SME
Remote Sensing
Active remote sensing (radar and lidar)
Use this for “A lidar pulse returns after 6.67 μs from the canopy top; the ground return comes 200 ns later. Find the range and the tree height.”
This is an example. Type your own values to start a new problem.
R (m)
=
299.8
m/μs
×
t (μs)
÷
2
h (m)
=
0.2998
m/ns
×
Δt (ns)
÷
2
Rg
Range to the ground
worked out (not typed)
τ
Pulse length
example · ns
ΔR
Range resolution
worked out (not typed) · m
Formulas
R
=
299.8
×
t
2
999.833
=
299.8
×
6.67
2
h
=
0.2998
×
Δt
2
29.98
=
0.2998
×
200
2
Rg
=
R
+
h
1,029.813 = 999.833 + 29.98
ΔR
=
0.2998
×
τ
2
0.7495
=
0.2998
×
5
2
Assumptions
•
The pulse goes out and comes back, so it covers the distance twice: halve the path to get the range.
•
Light travels at c = 2.998 × 10⁸ m/s in air, which is 299.8 m each microsecond or 0.2998 m each nanosecond.
•
The shot is nadir (straight down), so the canopy top and the ground under it lie on one line.
•
Two surfaces closer than the range resolution ΔR = cτ ÷ 2 send back echoes that overlap, so they cannot be told apart.
Step-by-step
WE KNOW
t = 6.67 μs, Δt = 200 ns, τ = 5 ns
FIND
range (R), canopy height (h), range to the ground (Rg), range resolution (ΔR)
Step 1 · Find range (R)
Use R = 299.8 × t ÷ 2
Light covers 299.8 m each microsecond; the pulse went down and back, so the range is half that path.
R
=
299.8
×
6.67
2
R
=
1,999.666
2
R
=
999.833
m
Step 2 · Find canopy height (h)
Use h = 0.2998 × Δt ÷ 2
The ground echo travels the canopy’s height down and back again, so the gap times 0.2998 m each nanosecond, halved, is the height.
h
=
0.2998
×
200
2
h
=
59.96
2
h
=
29.98
m
Step 3 · Find range to the ground (Rg)
Use Rg = R + h
The ground lies one canopy height beyond the canopy top on the same nadir line.
Rg = 999.833 + 29.98
Rg = 1,029.813 m
Step 4 · Find range resolution (ΔR)
Use ΔR = 0.2998 × τ ÷ 2
Echoes from two surfaces closer than half the pulse’s length in space overlap, so this is the smallest height step the lidar can split.
ΔR
=
0.2998
×
5
2
ΔR
=
1.499
2
ΔR = 0.7495 m
Check
Put the numbers back in. Both sides should match.
999.833
=
299.8
×
6.67
2
✓
29.98
=
0.2998
×
200
2
✓
1,029.813 = 999.833 + 29.98 ✓
0.7495
=
0.2998
×
5
2
✓
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