Measuring the Philippine Coastline.

Theoretical Framework

From Classical Geometry to Fractional Dimensions

In Euclidean geometry, dimensions are whole numbers: a point (D = 0), a line (D = 1), a plane (D = 2), and a volume (D = 3). Cut a line into pieces 1/s as long and you get N = s¹ copies. Do the same to a square and you get N = s² copies.

In 1918, Felix Hausdorff extended dimension to non-integers. If scaling an object by 1/s gives N self-similar pieces, its Hausdorff-Besicovitch dimension satisfies N = (1/s)^D, so D = log(N) / log(1/s).

The Coastline Paradox in Nature

Benoit Mandelbrot (1967) observed that natural boundaries such as islands, bays, and lakes behave as statistical fractals. A smooth curve has D = 1. A real coastline does not behave that way. A shorter ruler picks up bays, coves, and points the longer one skipped, so the measured length keeps growing. Laguna de Bay's box-counting D in the database below is about 1.13. If construction such as a lakeshore road straightened its margin, D would move toward 1. Measuring D before and during construction gives a way to check for that.

Algorithm Implementation

Estimator Pseudocode

The code is private for now. This pseudocode shows the steps all three estimators share.

FOR each measurement scale
    measure the shoreline with box, rigid-yardstick, or bending-yardstick rules
    record the scale and resulting count
END FOR

fit the slope of log(count) against log(inverse scale)
report the slope as the estimated fractal dimension

Accuracy on Curves with Known D

We tested the estimators on curves whose true dimension is known. On Koch and quadratic Koch curves, every estimator's error shrinks as detail increases. At the finest level tested, all three are within 1.6% of the true value. Box counting reads the Koch curve 2 to 3% high at middle levels, while the walks read both curves slightly low.

Real coasts are irregular, so we also used random Koch curves: each bump points to a random side, but the true D stays at 1.2619. Over 20 such curves per level, the bending yardstick had the smallest spread:

Preliminary. Bias is the mean error against the true D; spread is the standard deviation across 20 random curves, using the fixed 12-scale ladder. For detecting change, spread matters more than bias: a constant bias cancels when two dates are measured the same way, but spread does not. The bending yardstick's advantage is precision, not accuracy.

Computational Cost

For a shoreline of n points fitted over K = 12 scales, all three estimators should take time linear in n. The measurements agree. Each timing run is pinned to one CPU core, discards a warm-up run, and averages three wall-clock runs.

Exponents are log-log fits of kernel time against point count on a square densified from 1,000 to 64,000 points, ± standard error. Box counting used to measure 1.43. The cause was the hash function in its cell counter: every cell in a grid row landed in the same probe chain. After the fix, a 128,000-point input runs in 30 ms instead of 408 ms, and the box counts are unchanged. On real shorelines each estimator takes a few milliseconds.

Parallel Version (MPI)

All three estimators also run under MPI. Box counting splits the grid into vertical strips that hold about the same number of segments. Each process counts only the cells in its own strip, so the strip counts add up to the serial count without exchanging cells.

A yardstick walk cannot be split, because each step starts where the last one ended. The unit of parallel work is one complete walk for one start vertex and one scale. Walking from many start points also improves the estimate. On the Koch snowflake, averaging 32 starts moves the rigid estimate from 1.2545 to 1.2613, against a true value of 1.2619.

The parallel output matches the serial program exactly at every process count tested, and an automated test checks this. Walk speedups are measured against the same code on one process. Box counting gains less because the serial version is already fast and limited by memory bandwidth. These timings are preliminary: they come from one 8-core laptop (AMD Ryzen 7 5800H) that was also running other programs, and the ranges cover repeated runs, since the laptop's power settings changed timings by about 30%. We plan to repeat them on the ICS Compute Swarm.

Robustness of the Estimates

Each estimator hides choices that should not matter. A walk has a start point and a direction, a box grid has a position, and the shape has an orientation. We swept each one. None of them changes the running time, but all of them move D:

What this means: the absolute D of a real shoreline depends on these choices at the 0.01 to 0.04 level. Comparisons between dates or sites are valid only when one set of choices is applied to every input. The rotation sweep also exposed a bug in our scale ladder. It was sized by the bounding-box diagonal, which changes as a shape turns. Sizing it by the true diameter cuts the rigid walk's rotation spread on Laguna from 0.012 to 0; that fix is opt-in for now. On random fractal curves no window rule won everywhere, so we report both. The bending yardstick's bend budget saturates by K = 8 bends per stick, and headline tables use K = 1.

Review of Related Literature

Box counting is the standard baseline. Our rigid yardstick follows the digital compass walk of Shelberg, Moellering, and Lam (1982). Our experimental bending yardstick sits between that walking family and the constant-deviation variable-step (CDVS) convex-hull methods of Normant and Tricot (1991).

Philippine prior art is thin. Alova (2025) measured the Negros Island coastline at D = 1.018 by box counting on a GADM-derived coastline image. We found no published archipelago-wide values and no studies of Philippine lakes such as Laguna de Bay or Taal Lake from satellite-derived masks. To our knowledge, this is the first fractal study of Philippine shorelines that covers many features and compares algorithms.

Data Pipeline

Where Pretrained Vision Models Fell Short

Before settling on temporal compositing and morphology, we tried pretrained segmentation models and classical edge filters on our imagery. The 6-band UNet++ above matched our rule. The others failed in ways that point to gaps in current remote-sensing models:

Archipelago Database

Nationwide Shoreline Survey

We ran the three C estimators on vector boundaries of 122 Philippine features (108 islands and 14 lakes) across Luzon, the Visayas, and Mindanao. To our knowledge, no archipelago-wide set of fractal dimensions existed before.

These values are preliminary. They use the fixed 12-scale ladder, and the sources differ in vertex density. Absolute D can shift by 0.01 to 0.04 with the window rule, so compare features with each other rather than with outside studies.

Click a row to see the feature's simplified outline and its box-counting log-log plot.

Box-counting D for 122 Philippine shorelines (108 islands and 14 lakes). Orange: islands. Green: lakes. Reference lines mark a smooth curve (D = 1.0), the national mean (D = 1.1545), and the Koch snowflake (D = 1.2618).

Findings

Multi-Temporal Composite Verification

In the tropics, a single-date scene often has cumulus clouds that break up the extracted water boundary. Compositing matched dates reduces this. Pixels that were never clear stay marked as unknown, and we report them as a limitation.

Laguna de Bay, March 2026 (Sentinel-2B), with the traced boundary overlaid. The boundary follows the urban West Bay at left, where LLRN Phase I runs along the shore. It wraps around forested Talim Island in the center and continues into the eastern bay. The straight feature crossing the lower-left shore is the Napindan Channel crossing; the 150 m rule removes the narrow channel. Faint grids in the West Bay are fishpens, which read as water. Leftover clouds at upper right are excluded, not traced.

March 2026 median composite of seven observations. Black holes are pixels that were never cloud-free during the month, mostly over the eastern ridges and a persistent cloud cluster east of Talim Island. They count as unknown, not land. Sediment plumes show along the shallow western shore. Each pixel is the median of up to seven views. A single passing cloud or floating mat of vegetation does not reach the measured boundary.

LLRN Shoreline Monitoring

Mosquito-Borne Disease Data Readiness

This table shows which public records exist, not disease rates. It explains why we cannot yet estimate a shoreline–disease correlation without claiming more precision than the records allow.

Sources: DOH eFOI Laguna dengue municipality request; DOH eFOI Batangas dengue release; and DOH eFOI FHSIS vector-disease response. A valid test still requires case counts, population denominators, time lags, and a direct water-stagnation measure.

Shoreline Geometry Analysis

Open shoreline arcs kept at least 100 m from unknown pixels in both periods. Blue: 2025. Orange: 2026. March and January–March windows are reported separately. The robustness sweeps (24 start points, both walking directions, 16 grid placements) show how much of each change the measuring method alone can produce.

The full LLRN Phase I-adjacent corridor was clear in both January–March composites. Its walks rose by about 0.005 under every window and direction; box counting was less consistent. That rise is a quarter of the corridor's normal month-to-month variation, so we claim no construction signal.