions, and CitcomS Legendre polynomial normalization

[Robert Moucha]

Thanks for the notes Shijie.

While we are on the topic of spherical harmonics.  I noticed that
within CitComS 3.1.1 there is an option to compute the spherical
harmonics representation of the temperature field using the function
debug_sphere_expansion(). I myself have found this to be most useful
and I have added an output option in my version of CitComS to generate
the spherical harmonics of the temperature field at each monitoring
step. For L=64, the computing cost is small.  I have also done the
same for the dynamic topography (already calculated for the geoid).
Perhaps if others would find this to be useful, the spherical harmonic
output option could be included in future versions.

BTW, for my visualizations I use MATLAB, and even on my laptop I have
been able to easily and quickly generate cross-sections or maps for up
to L=512 -- something that would be impossible to do with equivalent
resolution using the spatial representation of the field.

Cheers
Rob

On Thu, Mar 4, 2010 at 2:37 PM, Shijie Zhong <Shijie.Zhong at colorado.edu> wrote:
>
> I am not sure how many people would be interested in this kind of stuff, but I
> send it to cig-mc anyway. Some of you may find it useful if you want to write
> your own spherical harmonic function/expansion code. While Press' Numerical
> Recipes book gives a nice code for computing the Legendre functions, it is not
> as straightforward to compute spherical harmonic functions.
>
>
>
> Shijie Zhong
> Department of Physics
> University of Colorado at Boulder
> Boulder, CO 80309
> Tel: 303-735-5095; Fax: 303-492-7935
> Web: http://anquetil.colorado.edu/szhong
>
> ---- Original message ----
>>Date: Wed, 3 Mar 2010 17:26:21 -0800
>>From: Magali Billen <mibillen at ucdavis.edu>
>>Subject: Re: [CIG-MC] CitcomS Legendre polynomial normalization
>>To: Shijie Zhong <Shijie.Zhong at Colorado.Edu>
>>Cc: "Joy Hines" <jmhines at ucdavis.edu>,cig-mc at geodynamics.org
>>
>>   Hello Shijie,
>>   Yes, its the factors in the recursion relationship
>>   that we were surprised to find, as the rest the
>>   recursion relationship steps are the same as that in
>>   Numerical recipes...I'm glad to know that this
>>   is stable for large L as we'll be going up to about
>>   L=360. Any chance that there's a reference
>>   for the recursion relationship (or maybe point us in
>>   the right direction, we've looked up several others
>>   and haven't found the same thing)? No hurry, but it
>>   seems like this is a good thing to have reference
>>   for general
>>   CitcomS background information too.
>>   Thanks for answering our e-mail so quickly,
>>   Magali
>>   On Mar 3, 2010, at 5:13 PM, Shijie Zhong wrote:
>>
>>     You are right that the recursion relation for the
>>     Legendre polynomials used in
>>     CitcomS is different from the standard in
>>     Numerical Recipes by Press. The
>>     standard one in Numerical Recipes blows up for
>>     spherical harmonic degree l >
>>     20 or something, even with double precision
>>     computation. I re-derived a
>>     recursion relation in 1997 that is stable for
>>     nearly all l's and is used in CitcomS.
>>
>>   -----------------------------
>>   Associate Professor, U.C. Davis
>>   Department of Geology/KeckCAVEs
>>   Earth & Physical Sciences Bldg, rm 2129
>>   Davis, CA 95616
>>   -----------------
>>   mibillen at ucdavis.edu
>>   (530) 754-5696
>>   --------------------------
>>   ** Note new e-mail, building, office
>>       information as of Sept. 2009 **
>>   -----------------------------
>
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