The active remote sensing equation

A
T
M
O
S

Active Remote Sensing Equation
- the basis of RADAR, LIDAR, and SODAR measurements -

Tobias Otto

Delft
University of
Technology Remote Sensing of the Environment (RSE)
1

A
T Content
M
O
S

• 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

Delft
University of
2

A
T The Active Remote Sensing Equation
M
O
S

• 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

Delft
University of
3

A
T The Active Remote Sensing Equation
M
O range R
S

active remote system target

Pr C G ( R) B R T R
mean received power

target transmission term
active remote sensing
characteristics (attenuation)
system constant
(backward-scattering)

range dependent
measurement geometry

Delft
University of
4

A
T Content
M
O
S

• active remote sensing equation



• calibration

Delft
University of
5

A
T Radar Equation for a Point Target
M
O
S antennae range R

Pt
transmitter Gt
target
σ
Pr
receiver Gr
backscattered power density
at receiving antenna

isotropic
antenna
Pt .. transmitted power (W)
Gt .. antenna gain on transmit Pt 1 Gr 2
R .. range (m) Pr Gt
σ .. radar cross section (m2) 4 R 2
4 R2 4
Gr .. antenna gain on receive
Pr .. received power (W)
power density incident effective area / aperture of
on the target the receiving antenna

Delft
University of
6

A
T Radar Equation for a Point Target
M
O
S antennae range R

Pt
transmitter Gt
target
σ
Pr
receiver Gr

radar constant target characteristics
Gt .. antenna gain on transmit 2
R .. range (m) Pt Gt Gr
σ .. radar cross section (m2) Pr 3 4
Gr
Pr
..
..
antenna gain on receive
received power (W)
4 R free-space
propagation

Delft
University of
7

A
T From Point to Volume Targets
M
O
S

- 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):

i V i
range bin unit volume

Delft
University of
8

A
T Radar Resolution
M
O
S

antenna beam-width

θ

range resolution volume
c
range R r
(range-bin) 2B
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
Delft
University of
9

A
T Range Resolution of a pulsed Active Remote Sensing Instrument
M
O
S

1 2 3

e.g. pulse duration 1 µs
c m
300 m

f0

Delft
University of
10

A
T Range Resolution of a pulsed Active Remote Sensing Instrument
M
O
S

1 2 3

c
2
• each target 1 consists of the sum of the
sample
backscattered signals of a volume with
response
the length c·τ/2
target 2
• for a pulsed active remote sensing instrument,
response
the optimum sampling rate of the
backscattered signal is 2/τ (Hz)
target 3
response

Now we sample
the backscattered signal.

Delft
University of
11

A
T Radar Equation for Volume Targets
M
O
S 2
PGt Gr
t
Pr 3 4
V i
4 R unit volume

2 c
V r
2B
2
c
r V R tan
θ/2
2 2B
R tan(α) ≈ α (rad) for small α
c 2
r rad c 1
2B 2
V R
4 2 B 2 ln 2
volume reduction factor due to
Gaussian antenna beam pattern
Delft
University of
12

A
T Radar Equation for Volume Targets
M
O
S 2
PGt Gr
t R2 2c
Pr 3 42 i
4 R 16 ln 2 B unit volume

2 c
V r
2B
2
c
r V R tan
θ/2
2 2B
R tan(α) ≈ α (rad) for small α
c 2
r rad c 1
2B 2
V R
4 2 B 2 ln 2
volume reduction factor due to
Gaussian antenna beam pattern
Delft
University of
13

A
T Isotropic Scattering Cross Section σ
M
O
S

Pbackscattered 2
(m )
Sincident
Pbackscattered .. backscattered power (W)
Sincident .. 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

Delft
University of
14

A
T Isotropic Scattering Cross Section σ
M
O
S Monostatic isotropic scattering cross section of a conducting (metallic) sphere:

a .. radius of the sphere
normalised radar cross section

.. wavelength

Rayleigh region: a <<

Resonance / Mie region:

electrical size
Optical region: a >>

Figure: D. Pozar, “Microwave Engineering”, 2nd edition, Wiley.
Delft
University of
15

A
T Radar Cross Section σ
M
O
S

 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:

5 2
K Di6 D .. hydrometeor diameter
.. radar wavelength
i 4 |K|2 .. dielectric factor depending on the
material of the scatterer

Delft
University of
16

A
T Radar Equation for Weather Radar
M
O
S
2 2 2 5 2
PGt Gr R c K
Pr t
3 4 4
Di6
4 R 16 ln 2 B unit volume

3
PGt Gr 2 c
t 2 6 1
Pr 2
K Di
1024ln 2 B unit volume R2

radar constant radar reflectivity factor z, solely a
property of the observed precipitation

Delft
University of
17

A
T Radar Reflectivity Factor z
M
O
S
6 mm6  spans over a large range; to compress it into a smaller
z i D range of numbers, engineers prefer a logarithmic scale
unit volume m3

1 m3
z one raindrop equivalent to
Z 10log10 dBZ
1mm6 m 3
D = 1mm 1mm6m-3 = 0 dBZ

raindrop diameter #/m3 Z water volume
per cubic meter
1 mm 4096 36 dBZ 2144.6 mm3
4 mm 1 36 dBZ 33.5 mm3

Knowing the reflectivity alone does not help too much.
It is also important to know the drop size distribution.

Delft
University of
18

A
T Raindrop-Size Distribution N(D)
M
O
S 6 6 N0
z D
i D N ( D)dD 7
6!
unit volume 0
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:

N D N 0 exp D
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)

Delft
University of
19

A
T Reflectivity – Rainfall Rate Relations
M
O
S
reflectivity (mm6m-3) z D 6 N ( D)dD
D

liquid water content (mm3m-3) LWC D 3 N ( D)dD
6 D raindrop volume

rainfall rate (mm h-1) R D 3v( D ) N ( D )dD
6 D
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 z aRb
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
Delft
University of
20

A
T Summary of the assumptions in the radar equation
M
O
S

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)

Delft
University of
21

A
T Content
M
O
S




• calibration

Delft
University of
22

A
T Lidar Equation for Volume Targets
M
O
S R
laser

AL c receiver field
receiver of view
2
telescope
area
O(R) 0 0 O( R) 1 O( R) 1
Pr .. received power (W)
AL .. laser beam cross section (m2)
c .. speed of light (ms-1)
τ .. temporal pulse length (s)
R .. range (m)
Pt 1
Pr i A O( R ) T ( R )
σ .. isotropic scattering cross section (m2) AL 4 R 2
i
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)
Delft
University of
23

A
M
O R
S laser

AL c receiver field
receiver of view
2
telescope
area

Pt 1
Pr i 2
A O( R ) T ( R )
AL i 4 R
c
i V i AL i
i unit 2 unit
volume volume

c 1
Pr Pt i 2
A O( R) T ( R)
2 unit 4 R
volume

Delft
University of
24

A
M
O R
S laser

AL c receiver field
receiver of view
2
telescope
area
differential scattering
c 1 cross section (m2sr-1)
Pr Pt i 2
A O( R) T ( R)
2 unit 4 R
volume number concentration

d i , sca
i 4 with the backscatter coefficient β (m-1sr-1): ( R, ) Ni ( R) ( , R)
unit unit d
volume volume

π indicating scattering
c 1 in the backward direction
Pr Pt 2
A O( R) T ( R)
2 R

Delft
University of
25

A
M
O
S c O( R)
Pr Pt A T ( R)
2 R2

backscatter transmission term
coefficient (attenuation)

lidar system constant range dependent
measurement geometry

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.

Delft
University of
26

A
T Summary: Radar and Lidar Equation
M
O R
S monostatic, i.e. co-located
active remote system target transmitter and receiver

C active remote sensing system constant
M(R) range dependent measurement geometry
Pr C G ( R) B R T R B(R) target characteristics
T(R) transmission term (attenuation)

Radar equation for volume targets
2 2
t t rPG G c 1
P
r i T ( R)
1024 ln 2 Bt R 2
2
unit volume

Lidar equation for volume targets
c O( R) 1
Pr Pt A ( R) T ( R) (m 1sr 1 ) i
2 R2 4 unit
volume

Delft
University of
27

A
T Summary: Radar and Lidar Equation
M
O
S
Radar:
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.

Delft
University of
28

A
T Measurement example from Cabauw, Netherlands
M
O UV-Lidar Transportable Atmospheric Radar
S Uncalibrated attenuated backscatter Calibrated reflectivity not corrected for propagation effects.

C active remote sensing system constant
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 http://www.cesar-database.nl
Delft
University of
29

A
T Content
M
O
S




• calibration

Delft
University of
30

A
T Calibration of Active Remote Sensing Measurements
M
O
S C active remote sensing system constant
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
Pr
B R T R
C G ( R)
Delft
University of
31

A
T Content
M
O
S




• calibration

Delft
University of
32

A
T Radar performance
M
O
S What is the minimum reflectivity detectable by a meteorological radar?
2
1024 ln 2 B
zmin
z 3 2 2
R 2 Pmds
r
PGt Gr
t cK
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 .. minimum detectable signal
S k .. Boltzmann constant
PMDS kTBr
N T .. noise temperature
min
Br .. receiver bandwidth

radar receiver noise expressed in terms of thermal
noise using the Rayleigh-Jeans approximation
which is valid at microwaves (not for lidar!)
Delft
University of
33

A
T Radar performance
M
O
S Result of radar performance calculation of an arbitrary weather radar:

2
1024 ln 2 B
zmin 3 2 2
R 2 Pmds
PGt Gr
t cK

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 is saturated in heavy precipitation)

Delft
University of
34

A
T IDRA reflectivity measurement of insects in summer
M
O Why are there only insects close to the radar, because the
S 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 http://www.cesar-database.nl
Delft
University of
35

A
T Summary
M
O
S 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

Delft
University of
36

A
T
M
O
S

Active Remote Sensing Equation
- the basis of RADAR, LIDAR, and SODAR measurements -

Tobias Otto

e-mail t.otto@tudelft.nl
web http://atmos.weblog.tudelft.nl

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.

Delft
University of
37

The active remote sensing equation

Recommandé

Recommandé

Contenu connexe

Tendances

Tendances (20)

En vedette

En vedette (20)

Similaire à The active remote sensing equation

Similaire à The active remote sensing equation (20)

Dernier

Dernier (20)

The active remote sensing equation