# An explanation of ORS (spire measure)

## Brief explanation                                (Back to the main ORS page)

The idea of omnidirectional relief and steepness (ORS), formerly known as spire measure, is to evaluate a point (often the summit of a mountain) by measuring its overall relief and steepness, using a sort of average over all points surrounding the chosen point.

Here's a quick description of the ORS calculation. It can be thought of as a measure of how impressive it is to stand at the top of the peak and look down. In fact, you can take any point on the Earth's surface and calculate an ORS value for it, which measures how impressive the view is from that point.

So, imagine that you are standing at some fixed reference point (say the top of Mt. Shuksan, which stands very high above nearby terrain and has a very impressive North Face). Look at all the points you can see, in all directions, at all distances. For each such point you will be looking down (or occasionally up) at some angle (or slope, which is easier to calculate). If there is a lot of area where that downward slope is large, you are at an impressive point on the Earth's surface. That's the idea.

For example, standing on top of Shuksan, looking down the North face, you have to look very steeply down to see the bottom of the face and the valley below. That gives a good contribution to ORS. Also, since it is high above the terrain in the range 5-15km away, when you look down at those points, you are looking down at a pretty fair angle--a steeper angle, anyway, than if you were standing on top of a gentler peak (for example, Mount Elbert in Colorado) looking down at the valleys near it. So again Shuksan gets a good contribution from that terrain. Summing over all directions and all distances gives the total ORS value for Shuksan, which is quite large (552 meters).

You can also look at more extreme examples: Yosemite's El Capitan (ORS=575 m) gets a huge contribution from very close-in terrain (to the south), since looking down from the brim of the cliff to the Valley floor is very steep indeed. But El Cap gets very little contribution from far-away terrain since it is not very high compared to that terrain (in fact it is lower than a lot of the surrounding high country). And if you look north from the brim you get no contribution since you are looking up, not down.

Mauna Kea (ORS=368) lies somewhat at the other extreme. If you stand atop Mauna Kea there is no point you can look at that forces you to look very steeply downward, since it has no cliffs or super-steep faces to speak of. But there is a large area where you have to look moderately downward, since Mauna Kea rises so high above the surrounding terrain. This gives it the moderately high ORS value that it has.

Often the point on a peak with the best ORS value is not the main (highest) summit. For example if we measure ORS from the "summit" of El Cap (a little knoll well away from the cliff edge) we will not get a very good value, since we get none of the super-high slopes which we would get from standing at the brim. The optimal reference point for El Cap is right at the brim of the cliff, above the very steep North America Wall. All of the ORS values in the tables on this website are calculated at the best reference point for the given peak--the point with the "most impressive view" downwards, or equivalently, the point which looks most impressive when looking up at it. Practically speaking, ORS rewards peaks which are high above surrounding terrain, steep, and pointy. (If a peak is not pointy then there will be some directions from the reference point in which you do not go down steeply. El Cap is an extreme example of this, although it is so good in some directions that it still gets a very high ORS value.) In other words tall, steep spires are favored. This is why it used to be called "spire measure".

List-making and Reduced ORS In terms of making lists of the "Top N" peaks in a region, ORS has one drawback. Like height, it assigns a number to every point on the Earth's surface in a continuous fashion. So if a point (e.g. the summit of Mt. Shuksan) gets a high spire measure rating, then all of the nearby points will get a high rating as well. So we cannot simply make a list of all of the points with ORS above some certain value, since that list will be infinite and will mostly be trivial subpeaks and points near to real peaks.

For listmaking we need a measure that includes at least some degree of reward for independence, or conversely, some degree of reduction for being very close to a superior peak. Such a measure will automatically exclude trivial subpeaks from a "Top N" list. Of course it changes the nature of what we are measuring: instead of trying to quantify simply how dramatically a peak rises above its surrounding terrain, we are factoring in some reward for independence as well. Edward Earl and I have a variant of spire measure called reduced ORS which does this job, in what we feel is a natural way. So most of the lists rank peaks by reduced ORS. However each table also includes the rating of the peak by unreduced ORS for those who prefer to see how the peak compares when independence is not taken into account.

### A note on the units of ORS

ORS is measured in units of height. In these tables I have used meters; one could also use feet if desired. The units are not too important, since what is most meaningful is comparisons between two peaks. However ORS is normalized so that a "flagpole" of height H (and infinite slope) will get ORS = H, exactly. For a more realistic comparison, a perfectly conical peak of height H and angle 45 degrees (slope 1) will get ORS = H/2. So for example, a conical peak of height 1654m (5426ft) and slope 1 would get the same SM value as Mount Rainier, 827m.

## More detailed, technical explanation

The first problem is to give a number for the "impressiveness" of a certain drop of height H and horizontal distance B. This is not terribly well-defined, and I would argue that you'll never get a number that everyone will agree with, since different people have different tradeoffs between height and steepness. (In particular the word "impressiveness" is only supposed to be a handy, suggestive shorthand, not implying that this is some magic, universally agreed-upon number.) However it's still tempting to come up with a simple formula, with nice properties, that gives a reasonable answer. We call this point-to-point relief and steepness, or PPRS.

The simplest formula one can come up with is "relief times slope": PPRS(H,B) = H * (H/B) = H^2/B, but this becomes infinite for any vertical cliff, no matter how small. So it is not a good formula. There is a simple variant that is much better, namely "relief times angle", H*arctan(H/B). We prefer to normalize it so that if the angle is 90 degrees (vertical), we will get H, so we use PPRS(H,B) = (2/pi)*H*arctan(H/B). Note that a 45 degree slope gets PPRS(H,H) = H/2.

NOTE: until recently we used the following variant, due to Edward Earl: PPRS(H,B) = H * S/(S+1) = (H^2)/(H+B). A few older lists on this website are still calculated using this older variant, which can differ from the newer "angle" version by up to 25%. We are in the process of recalculating and retabulating the lists.

What omnidirectional relief and steepness (ORS) does is apply these ideas not to a single drop, but to a whole mountain. Here's how the story goes. First, we take a simple test case: a perfectly conical peak of height H and base radius B (hence slope S = H/B), on a flat plain. Wen use PPRS to suggest what "impressiveness" number to assign to this peak. Namely, we assign the cone an ORS of ORS(H,B) = (2/pi)*H*arctan(H/B).

Now the question is, given an arbitrary peak, how should we assign such a number, taking into account how "impressive" the peak is in all directions? I want a measurement, call it "ORS", that has the following properties:

1. it gathers information about drops in all directions *and* at all distances and takes some kind of average. It should emphasize local relief, but not ignore height above further-away terrain.
2. it scales nicely: if I scale both vertical and horizontal dimensions by the same amount K (thereby increasing all heights but keeping slopes constant), then ORS also increases by a factor of K. Note that this is already a feature of PPRS.
3. it gives the answer we already have chosen for a conical peak.
4. it does not require one to determine where the mountain stops and the rest of the terrain begins.

(1) is the most important feature, saying that SM will be an all-around measure of impressiveness. For example for Mount Rainier it will somehow combine the fact that it looks very impressive from the Carbon Glacier (medium height but very good slope) and also very impressive from Seattle (large height but smaller slope).

As to (4), I have never seen a definition of how to delineate where a mountain stops that does not have a lot of choices that are open to argument. So to me (4) is an important feature.

Now it turns out that these requirements lead us to the following definition. First of all, because of (4), we will not actually directly define the ORS of a particular mountain. We will instead define an ORS value attached to any point on the Earth's surface. The basic idea is this: fix a reference point for which we want to measure ORS, for example, the summit of Liberty Cap. Now look in all directions and at all distances. For any particular direction and distance, we are looking at some "sample point" of terrain, say a spot on the Carbon Glacier.

Now how will we know that we are standing at an impressive spot? It will be because there are a lot of sample points which are steeply below us, that is, for which the line of between the reference point and the sample point has a steep slope. To put it another way, the reference point is impressive if, from a lot of possible viewpoints, you have to look steeply up to see the reference point. To me that's a pretty reasonable definition of "impressive" in a relief-combined-with-slope sense.

So perhaps the answer is to add up all those slopes over all the sample points. However that runs up against the problem we saw before, that if we are standing atop a vertical cliff, the slopes will diverge, and we will get an infinite answer. The fix here is similar: we take some modified slope, which does not diverge, or at least does not diverge too badly, for large slopes. Call this modified slope function f(s). We do not yet know which one to take, but soon we will, hold on.

So now here is our first attempt at the real formula. Say the function (height of terrain above sea level) is given by h(x). We fix our reference point p and we want to calculate ORS(p). First consider a fixed direction and just look at all the sample points along that direction. For any sample point x we calculate the relief y = h(p) - h(x) and the distance r = dist(p,x), and hence the slope s = y/r. Then we integrate the modified slope:

ORS(p,this direction) = int { f(y/r) dr }      (provisional version)
where the integral is from r=0 out to infinity. (We will make sure that our f is nice enough so that this is finite.) This defines the "total impressiveness of this reference point with respect to this direction." Now to get the final answer, we just average all these directional answers. That's an integral too:
ORS(p) = (1/(2 pi)) * int { int { f(y/r) dr } dtheta } = (1/(2 pi)) * int { f(y/r)/r dA }.      (provisional version)
This is close to the actual formula. However there is one problem with the above formula--it is very sensitive to the height function right around the reference point. For example, simply taking a stepladder and standing on top of the stepladder instead of on the ground can change the ORS value measurably. So we need to take a different kind of average, one that concentrates attention in the middle-distance region. This is accomplished by taking a "root-mean-square" (RMS) integral instead of the simple integral above. The resulting formula is
ORS(p) = sqrt( (1/(2 pi)) * int { (f(y/r))^2 dA }).      (final version)
Remember that here y = h(p)-h(x), the difference in height between the fixed reference point p and the varying sample point x. Now I can say how we are supposed to choose the function f. We choose it in such a way that the ORS of a conical peak of height H and base B, measured from its summit, is SMR(H,B) = H^2/(H+B). It's possible to write down the function f explicitly, but it's not particularly instructive; the important thing is the result for a conical peak.

So this is the proposal for "the impressiveness of the point p", figured out by myself and Edward Earl. Now if you want, you can take a peak like Rainier, look at all the possible reference points everywhere in the area, and take the best one. This would be the candidate for "the most impressive point on Mount Rainier," and its SM value would be a reasonable choice for "the SM of Mount Rainier." Note that this is very insensitive to what you define as "Mount Rainier," since any point near the "base" of Rainier, where there might be disagreement about whether it really is part of Rainier or not, would not have a terribly high ORS value anyway. So you can get around the issue of "where does the mountain stop."

In fact, for the case of Rainier, it should be no surprise that the optimal value comes at the top of Liberty Cap. (In fact this is the optimal point in the lower 48, see below.) Liberty Cap is better than Columbia Crest since it sits atop the Willis Wall and the Sunset Amphitheater, so there are a lot of sample points with steep corresponding slopes. Columbia Crest is pretty good, since the slope values for distant sample points are very similar to or better than those for Liberty Cap, but the lack of those nearby high-slope values means it gets a lower ORS value than LC, despite being higher.

I claim this is eminently reasonable, that in fact Liberty Cap is a more impressive point to stand on than Columbia Crest. Rainier is a great mountain to climb but the summit itself is a bit anticlimactic. For example, the last time I was there it was moderately whited out, so that we could only see barely across the crater and barely to the saddle with LC. In such conditions CC is not very impressive. However in the same conditions LC would have still been impressive, since we still could have seen a good way down the steep walls.

So in fact this calculation could be used to say things like "when you climb Rainier, you should actually climb Liberty Cap instead of Columbia Crest." That's an extreme version, perhaps, but it's in the spirit of alternative mountain measures in general: they tell you more interesting places to go than just the highest points in terms of absolute elevation.

Also, in doing a lot of these calculations, I have found that often the optimal reference point (if different from the summit) sits atop a classic route on the mountain, like LC sitting atop Liberty Ridge and Ptarmigan Ridge. For a decently pointy peak, like the Grand Teton, it's not an issue, as the true summit is the best reference point.

In any case, issues of Liberty Cap versus Columbia Crest and of what point exactly one should climb aside, ORS gives a way to attach a number to any peak that records how high and how steeply it rises above the surrounding terrain. It naturally emphasizes more strongly the nearby terrain (since that is the only place you can get really high slopes) but it does take into account the more distant terrain, albeit with lower and lower weighting as you go out. (This isn't completely obvious until you investigate the function f a little bit, but it's true). In short, it satisfies all the properties (1)--(4) mentioned above, and has a number of other nice properties too.

Like prominence, ORS serves as a good complement and/or alternative to measuring mountains by absolute elevation, and directs ones attention to interesting, "impressive," but perhaps not incredibly high, peaks. Take a look at the tables to get a better sense of spire measure in the real world. In judging whether you like the ORS idea it's best to use it to compare real peaks and see if you think it gives a reasonable comparison.

Hozomeen South and Kinnerly Peak (see the Contiguous US list) are good examples of how ORS picks out impressive peaks which are totally ignored by elevation-based lists. Both of these peaks have very large, steep faces. Steve Fry has determined that Hozomeen South has the largest drop in 0.1 horizontal mile in the lower 48, and Kinnerly has the largest drop in 1 mile. This steepness is why these peaks have such good ORS values, despite their elevations: Hozomeen at only about 2450m, Kinnerly at just over 3000m.

Going further down the contiguous US list, one finds that the North Cascades and Glacier National Park contribute many high-ORS peaks, whereas the Colorado Rockies, which do not rise as high above the ambient terrain and are not as steep, get much lower numbers. (The best in Colorado are Pyramid Peak in the Elks, 394m; Longs Peak, 373m; and West Spanish Peak, 415m; and Pigeon Peak in the Needle Mountains, 362m.)

Of course, as I have noted, ORS is not meant to be a magic number that everyone will agree measures true "impressiveness." That is such a slippery, subjective concept, and necessarily involves so many features of a mountain, that there can't be such a perfect measure. In particular it's important to view these lists by ORS (or RORS) as being complementary to lists by absolute elevation or by prominence, not as replacements for those lists.

But I think that ORS is nice because one is led to it almost inexorably by the simple criteria (1)-(4) above. So it is not very arbitrary, once you agree that you want the kind of measure that satisfies those properties.

The main arbitrariness is in the choice of the original point-to-point function, that is, in your choice of "how should slope trade off against height?" Deciding this tradeoff is crucial to any calculation that assigns a single number to peaks of varying height and slope. I noted above that the lists on this site are being changed from using

PPRS(H,B) = H^2/(H+B)
to using
PPRS(H,B) = (2/pi)*H * arctan(H/B).
This change turns out to favor smaller, steeper peaks (like Hozomeen) over larger, gentler peaks (like Baker). We feel that the new measure is a bit more elegant mathematically, and provides a better complementary measure to height and prominence.

However, once this choice has been made, you don't have any more arbitrary parameters to put in. In particular, you don't have an arbitrary choice of horizontal scale, like 0.1 mile, 1 mile, 10 mile, etc.

The real drawback of ORS is that it is a pain to calculate. After first defining it, I did a number of peaks "by hand" (actually by TopoUSA and a spreadsheet), but now I calculate it automatically using downloaded digital data. Once one cranks some numbers, it is possible to get a feel for which peaks will turn out well and which will not (in fact, you can just look at the peak and measure your "ooh-aah" factor, pretty much--a sign that it really is a good measure). But I have had some surprises (actually another good sign, since otherwise this would all be obvious). And to get an accurate number, some crunching is required, which makes it a little harder to convince people of its usefulness. Prominence is nice in that in principle it just requires a map (or a lot of maps) and patience. ORS requires a bit more work, but I think it's worth it.

### Reduced ORS (RORS)

Soon (?) I will post a detailed explanation of reduced ORS (formerly "reduced spire measure") or RORS here. Stay tuned.

## Back to the main ORS pageOn to the lists

Any comments or questions are very welcome:

metzler (at) aa (dot) edu