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Re: [prominence] Re: prominence analysis - additional comments

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  • Adam Helman
    Andrew - It depends on one s interpretation of self-similar. Imagine a sea of 1,000 foot prominence peaks in a square region. Imagine the same region is
    Message 1 of 11 , May 2 11:13 AM
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      Andrew -

      It depends on one's interpretation of self-similar.
      Imagine a sea of 1,000 foot prominence peaks in a square region.
      Imagine the same region is overlayed with a set of 2000 foot prominence
      peaks.

      Given identical slopes, there is enough "room" for only 1/4 as many
      of the 2,000 foot prominences are there are for the 1,000 foot prominences -
      implying an inverse square relationship - and this is the "2.00" that I
      have
      been referring to.

      It is only when you assign prominence bins to these peaks that
      the question arises of how the DENSITY scales.

      When the bin widths scale as the peak size then the number of peaks
      per bin still scales as the inverse square of the prominence.
      Bin widths linear in their contained prominences is wholly appropriate
      as they too then "scale" with the objects under consideration.

      For example, for self-similarity the number of peaks from
      995 to 1005 feet is four times as many as the number of peaks from
      1990 to 2010 feet. Note here how the bin width enlarges, appropriately
      I feel, with the prominence.

      When the bin widths are fixed (so corresponding to the formula
      for rho (p)), then the numbers of peas per bin scales as the inverse cube
      as you indicate.

      I vastly prefer to report the numbers of prominences in groups ("bins")
      of ever-increasing width as the prominence rises.

      Ideally the bin width should be a constant fraction of the
      prominence value.
      When this is done, the number of peaks per bin scales as the inverse
      square of the prominence for a self-similar landscape.

      ******************************************

      So, in review, although sensu stricto rho (p) = p**-3 yields
      a self-similar terrain, I prefer for practical (reporting) purposes to
      think of this
      relationship as f p * rho(p) = p**-2 where f is the fraction of p
      corresponding to the bin width AT prominence p.

      Adam Helman


      k_over_hbarc wrote:

      > --- In prominence@yahoogroups.com
      > <mailto:prominence%40yahoogroups.com>, Adam Helman <helman@...> wrote:
      >
      > OK, we agree on the derivative.
      >
      > > Edward Earl can provide a nice proof of why
      > > the choice a = 2.00 in a simple power-law model,
      > >
      > > rho (p) = p ** -a
      > >
      > > yields a mountain "range" that is self-similar
      > > on all length (i.e. prominence) scales - and it is
      > > well to see that 2.00 is indeed a typical "alpha equivalent"
      > > (which is what I term the sum a0 + 2 a1 log p)
      > > for mid-range prominences such as 500-1000 feet
      > > (150 - 300 meters).
      >
      > No, I'm quite sure it's a = 3 in your equation that gives self-similarity.
      >
      > Look, the number of prominencess >= p in an x*x area should equal the
      > number of prominences >= kp in a kx*kx area. That is, R should drop as
      > the inverse square. Since r (= rho(p)) is the derivative of that, it
      > must go as the inverse cube.
      >
      > Andrew Usher
      >
      >




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    • JOHN D KIRK
      Getting close to finishing up Montana to P300. Noticed this one not listed anywhere as a 2K. Goat Mountain (8,191):
      Message 2 of 11 , May 2 3:07 PM
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        Getting close to finishing up Montana to P300. Noticed this one not listed
        anywhere as a 2K.
        Goat Mountain (8,191):
        http://www.topozone.com/map.asp?z=12&n=5342843&e=340917&s=50&size=l&datum=nad83

        Goat Mountain Saddle (5,974) :
        http://www.topozone.com/map.asp?z=12&n=5334454&e=345597&s=50&size=l&datum=nad83
        Line Parent is Pondera HP.

        Lots of corrections to MT eastern County Prom Points and another error range
        2k:
        Mount Gould 9553' saddle@ 7560 (+/- 40).

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      • k_over_hbarc
        ... fraction of p ... So we agree here too. I like to think of it primariliy in terms of R (the density of prominence = p), because we normally talk in terms
        Message 3 of 11 , May 2 5:34 PM
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          --- In prominence@yahoogroups.com, Adam Helman <helman@...> wrote:

          > So, in review, although sensu stricto rho (p) = p**-3 yields
          > a self-similar terrain, I prefer for practical (reporting) purposes to
          > think of this relationship as f p * rho(p) = p**-2 where f is the
          fraction of p
          > corresponding to the bin width AT prominence p.

          So we agree here too. I like to think of it primariliy in terms of R
          (the density of prominence >= p), because we normally talk in terms of
          prominence thresholds. For example, we can say that self-similar
          terrain has exactly 4x as many P1Ks as P2Ks.

          (Why do you put double spaces everywhere? I fix them in quoting you.)

          Andrew Usher
        • Adam Helman
          ... You refer to a cumulative prominence distribution function; as the definite integral from the lower threshold to infinity for rho(p). ... Integration of
          Message 4 of 11 , May 2 5:51 PM
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            Andrew wrote:
            >because we normally talk in terms of
            >prominence thresholds. For example, we can say that self-similar
            >terrain has exactly 4x as many P1Ks as P2Ks.

            You refer to a cumulative prominence distribution function;
            as the definite integral from the lower threshold to infinity for rho(p).

            >terrain has exactly 4x as many P1Ks as P2Ks.
            Integration of p**-3 indeed yields p**-2 apart from a constant of
            integration.

            FYI, the definite integral of p**-(a0 + a1 ln p) is also analytically
            evaluatable.
            However it involves the error function of the prominence threshold -
            which can then be numerically approximated.
            Derivation of this integral is a real mess, and, of course, blows-up for
            a1 < 0.

            Perhaps Maple could generate an integral easier than what I've
            found by hand??

            In practice one sets an upper limit as well since real mountains
            don't extend to infinity.

            Adam Helman

            PS Never mind double spaces: all of my recent messages
            have taken a seriously long FOUR HOURS to get posted.

            There's may e a "disconnect" between my service provider
            and Yahoo!! that precludes more timely messaging.

            k_over_hbarc wrote:

            > --- In prominence@yahoogroups.com
            > <mailto:prominence%40yahoogroups.com>, Adam Helman <helman@...> wrote:
            >
            > > So, in review, although sensu stricto rho (p) = p**-3 yields
            > > a self-similar terrain, I prefer for practical (reporting) purposes to
            > > think of this relationship as f p * rho(p) = p**-2 where f is the
            > fraction of p
            > > corresponding to the bin width AT prominence p.
            >
            > So we agree here too. I like to think of it primariliy in terms of R
            > (the density of prominence >= p), because we normally talk in terms of
            > prominence thresholds. For example, we can say that self-similar
            > terrain has exactly 4x as many P1Ks as P2Ks.
            >
            > (Why do you put double spaces everywhere? I fix them in quoting you.)
            >
            > Andrew Usher
            >
            >
            >




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