Equivalent Tower Height (ETH): Which is the most magnificent of peak on the Earth?

Introduction

When talking about the magnificence of a peak, people usually either talk about its elevation, relative height (rise), or use photos and videos taken nearby. However, none of these traditional factors is both accurate and objective enough, to convince one peak is more magnificent than another.

Kai Xu introduced the concept of "jut" (described in more detail below), to measure its maximum degree of rise from a nearby terrain. While jut provides a better measurement of the peak's impressiveness from a ground-observer's perspective, it usually favors tall cliffs or walls over peaks itself. This property sometimes diverts jut from the traditional definition of mountain peaks, especially in such cases:

• Peaks with high jut are not necessarily hard to climb. There usually exist routes that go around the steep cliff.
• Peaks with high jut are not necessarily noticeable from height, if they form relatively flat ridges with nearby peaks.

Equivalent Tower Height (ETH), arisen from the concept of "jut", works better in those aspects. Peaks with high ETH is not only guaranteed to look impressive from any direction, but also are hard to summit. Moreover, as an "averaged" version of jut, ETH is more resistant to noise in local terrain height data, allowing the computation from civil-precision data.

Jut: how much a peak rises from a ground observer

Jut of a peak P is defined as the maximum angle-reduced height of P above all surrounding points:

jut(P) = max[h * sin(θ)] for any chosen Q

ETH: Jut applied to a cone-fitted peak

By definition, jut depends on the optimal observation location. To overcome this, we first fit a mountain peak to a perfect cone, then compute the cone's jut. As a cone is fully symmetric by rotation, the problem is reduced from 2-dimensional to a 1-dimensional.

Now the question is, how do we fit a peak (precisely speaking, a certain section below the summit) to a cone?

There exist many methods where an irregular shape fits into a ideal geometry, such as volume-based, cross-section-based, rotational-inertia-based. For simplicity, cross-section-based fitting is adopted in this work.

    Init: Eth <- 0
    for vertical_drop := 1..elevation below the summit,
        A <- Maximum connected area (including peak) with elevation >= (summit_elevation - vertical_drop)
        S <- area(A)
        R <- sqrt(S / PI)
        Alpha <- arctan(vertical_drop / R)
        AngleReducedHeight <- vertical_drop * sin(θ)
        Eth <- max(Eth, AngleReducedHeight)
    Return: Eth 

The name "Equivalent Tower Height", comes from its close relationship with human-made towers. A perfectly-slim tower, with nearly-zero cross-section along its entire height, receives ETH identical to its height. Therefore, some of the tallest human-made structures have ETH comparable to major mountain peaks.

Evaluation with real-world data

Elevation dataset

We choose ASTER Global Digital Elevation Model V003 due to its public accessibility at earthdata.nasa.gov. The dataset features a spatial resolution of 1 arc second (~30m). Although it claims to have a elevation precision of 16 meters, it's observed to be much poorer in areas with rugged terrian. For instance, the highest point in the dataset is only 8802m, about 50m below Mt Everest's true elevation 8849m. If a peak measures 5% lower than its true elevation, we will skip its evaluation to avoid wrong results.

Compute maximum connected area from grid samples

We perform a DFS or BFS starting from the peak point to get set S, where every point in S is higher or equal to elevation_threshold = peak_elevation - elevation_drop.

All quads Q containing at least 1 vertex in S is taken into consideration:

1. If all 4 vertice are in S, count Q as a full grid.
2. Otherwise, perform an interpolation on two of its edges and form a straight line across Q with elevation == elevation_threshold. Count Q as a partial grid sliced by the straight line.

This figure shows how we define Mt Everest's area above 8750m by interpolating a contour. Elevation data is for illustration purpose only.

Once we know the number of grids, we can multiply it by a single grid (1 by 1 arc second)'s area, which only depends on latitude.

This calculation method provides relatively accurate estimation at reasonable implementation complexity.

Peak selection

We use two lists on wikipedia, providing 213 peaks combined.

1. https://en.wikipedia.org/wiki/List_of_mountain_peaks_by_prominence
2. https://en.wikipedia.org/wiki/List_of_highest_mountains_on_Earth

Cut-off

Although there's no limit how far vertical drop can extend to, we are not interested in areas too far from the summit, where the cone has become too flat to result in high ETH values. We increase vertical drop until the following criteria is reached:

1. Connected area reaches 30 kilometers from peak;
2. sin(Alpha) < 0.30 AND vertical_drop >= 1000 meters.

Additionally, vertical drop starts from 300 meters to lower the effect of data noise around the summit, and increases at the step of 200 meters. These thresholds work well for ultra prominent and high peaks, but need be different for lower and more flat ones.

Result

Tier Zero Club (Overall Maximum)

Noticeably, Alaska stands out from the rest of ~8000m giants.

Front runner within each elevation drop range

Those peaks are the steepest among those of similar scale.

Steepest @ last 700m till summit

These peaks only welcome the most skilled climbers. No surprise, Karakoram dominates.

How steepness changes with elevation drop

The following plot has X axis inversed: 1 / sin(θ). Points with greater slope, i.e. closer to top-left area, reaches greater ETH.

Peaks less than 900 ETH are partially shown due to tidiness consideration. Please refer to the appendix for the full list.

Appendix

Maximum connected area evaluation for Mt.Everest at different elevations

Full list of calculated ETH, ranked by elevation

Note:

• E: Error in data (observed height far from truth height)
• F: Too flat / too close to its parent peak to be considered

Reference

Beyond Elevation: New Metrics to Quantify the Relief of Mountains and Surfaces of Any Terrestrial Body (Kai Xu, Yale University)