What can you see from this location? Visibility analysis on terrain
2026-02-27
The fire lookout towers that dot the ridges of western North America were placed, before any of them were built, by surveyors who climbed candidate peaks and looked. They were searching for locations that could see the greatest area of forested slope with the fewest blind spots — natural terrain convexities or intervening ridges that would hide a smoke column from view until it was too large to suppress. That process of visual survey, repeated at hundreds of sites across BC, Alberta, Montana, and Idaho, is now done in an afternoon using a DEM and a viewshed algorithm. The mathematics is the same; the speed is not.
Viewshed analysis answers a question that appears in an surprisingly large number of practical contexts: from a given observer location at a given height, which cells in the surrounding terrain are visible and which are hidden by intervening topography? The calculation is fundamentally a ray-tracing problem — cast a line from the observer to each target cell, sample the DEM along the ray, and check whether any intervening point is higher than the line of sight. Efficiency matters because a single viewshed for a 10-metre DEM of a typical mountain region involves testing line-of-sight to millions of cells. This model derives the geometry of the problem, implements the reference-height comparison algorithm, and extends it to cumulative viewshed (which observers can see a given cell?) and visual impact assessment for towers and turbines.
From the top of this mountain, which valleys can you see?
Viewshed (also called “intervisibility”) determines visible areas from an observer location:
Applications: - Fire towers: Where can rangers see smoke? - Cell towers: Which areas have signal coverage? - Wind turbines: Where will they be visible (visual impact)? - Real estate: Which properties have ocean/mountain views? - Military: Where can this position provide surveillance? - Solar farms: Where will there be shadow/glint?
The mathematical question: Given an observer at position (x_0, y_0, z_0) on a terrain, which cells in the DEM are visible (not blocked by intervening terrain)?
Challenge: Must test line-of-sight to potentially millions of cells efficiently.
Two points have line-of-sight if:
The straight line connecting them doesn’t intersect terrain between them.
Observer: O = (x_0, y_0, z_0)
Target: T = (x_1, y_1, z_1)
Line-of-sight exists if:
\forall t \in (0, 1): \quad z_{\text{line}}(t) > z_{\text{terrain}}(t)
Where: - z_{\text{line}}(t) = z_0 + t(z_1 - z_0) (elevation on sightline) - z_{\text{terrain}}(t) = terrain elevation at position (x_0 + t(x_1 - x_0), y_0 + t(y_1 - y_0))
If line dips below terrain at any point → blocked!
Total observer elevation:
z_{\text{observer}} = z_{\text{ground}} + h_{\text{observer}}
Examples: - Person: h = 1.7 m (eye level) - Fire tower: h = 20 m (platform height) - Cell antenna: h = 30 m (tower height)
Similarly, target height:
z_{\text{target}} = z_{\text{ground}} + h_{\text{target}}
Person standing: h_{\text{target}} = 1.7 m
Building: h_{\text{target}} = 10 m (roof height)
Binary viewshed:
V(x, y) = \begin{cases} 1 & \text{if visible from observer} \\ 0 & \text{if not visible} \end{cases}
Cumulative viewshed (multiple observers):
V_{\text{total}}(x, y) = \sum_{i=1}^{n} V_i(x, y)
Counts how many observers can see each location.
For each target cell: 1. Sample points along line from observer to target 2. Interpolate terrain elevation at each sample point 3. Check if sightline passes above terrain 4. If all samples clear → visible
Problem: For n \times n DEM, testing n^2 cells × k samples per line = O(n^2 k) complexity.
Example: 1000×1000 DEM, 100 samples per line = 100 million tests!
Key insight: Use angle of elevation instead of sampling.
Elevation angle from observer to point (x, y):
\alpha(x, y) = \arctan\left(\frac{z(x,y) - z_0}{d(x,y)}\right)
Where d(x,y) = \sqrt{(x - x_0)^2 + (y - y_0)^2} is horizontal distance.
Visibility rule:
Point is visible if its elevation angle exceeds maximum angle seen so far along that ray.
Algorithm: 1. For each radial direction from observer 2. March outward along ray 3. Track maximum elevation angle seen 4. If current point’s angle > max → visible, update max 5. If current point’s angle ≤ max → blocked
Complexity: O(n^2) but with much lower constant (single pass per cell).
Pseudocode:
For each azimuth angle θ from 0° to 360°:
max_angle = -∞
For distance d from 0 to max_distance:
x = x0 + d * cos(θ)
y = y0 + d * sin(θ)
z_terrain = interpolate(DEM, x, y)
z_target = z_terrain + target_height
angle = atan((z_target - z_observer) / d)
if angle > max_angle:
mark(x, y) as visible
max_angle = angle
else:
mark(x, y) as not visibleKey: Maximum angle acts as “horizon” - anything below it is blocked.
Process cells in rings around observer:
Ring 0: Observer cell (always visible)
Ring 1: 8 neighbors
Ring 2: 16 cells at distance 2
…
For each cell in ring k: - Compute elevation angle to cell - Compare to maximum angle seen in that direction - Update visibility and max angle
Advantage: Processes each cell exactly once, cache-friendly.
Problem: Determine viewshed from observer at (2, 2) with eye height 2m.
DEM (meters):
j=0 j=1 j=2 j=3 j=4
i=0 10 12 14 16 18
i=1 12 14 16 18 20
i=2 14 16 18 20 22
i=3 12 14 16 18 20
i=4 10 12 14 16 18Observer at (2, 2): ground elevation 18m, observer height 2m → total 20m
Check visibility of cells along row 2 (eastward from observer).
Cell (2,3): 1 cell east
Ground elevation: 20m
Horizontal distance: d = 1 cell = 30m (assume 30m cell size)
Elevation difference: $20 - 20 = 0$ m
Angle: \alpha = \arctan(0/30) = 0°
Visible (no obstruction yet)
Max angle: $0°$
Cell (2,4): 2 cells east
Ground elevation: 22m
Distance: d = 2 \times 30 = 60 m
Elevation difference: $22 - 20 = 2$ m
Angle: \alpha = \arctan(2/60) = 1.91°
Compare to max angle: $1.91° > 0°$ → Visible
Update max angle: $1.91°$
Cell (3,3): 1 cell southeast
Ground elevation: 18m
Distance: d = \sqrt{30^2 + 30^2} = 42.4 m
Elevation difference: $18 - 20 = -2$ m
Angle: \alpha = \arctan(-2/42.4) = -2.70°
Compare to max in SE direction: Assume max was $0°$ (flat)
-2.70° < 0° → Not visible (below horizon)
Continue for all cells…
Result: Cells at same or higher elevation with positive angles are visible. Lower cells may be blocked.
Below is an interactive viewshed simulator.
<label>
Observer height (m):
<input type="range" id="obs-height" min="0" max="20" step="1" value="5">
<span id="obs-height-val">5</span> m
</label>
<label>
Target height (m):
<input type="range" id="target-height" min="0" max="5" step="0.5" value="1.7">
<span id="target-height-val">1.7</span> m
</label>
<label>
Show terrain:
<input type="checkbox" id="show-terrain" checked>
</label>
<label>
Show 3D profile:
<input type="checkbox" id="show-profile">
</label><canvas id="viewshed-canvas" width="700" height="350" style="border: 1px solid #ddd;"></canvas><p><strong>Visible cells:</strong> <span id="visible-count">--</span></p>
<p><strong>Total cells:</strong> <span id="total-cells">--</span></p>
<p><strong>Visibility:</strong> <span id="visibility-pct">--</span>%</p>Try this: - Increase observer height: See more terrain (taller tower = bigger viewshed) - Increase target height: People standing visible farther than ground - Show 3D profile: See elevation cross-section with sightlines - Green areas: Visible from observer (red dot) - Dark areas: Blocked by intervening terrain - Notice: Higher observer = exponentially more visibility
Key insight: Small height changes dramatically affect viewshed—why towers are tall!
Cell tower placement:
1. Compute viewshed from potential tower location
2. Visible area = potential coverage
3. Optimize tower locations to maximize coverage
4. Account for signal attenuation with distanceCoverage gaps = areas not visible from any tower.
Overlap = redundant coverage (good for reliability).
Wind turbine visibility:
For each turbine location:
Compute viewshed to distance 10 km
Count houses/roads in viewshed
Assess visual impact scoreHigh impact: Visible from many populated areas
Low impact: Visible only from remote locations
Optimal fire tower placement:
Goal: Maximize forest visibility with minimum towers
1. Compute viewshed from each candidate location
2. Solve set cover problem:
- Select minimum towers to cover all forest
3. Account for atmospheric visibility limitsTypical: 20-30 km visibility in clear conditions.
Surveillance and targeting:
Inverse viewshed: From which locations is this point visible? (exposure analysis)
For long distances, Earth curves away.
Horizon distance (for observer at height h):
d_{\text{horizon}} = \sqrt{2Rh}
Where R = 6371 km (Earth radius).
Example: Observer at 100m height:
d = \sqrt{2 \times 6371000 \times 100} = 35.7 \text{ km}
Beyond 36 km: Target below horizon even on flat terrain.
Solution: Adjust terrain elevation for curvature:
z_{\text{adjusted}} = z_{\text{DEM}} - \frac{d^2}{2R}
Light bends in atmosphere (increases visible distance).
Effective Earth radius: R_{\text{eff}} = \frac{4}{3}R
Refraction extends horizon ~7%.
Solution: Use adjusted radius in curvature calculations.
Coarse DEM misses small obstructions:
30m DEM: Can’t represent 2m wall.
Solution: - Use high-resolution DEM (1m LiDAR) - Add separate obstruction layer (buildings, vegetation)
Full viewshed for large DEM:
1000×1000 DEM → 1 million line-of-sight tests
Each test: ~100 samples → 100 million operations
Solution: - Limit search radius - Use hierarchical algorithms - GPU acceleration - Approximate methods (sector-based)
Multiple observers: Which areas are visible from ≥ k observers?
Applications: - Surveillance networks: Ensure double coverage - Tourism: Scenic viewpoints (visible from multiple spots) - Privacy: Areas visible from many locations (lack privacy)
Algorithm:
cumulative_viewshed = zeros(rows, cols)
for each observer location:
viewshed = compute_viewshed(observer)
cumulative_viewshed += viewshed
# Result: count of observers that can see each cellQuery: Find cells visible from ≥ 3 observers:
high_visibility = (cumulative_viewshed >= 3)Ray from point O in direction \mathbf{d}:
\mathbf{r}(t) = O + t\mathbf{d}, \quad t \geq 0
Components:
x(t) = x_0 + t d_x y(t) = y_0 + t d_y z(t) = z_0 + t d_z
Terrain surface: z = f(x, y) (DEM function)
Intersection: Solve z(t) = f(x(t), y(t))
For simple case (linear terrain segment):
z_0 + t(z_1 - z_0) = z_{\text{terrain}}
t = \frac{z_{\text{terrain}} - z_0}{z_1 - z_0}
If $0 < t < 1$: Ray intersects terrain → blocked
If t < 0 or t > 1: No intersection → clear
From observer O at height z_0 to target at distance d and height z_1:
\alpha = \arctan\left(\frac{z_1 - z_0}{d}\right)
Positive angle: Target above observer
Negative angle: Target below observer
Zero angle: Same elevation