1. Active Remote Sensing Equation - the basis of RADAR, LIDAR, and SODAR measurements -
Tobias Otto

2. Content the active remote sensing equation
derivation of the radar equation
derivation of the lidar equation
how to apply the active remote sensing equation for
calibration
system performance analysis

3. The Active Remote Sensing Equation
is an analytical expression for the power received by an active remote sensing system, i.e. RADAR, LIDAR or SODAR (RAdio / LIght / SOnic Detection and Ranging)
merges all the knowledge about
the system (relevant system parameters),
the propagation path, and
the targets that are remotely sensed
is frequently applied for active remote sensing instrument:
design and performance analysis,
calibration, conversion of the received power into a meaningful measurement, i.e. an observable that ideally solely depends on the targets itself

4. The Active Remote Sensing Equation
range
active remote system
target
mean received power
active remote sensing
system constant
range dependent
measurement geometry
target
characteristics
(backward-scattering)
transmission term
(attenuation)

5. Content active remote sensing equation
derivation of the radar equation
derivation of the lidar equation
how to apply the active remote sensing equation for
calibration
system performance analysis

6. Radar Equation for a Point Target
transmitter
receiver
antennae
range R
target Pt Gt σ Pr Gr
backscattered power density
at receiving antenna
isotropic
antenna
Pt .. transmitted power (W)
Gt .. antenna gain on transmit
R .. range (m)
σ .. radar cross section (m2)
Gr .. antenna gain on receive
Pr .. received power (W)
power density incident
on the target
effective area / aperture of
the receiving antenna

7. Radar Equation for a Point Target
transmitter
receiver
antennae
range R
target Pt Gt σ Pr Gr
target characteristics
radar constant
Pt .. transmitted power (W)
Gt .. antenna gain on transmit
R .. range (m)
σ .. radar cross section (m2)
Gr .. antenna gain on receive
Pr .. received power (W)
free-space
propagation

8. From Point to Volume Targets
the radar equation for a point target needs to be customised and expanded to fit the needs of each radar application (e.g. moving target indication, synthetic aperture radar, and also meterological radar)
active remote sensing instruments have a limited spatial resolution, they do not observe single targets (raindrops, ice crystals etc.), instead they always measure a volume filled with a lot of targets
 volume target (distributed target) instead of a point target
to account for this, the radar cross section is replaced with the sum of the radar cross sections of all scatterers in the resolution volume V (range-bin):

9. range resolution volume
Radar Resolution θ
antenna beam-width
range R
range resolution volume
(range-bin)
c .. speed of light
B .. bandwidth of the transmitted signal
(the bandwith of a rectangular pulse is the inverse its duration B=1/τ)
 Δr is typically between 3m - 300m, and the antenna beam-width is between 0.5° - 2° for weather radars

10. Range Resolution of a pulsed Active Remote Sensing Instrument

11. Range Resolution of a pulsed Active Remote Sensing Instrument
e.g. pulse duration 1 µs
300 m f0

12. Range Resolution of a pulsed Active Remote Sensing Instrument
each sample consists of the sum of the backscattered signals of a volume with the length c·τ/2
target 1
response
target 2
response
for a pulsed active remote sensing instrument, the optimum sampling rate of the backscattered signal is 2/τ (Hz)
target 3
response
Now we sample
the backscattered signal.

13. Radar Equation for Volume Targets2 r R
θ/2
tan(α) ≈ α (rad) for small α
volume reduction factor due to
Gaussian antenna beam pattern

14. Isotropic Scattering Cross Section σ
Pbackscattered
Sincident
.. backscattered power (W)
.. incident power density (Wm-2)
Depends on:
frequency and polarisation of the electromagnetic wave
scattering geometry / angle
electromagnetic properties of the scatterer
target shape
 hydrometeors can be approximated as spheres

15. Isotropic Scattering Cross Section σ
Monostatic isotropic scattering cross section of a conducting (metallic) sphere: a 
.. radius of the sphere
.. wavelength
Rayleigh region: a << 
normalised radar cross section
Resonance / Mie region:
electrical size
Optical region: a >> 
Figure: D. Pozar, “Microwave Engineering”, 2nd edition, Wiley.

16. Radar Cross Section σ
hydrometeors are small compared to the wavelengths used in weather radar observations: weather radar wavelength 10cm  max. 6mm raindrop diameter
Rayleigh scattering approximation can be applied; radar cross section for dielectric spheres: D 
|K|2
.. hydrometeor diameter
.. radar wavelength
.. dielectric factor depending on the material of the scatterer

17. Radar Equation for Weather Radar
radar constant
radar reflectivity factor z, solely a property of the observed precipitation

18. Radar Reflectivity Factor z
spans over a large range; to compress it into a smaller range of numbers, engineers prefer a logarithmic scale
1 m3
one raindrop
D = 1mm
equivalent to
1mm6m-3 = 0 dBZ
raindrop diameter
#/m3 Z
water volume
per cubic meter
1 mm
4096
36 dBZ
mm3
4 mm 1
33.5 mm3
Knowing the reflectivity alone does not help too much.
It is also important to know the drop size distribution.

19. Raindrop-Size Distribution N(D)
where N(D) is the raindrop-size distribution that tells us how many drops of each diameter D are contained in a unit volume, i.e. 1m3.
Often, the raindrop-size distribution is assumed to be exponential:
concentration (m-3mm-1) slope parameter (mm-1)
Marshall and Palmer (1948):
N0 = 8000 m-3mm-1
Λ = 4.1·R-0.21
with the rainfall rate R (mm/h)

20. Reflectivity – Rainfall Rate Relations
reflectivity (mm6m-3)
liquid water content (mm3m-3)
raindrop volume
rainfall rate (mm h-1)
terminal fall velocity
the reflectivity measured by weather radars can be related to the liquid water content as well as to the rainfall rate:
power-law relationship
the coefficients a and b vary due to changes in the raindrop-size distribution or in the terminal fall velocity.
Often used as a first approximation is a = 200 and b = 1.6

21. Summary of the assumptions in the radar equation
In the derivation of the radar equation for weather radars, the following
assumptions are implied:
the hydrometeors are homogeneously distributed within the range-bin
the hydrometeors are dielectric spheres made up of the same material with diameters small compared to the radar wavelength
multiple scattering among the hydrometeors is negligible
incoherent scattering (hydrometeors exhibit random motion)
the main-lobe of the radar antenna beam pattern can be approximated by a Gaussian function
far-field of the radar antenna, using linear polarisation
so far, we neglected the transmission term (attenuation)

22. Content active remote sensing equation
derivation of the radar equation
derivation of the lidar equation
how to apply the active remote sensing equation for
calibration
system performance analysis

23. Lidar Equation for Volume Targets
laser
laser beam
receiver
receiver field
of view
telescope
area
Pr .. received power (W)
Pt .. transmitted power (W)
AL .. laser beam cross section (m2)
c .. speed of light (ms-1)
τ .. temporal pulse length (s)
R .. range (m)
σ .. isotropic scattering cross section (m2)
A .. area of the primary receiver optics (m2)
backscattered power density
at the telescope
power density incident
on the target
telescope aperture

24. Lidar Equation for Volume Targets
laser
laser beam
receiver
receiver field
of view
telescope
area
Pr .. received power (W)
Pt .. transmitted power (W)
AL .. laser beam cross section (m2)
c .. speed of light (ms-1)
τ .. temporal pulse length (s)
R .. range (m)
σ .. isotropic scattering cross section (m2)
A .. area of the primary receiver optics (m2)
η .. receiver efficiency (how many of the
incoming photons are detected)
O(R) .. receiver-field-of-view overlap function
T(R) .. transmission term (attenuation)

25. Lidar Equation for Volume Targets
laser
laser beam
receiver
receiver field
of view
telescope
area

26. Lidar Equation for Volume Targets
laser
laser beam
receiver
receiver field
of view
telescope
area
differential scattering
cross section (m2sr-1)
number concentration
with the backscatter coefficient β (m-1sr-1):
π indicating scattering
in the backward direction

27. Lidar Equation for Volume Targets
range dependent
measurement geometry
lidar system constant
backscatter
coefficient
transmission term
(attenuation)
Both the backscatter coefficient and the transmission term (attenuation) contain significant contributions from
molecular scattering (gases like oxygen, nitrogen)  Rayleigh scattering
and
particle scattering (liquid and solid air pollution particles such as sulfates, mineral dust, sea-salt, pollen but also larger hydrometeors as rain, ice, hail and graupel)
 resonance or optical scattering
Difficult to differentiate with power measurements only.

28. Summary: Radar and Lidar Equation
target
monostatic, i.e. co-located
transmitter and receiver
active remote system
C active remote sensing system constant
M(R) range dependent measurement geometry
B(R) target characteristics
T(R) transmission term (attenuation)
Radar equation for volume targets
Lidar equation for volume targets

29. Summary: Radar and Lidar Equation
Radar observations of the atmosphere mainly contain contributions from hydrometeors which are Rayleigh scatterers at radar frequencies. This allows the definition of the reflectivity z, a parameter that is
only dependent on the hydrometeor microphysics and independent on the radar wavelength, i.e. the reflectivity within the same radar resolution volume measured by different radars should be equal
Lidar:
Both the backscatter coefficient β and the transmission term T contain significant contributions from
molecular scattering (gases like oxygen, nitrogen)  Rayleigh scattering
and
particle scattering (liquid and solid air pollution particles such as sulfates, mineral dust, sea-salt, pollen but also larger hydrometeors as rain, ice, hail and graupel)
 resonance or optical scattering
Lidar measurements of the atmosphere comprise contributions from all three scattering regimes Rayleigh, resonance and optical scattering  it requires more than a simple power measurement to separate them.
For this reason, lidar measurements are also strongly dependent on the lidar frequency and can not be easily compared to each other.

30. Measurement example from Cabauw, Netherlands
UV-Lidar
Transportable Atmospheric Radar
Uncalibrated attenuated backscatter
Calibrated reflectivity not corrected for propagation effects.
C active remote sensing system constant
M(R) range dependent measurement geometry
B(R) target characteristics
T(R) transmission term
Which terms of the active remote sensing equation contribute the figures of lidar backscatter and radar reflectivity shown above?
data available at

31. Content active remote sensing equation
derivation of the radar equation
derivation of the lidar equation
how to apply the active remote sensing equation for
calibration
system performance analysis

32. Calibration of Active Remote Sensing Measurements
C active remote sensing system constant
M(R) range dependent measurement geometry
B(R) target characteristics
T(R) transmission term (attenuation)
AMS Glossary of Meteorology:
The process whereby the magnitude of the output of a measuring instrument (e.g., the level of mercury in a thermometer or the detected backscatter power of a meteorological radar) is related to the magnitude of the input force (e.g., the temperature or radar reflectivity) actuating that instrument.
For the calibration of a radar / lidar measurement (output: mean received power), we need to know
- the range dependent measurement geometry (range normalisation, easy and accurate)
- the active remote sensing system constant ∙ can be determined analytically using the system specifications, however for an accurate calibration, extensive measurements of the system are needed ∙ because it can vary e.g. due to aging of hardware components, hardware changes it needs to be constantly monitored

33. Content active remote sensing equation
derivation of the radar equation
derivation of the lidar equation
how to apply the active remote sensing equation for
calibration
system performance analysis

34. signal-to-noise ratio
Radar performance
What is the minimum reflectivity detectable by a meteorological radar?
Determined by the minimum received power that can be discerned from the noise floor, i.e. the minimum detectable signal (Pmds).
radar receiver
signal-to-noise ratio
PMDS k T Br
.. minimum detectable signal
.. Boltzmann constant
.. noise temperature
.. receiver bandwidth
radar receiver noise expressed in terms of thermal noise using the Rayleigh-Jeans approximation
which is valid at microwaves (not for lidar!)

35. Radar performance
Result of radar performance calculation of an arbitrary weather radar:
How could we increase the sensitivity?
 reduce the range resolution (B )
 increase transmit power (Pt )
 reduce the noise floor of the system (Pmds )
 reduce the radar wavelength (λ )
If we use a small wavelength (e.g. cloud radar at 35 GHz), we are able to detect very
weak echoes (e.g. fog). Are those radars also suited for the observation of heavy rain?
 attenuation by rain increases with frequency
 radar has a limited dynamic range, i.e. there is a zmin but also a zmax given by the dynamic range of the receiver (a cloud radar receiver can be saturated by heavy precipitation)

36. IDRA reflectivity measurement of insects in summer
Why are there only insects close to the radar, because the radar microwaves are keeping them warm and cosy?
Of course not, insects are weak echoes. The radar can not detect them at far ranges because the echo is from a certain range on below the sensitivity (zmin) of the radar.
data available at

37. Summary
The active remote sensing equation is an expression for the mean received power only.
But beside power (amplitude), electromagnetic waves are also characterised by their frequency, phase and polarisation. Those are the properties that are exploited to gather more independent measurements of the atmosphere in order to separate e.g. transmission from backward-scattering, or for lidar particle from molecular scattering.
Advanced active remote sensing instruments:
 Doppler radar / lidar
 dual-polarisation radar / lidar
 multi-frequency radar / lidar
 Raman lidar, taking advantage of the inelastic / Raman scattering which leads to a change of the molecules quantum state (the energy level), such that the frequency of the scattered photon is shifted
a Raman lidar needs a high average laser power and has additional receiver chanels for the Raman backscatter spectrum of gases such as N2 or H2O

38. Active Remote Sensing Equation - the basis of RADAR, LIDAR, and SODAR measurements -
Tobias Otto
web references R. E. Rinehart, “Radar for Meteorologists”, Rinehart Publications, 5th edition, 2010.
R. J. Doviak and D. S. Zrnić, “Doppler Radar and Weather Observations”, Academic Press, 2nd edition, 1993.
V. N. Bringi and V. Chandrasekar, “Polarimetric Doppler Weather Radar: Principles and Applications”, Cambridge University Press, 1st edition, 2001.
C. Weitkamp, “Lidar: Range-Resolved Optical Remote Sensing of the Atmosphere”, Springer, 2005.