modelling Level 4

Avalanche Terrain Analysis

Which slopes are dangerous? Where should backcountry routes go? Avalanche terrain analysis combines slope angle, aspect, elevation, and terrain shape to identify avalanche-prone areas. This model derives slope thresholds, implements terrain classification systems (ATES), and shows how to map runout zones using DEM analysis.

Prerequisites: slope analysis, terrain classification, hazard modelling, probability

27 Feb 2026 Updated 12 Mar 2026 13 min read

1. The Question

Is this slope safe to ski, or will it avalanche?

Avalanche terrain analysis identifies hazardous areas based on:

Terrain factors (unchanging):

Slope angle (30-45° = avalanche terrain)
Aspect (wind loading, solar exposure)
Elevation (snowpack depth/stability varies)
Terrain shape (convex vs. concave)
Anchors (trees, rocks that hold snow)

Snowpack factors (time-varying):

Recent snowfall (loading)
Wind (slab formation)
Temperature (weak layer formation)
Layer structure (buried weak layers)

Human factors:

Route choice
Group management
Decision-making

The mathematical question: Given a DEM and snowpack conditions, how do we classify terrain by avalanche risk and predict where avalanches might run?

2. The Conceptual Model

Critical Slope Angles

Avalanche release zones:

\[\theta_{\text{start}} = 30-45°\]

Why this range?

Below 30°: Snow doesn’t slide (friction too high)
30-35°: Common for wet loose avalanches
35-40°: Peak slab avalanche frequency
40-45°: Still avalanche, but less snow accumulates (sloughs off)
Above 50°: Too steep to hold deep snowpack

Most dangerous: 38° (statistically most fatal avalanches)

Avalanche Types

1. Loose Snow Avalanche (Sluff)

Point release, fan-shaped
Surface snow only
Low danger unless terrain trap

2. Slab Avalanche

Cohesive layer breaks as unit
Wide fracture line
Most deadly (accounts for 90% of fatalities)
Requires weak layer beneath slab

3. Wet Avalanche

Warm temperatures or rain
Full-depth release (to ground)
Heavy, destructive

ATES Classification

Avalanche Terrain Exposure Scale:

Simple:

Primarily < 30° slopes
Some avalanche terrain nearby
Multiple route options
Risk: Low

Challenging:

30-35° slopes common
Exposure to avalanche paths
Limited route options
Risk: Moderate

Complex:

Multiple 35-45° slopes
Overhead hazard (terrain traps below steep slopes)
Few safe zones
Risk: High

3. Building the Mathematical Model

Slope Angle from DEM

Gradient magnitude (from Model 8):

\[|\nabla z| = \sqrt{\left(\frac{\partial z}{\partial x}\right)^2 + \left(\frac{\partial z}{\partial y}\right)^2}\]

Slope angle:

\[\theta = \arctan(|\nabla z|)\]

In degrees:

\[\theta = \arctan(|\nabla z|) \times \frac{180}{\pi}\]

Finite difference (3×3 window):

\[\frac{\partial z}{\partial x} \approx \frac{(z_{i,j+1} - z_{i,j-1})}{2\Delta x}\] \[\frac{\partial z}{\partial y} \approx \frac{(z_{i+1,j} - z_{i-1,j})}{2\Delta y}\]

Avalanche Hazard Classification

Based on slope angle:

\[\text{Hazard} = \begin{cases} \text{Non-avalanche} & \theta < 30° \\ \text{Low hazard} & 30° \leq \theta < 35° \\ \text{Moderate hazard} & 35° \leq \theta < 40° \\ \text{High hazard} & 40° \leq \theta < 45° \\ \text{Extreme steep} & \theta \geq 45° \end{cases}\]

Alpha Angle (Runout)

Maximum runout distance:

Alpha angle from avalanche start to stop:

\[\alpha = \arctan\left(\frac{z_{\text{start}} - z_{\text{stop}}}{d_{\text{horizontal}}}\right)\]

Empirical:

Small avalanches: $\alpha \approx 28-30°$
Medium avalanches: $\alpha \approx 25-27°$
Large avalanches: $\alpha \approx 18-22°$

Runout zone: All terrain downslope from start zone where $\alpha > \alpha_{\text{threshold}}$

Example: Avalanche starts at 3000m elevation

Alpha = 25°
Stop when: $\arctan(\Delta z / d) = 25°$
If horizontal distance 1000m: $\Delta z = 1000 \times \tan(25°) = 466$ m
Stops at: 3000 - 466 = 2534m elevation

Terrain Traps

Features that increase consequence:

Gullies/Couloirs:

Funneling effect (deep debris)
Burial depth increases
Hard to escape

Cliffs:

Trauma risk
Carried over cliff by avalanche

Trees (dense):

Trauma from impact
Difficult rescue

Flat areas below steep slopes:

Appears safe but runout zone
“Terrain trap”

Risk multiplication:

\[R_{\text{total}} = P_{\text{avalanche}} \times C_{\text{terrain trap}}\]

Where $C > 1$ for terrain traps (amplifies consequence).

4. Worked Example by Hand

Problem: Classify avalanche terrain for ski route planning.

DEM elevations (meters, 30m cell size):

      j=0   j=1   j=2   j=3   j=4
i=0  2900  2880  2850  2810  2760
i=1  2920  2900  2870  2820  2770
i=2  2940  2920  2890  2840  2790
i=3  2960  2940  2910  2860  2810

Calculate slope angle at cell (1,1).

Solution

Step 1: Finite differences

At (1,1):

\[\frac{\partial z}{\partial x} \approx \frac{z[1,2] - z[1,0]}{2 \times 30} = \frac{2870 - 2920}{60} = \frac{-50}{60} = -0.833\] \[\frac{\partial z}{\partial y} \approx \frac{z[2,1] - z[0,1]}{2 \times 30} = \frac{2920 - 2880}{60} = \frac{40}{60} = 0.667\]

Step 2: Gradient magnitude

\[|\nabla z| = \sqrt{(-0.833)^2 + (0.667)^2} = \sqrt{0.694 + 0.445} = \sqrt{1.139} = 1.067\]

Step 3: Slope angle

\[\theta = \arctan(1.067) = 46.9° \times \frac{180}{\pi} = 46.9°\]

Wait, that’s too high. Recalculate:

\[\theta = \arctan(1.067) \text{ radians} = 0.818 \text{ rad}\] \[\theta = 0.818 \times \frac{180}{\pi} = 46.9°\]

Actually this is correct! The slope is 46.9° - extremely steep, minimal snow accumulation.

Step 4: Classification

$\theta = 46.9° > 45°$ → Extreme steep

Interpretation: This slope is:

Too steep for most slab avalanches (snow sloughs off)
Potential loose snow avalanches
Not prime avalanche terrain (not enough snow accumulates)
But dangerous for climbing/skiing (rockfall, sluffs)

Step 5: Calculate for (2,2) as well

\[\frac{\partial z}{\partial x} = \frac{2840 - 2940}{60} = -1.667\] \[\frac{\partial z}{\partial y} = \frac{2910 - 2870}{60} = 0.667\] \[|\nabla z| = \sqrt{2.778 + 0.445} = 1.796\] \[\theta = \arctan(1.796) = 60.9°\]

Extremely steep cliff!

5. Computational Implementation

Below is an interactive avalanche terrain classifier.

Terrain complexity: Show hazard zones only: Show aspect:

Hazard distribution:

Non-avalanche: --%

Avalanche terrain: --%

Extreme steep: --%

Colors: ■ Safe (<30°) ■ Low (30-35°) ■ Moderate (35-40°) ■ High (40-45°) ■ Extreme (>45°)

Try this:

Simple terrain: Mostly green (safe slopes)
Moderate terrain: Mix of safe and hazard zones
Complex alpine: More red/purple (steep avalanche terrain)
Show hazard only: Gray out safe terrain, highlight danger zones
Show aspect: Right panel shows slope direction (N=blue, S=red, E=yellow)
Color code: Green=safe, Yellow/Orange/Red=avalanche terrain, Purple=extreme
Notice: Most avalanche terrain clusters in gullies and ridge flanks!

Key insight: Terrain classification enables route planning—stay in green zones, minimize time in red zones, avoid terrain traps!

6. Interpretation

Route Planning

Decision matrix:

Terrain	Hazard Level	Action
Simple (<30°)	Low	Safe travel
Challenging (30-35°)	Moderate	Assess snowpack, spacing
Complex (35-45°)	High	Expert only, stability tests
Extreme (>45°)	Variable	Sluff risk, limited accumulation

Spacing:

Simple: Group together
Challenging: 50m spacing
Complex: One at a time, safe zones

Aspect Considerations

North aspects (0-45°, 315-360°):

Cold, persistent weak layers
Longer avalanche season
Higher danger in cold climates

South aspects (135-225°):

Warm, quicker stabilization
Wet avalanches in spring
Lower danger in winter, higher in spring

East/West (45-135°, 225-315°):

Moderate
Wind effects important

Lee slopes (downwind):

Wind-loaded slabs
Very dangerous

Avalanche Bulletin Integration

Danger scale:

Low
Moderate
Considerable
High
Extreme

Terrain selection by danger:

Danger	Terrain
Low	All terrain OK (with normal caution)
Moderate	Avoid wind-loaded slopes >35°
Considerable	Simple terrain only
High	Avoid all avalanche terrain
Extreme	Stay home

7. What Could Go Wrong?

DEM Resolution Issues

Coarse DEM (30m):

Misses small gullies (terrain traps)
Smooths cliffs
Underestimates slope on convex features

Example: 30m DEM shows 35° slope, reality is 42° rollover.

Solution: Use highest resolution DEM (1-3m LiDAR ideal).

Wind Effects Not Captured

Terrain alone insufficient:

Wind loads lee slopes with thick slabs.

Example:

West aspect, 38° slope
Westerly winds → not wind-loaded (windward)
Lower danger than expected from slope alone

vs:

East aspect, 38° slope
Westerly winds → heavily wind-loaded (leeward)
Higher danger than expected

Solution: Wind models, weather data, field observation.

Human Factor

Most avalanche fatalities:

Victim or group member triggered avalanche (90%+).

Terrain selection just first step.

Also need:

Snowpack assessment
Decision-making skills
Communication
Rescue skills

Terrain analysis ≠ safety guarantee.

False Sense of Security

Green zones below steep slopes = terrain traps.

Example:

Flat bench at 2500m (10° slope, “safe”)
Above: 600m vertical of 38° terrain
Runout zone extends to bench
Extremely dangerous despite being “safe” slope angle

Solution: Alpha angle runout modelling.

8. Extension: Avalanche.ca Terrain Ratings

Canadian system:

Simple (Green):

Exposure to low-angle or primarily forested terrain
Some forest openings may involve steeper terrain
Many options for route finding with overhead hazard minimal
Avalanche terrain is avoidable

Challenging (Blue):

Exposure to well-defined avalanche paths, starting zones, or terrain traps
Options exist to reduce or eliminate exposure with careful route finding
Requires knowledge of avalanche terrain and winter travel skills

Complex (Black):

Exposure to multiple overlapping avalanche paths or large expanses of steep, open terrain
Multiple avalanche starting zones and terrain traps below
Minimal options to reduce exposure
Requires extensive avalanche knowledge and winter travel skills

Implementation: Requires expert judgment + terrain analysis.

9. Math Refresher: Slope Stability

Mohr-Coulomb Failure

Shear stress vs. shear strength:

\[\tau = c + \sigma \tan\phi\]

Where:

$\tau$ = shear strength
$c$ = cohesion
$\sigma$ = normal stress
$\phi$ = internal friction angle

On a slope:

Driving stress (downslope):

\[\tau_d = \rho g h \sin\theta\]

Resisting stress:

\[\tau_r = c + \rho g h \cos\theta \tan\phi\]

Failure when: $\tau_d > \tau_r$

Factor of safety:

\[FS = \frac{\tau_r}{\tau_d} = \frac{c + \rho g h \cos\theta \tan\phi}{\rho g h \sin\theta}\]

Stable: $FS > 1$
Failure: $FS < 1$

Critical Slope Angle

For cohesionless material ($c = 0$):

\[FS = \frac{\tan\phi}{\tan\theta}\]

Failure when: $\theta > \phi$

Dry snow: $\phi \approx 30-40°$ → slopes >40° unstable
Wet snow: $\phi \approx 20-30°$ → slopes >30° unstable

This explains why avalanches occur on 30-45° slopes!

Summary

Avalanche terrain: 30-45° slopes most dangerous (slab avalanches)
ATES classification: Simple/Challenging/Complex based on exposure and options
Slope angle from DEM: Calculate gradient, convert to degrees via arctan
Aspect matters: North (cold, persistent), South (warm, wet avalanches)
Alpha angle: Predicts runout distance (typically 18-30° from start to stop)
Terrain traps: Gullies, cliffs, trees amplify consequences
Route selection: Minimize time in hazard zones, use safe travel protocols
Limitations: DEM resolution, wind effects, human factors all critical
Applications: Backcountry skiing, snowmobiling, forecasting, infrastructure
Multi-factor: Terrain + snowpack + weather + human = avalanche risk
Foundation for avalanche safety and terrain-based decision making