- 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

[Non-text portions of this message have been removed]

> <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

>

>

- 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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http://im.live.com/messenger/im/home/?source=TAGHM_APR07 - --- In prominence@yahoogroups.com, Adam Helman <helman@...> wrote:

> So, in review, although sensu stricto rho (p) = p**-3 yields

fraction of p

> 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

> 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 - Andrew wrote:
>because we normally talk in terms of

You refer to a cumulative prominence distribution function;

>prominence thresholds. For example, we can say that self-similar

>terrain has exactly 4x as many P1Ks as P2Ks.

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

[Non-text portions of this message have been removed]

> <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

>

>

>