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

[Magali Billen]

Hello Shijie,

Yes this is very helpful. Its wasn't clear to me that by including the  
normalization factor
in the recursion relation that would lead to the extra factors. I  
tried to back this
out from the code, but its sometimes harder to go backwards,  
especially with the factorials
and the !! terms.

Thank you for spending the time on this. I'll work through the math to  
make sure I follow each
step. I also think it would be good to have CIG add you one page  
explanation to the documentation for the geoid in CitcomS since it is  
not derived elsewhere. Along with a note specifying to the user that  
they must use the same modified legendre polynmials in expanding the  
geoid coefficients from CitcomS to make a map of the geoid (i.e.,  
reading these into matlab and using their legendre functions won't  
work).

Thanks again,
Magali


On Mar 4, 2010, at 11:48 AM, Shijie Zhong wrote:

>
> Magali and Joy,
>
> I could not find my original note on computing spherical harmonic  
> functions
> anymore this morning. The original note was made when I was working on
> CitcomS in 1997 at ANU on a visit to Geoff Davies. I spent sometime  
> this
> morning to re-derive them, and I compared with the code in CitcomS.
>
> I once ran into Bill Press (the author for Numerical Recipes) at a  
> small
> conference in 2006. I mentioned to him about computing harmonic  
> functions.
> He asked me to send him my notes. I remembered having done something  
> after
> the meeting. It seems that what I did was to lose my original notes  
> (buried
> somewhere in my office) -- I did not get back to Press for this for  
> some
> reasons.
>
> Hopefully, you and Joy will find the note useful in reading the code.
>
>
> Shijie
>
> 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: Thu,  4 Mar 2010 12:37:44 -0700 (MST)
>> From: Shijie Zhong <Shijie.Zhong at Colorado.Edu>
>> Subject: Re: [CIG-MC] computing spherical harmonic functions, and  
>> CitcomS
> Legendre polynomial normalization
>> To: "Magali Billen" <mibillen at ucdavis.edu>,"Shijie Zhong"
> <Shijie.Zhong at Colorado.Edu>
>> Cc: "Joy Hines" <jmhines at ucdavis.edu>,cig-mc at geodynamics.org
>>
>>
>> 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 **
>>>  -----------------------------
>> ________________
>> modified_Legendre_functions.pdf (421k bytes)

-----------------------------
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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