The active remote sensing equation

Remote Sensing of the Environment (RSE)

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DelftUniversity ofTechnology 1

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

Tobias Otto


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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


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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


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The Active Remote Sensing Equation

active remote system target

( )rP R R R C G B T

active remote sensingsystem constant

range dependentmeasurement geometry

targetcharacteristics

(backward-scattering)

transmission term(attenuation)

Rrange

mean received power


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Content

• active remote sensing equation





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Radar Equation for a Point Target

Pt

Pr

Gt

Gr

transmitter

receiver

antennae range R

targetσ

Pt .. transmitted power (W)Gt .. antenna gain on transmitR .. range (m)σ .. radar cross section (m2)Gr .. antenna gain on receivePr .. received power (W)

44

1

4

2

22r

tt

r

G

RG

R

PP

backscattered power densityat receiving antenna

isotropicantenna

power density incidenton the target

effective area / aperture ofthe receiving antenna


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Radar Equation for a Point Target

Pt

Pr

Gt

Gr

transmitter

receiver

antennae range R

targetσ

Pt .. transmitted power (W)Gt .. antenna gain on transmitR .. range (m)σ .. radar cross section (m2)Gr .. antenna gain on receivePr .. received power (W)

43

2

4

R

GGPP rttr

radar constant target characteristics

free-spacepropagation


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From Point to Volume Targets

- the radar equation for a point target needs to be customised and expandedto 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):

eunit volumbinrange

ii V


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Radar Resolution

range Rrange resolution volume

(range-bin) B

cr

2

c .. speed of lightB .. 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

θ

antenna beam-width


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mc

Range Resolution of a pulsed Active Remote Sensing Instrument

e.g. pulse duration 1 µs300 m

1 2 3

f0


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Range Resolution of a pulsed Active Remote Sensing Instrument

1 2 3

target 1response

target 2response

target 3response

Now we samplethe backscattered signal.

2

c

• each sample consists of the sum of the backscattered signals of a volume withthe length c·τ/2

• for a pulsed active remote sensing instrument, the optimum sampling rate of the backscattered signal is 2/τ (Hz)


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Radar Equation for Volume Targets

2

3 4unit volume

4

t t rr i

PGGP V

R

B

cr

2

R

θ/2

B

crV

22

r B

cRV

22tan

2

tan(α) ≈ α (rad) for small α

2ln2

1

24

rad 22

B

cRV

volume reduction factor due toGaussian antenna beam pattern


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2 2 2

3 4unit volume

16 ln 24t t r

r i

PGG R cP

BR

Radar Equation for Volume Targets

B

cr

2

R

θ/2

B

crV

22

r B

cRV

22tan

2

tan(α) ≈ α (rad) for small α

2ln2

1

24

rad 22

B

cRV

2

volume reduction factor due toGaussian antenna beam pattern


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Isotropic Scattering Cross Section σ

)m( 2

incident

redbackscatte

S

P

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


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Isotropic Scattering Cross Section σ

Monostatic isotropic scattering cross section of a conducting (metallic) sphere:

a

.. radius of the sphere.. wavelength

Rayleigh region: a <<

Resonance / Mie region:

Optical region: a >>

Figure: D. Pozar, “Microwave Engineering”, 2nd edition, Wiley.

norm

alis

ed r

adar

cross

sect

ion

electrical size


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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:

4

625

ii

DK

D|K|2

.. hydrometeor diameter

.. radar wavelength

.. dielectric factor depending on the material of the scatterer


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Radar Equation for Weather Radar

252 2 26

3 44unit volume

16 ln 24t t r

r i

KPGG R cP D

BR

3 22 6

2 2unit volume

1

1024ln 2t t r

r i

PGG cP K D

B R

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


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Radar Reflectivity Factor z

10 6 310log dBZ

1mm m

zZ

3

6

eunit volum

6

m

mmiDz 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 volumeper 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.


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Raindrop-Size Distribution N(D)

!6)(70

0

6

eunit volum

6

N

dDDNDDz i

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:

DNDN exp0

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)


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Reflectivity – Rainfall Rate Relations

D

dDDNDz )(6reflectivity (mm6m-3)

D

dDDND )(6

LWC 3liquid water content (mm3m-3)

D

dDDNDvDR )()(6

3rainfall rate (mm h-1)

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

terminal fall velocity

baRz

raindrop volume


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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 approximatedby a Gaussian function

• far-field of the radar antenna, using linear polarisation

• so far, we neglected the transmission term (attenuation)


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Content






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Lidar Equation for Volume Targets

laserlaser beam

receiver fieldof view

telescopearea

2

c

R

receiver LA

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 functionT(R) .. transmission term (attenuation)

2

1( ) ( )

4t

r iiL

PP A O R T R

A R

0)( RO 0 ( ) 1O R ( ) 1O R


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2

1( ) ( )

4t

r iiL

PP A O R T R

A R

laserlaser beam


telescopearea

2

c

R

receiver LA

unit unitvolume volume

2i i L ii

cV A

2unit

volume

1( ) ( )

2 4r t i

cP P A O R T R

R


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laserlaser beam


telescopearea

2

c

R

receiver LA

2unit

volume

1( ) ( )

2 4r t i

cP P A O R T R

R

unitvolume

4i with the backscatter coefficient β (m-1sr-1):,

unitvolume

( , ) ( ) ( , )i scai

dR N R R

d

2

1( ) ( )

2r t

cP P A O R T R

R

number concentration

differential scatteringcross section (m2sr-1)

π indicating scatteringin the backward direction


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2

( )( )

2r t

c O RP P A T R

R

backscattercoefficient

transmission term(attenuation)

lidar system constant range dependentmeasurement 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 scatteringDifficult to differentiate with power measurements only.


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( )rP R R R C G B T

Summary: Radar and Lidar Equation

active remote system target

C active remote sensing system constant

M(R) range dependent measurement geometry

B(R) target characteristics

T(R) transmission term (attenuation)

R

monostatic, i.e. co-locatedtransmitter and receiver

Radar equation for volume targets

Lidar equation for volume targets

2 2

2 2unit volume

1 ( )

1024 ln 2t t r

r it

PGG cP T R

B R

2

( )( ) ( )

2r t

c O RP P A R T R

R

1 1

unitvolume

1 ( )

4 im sr


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Summary: Radar and Lidar Equation

Radar:

Radar observations of the atmosphere mainly contain contributions from hydrometeors which areRayleigh 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 radarsshould 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 beeasily compared to each other.


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Measurement example from Cabauw, Netherlands

Uncalibrated attenuated backscatter

UV-Lidar Transportable Atmospheric RadarCalibrated reflectivity not corrected for propagation effects.

Which terms of the active remote sensing equation contribute the figures oflidar backscatter and radar reflectivity shown above?

( )rP R R R C G B TC active remote sensing system constant



T(R) transmission termdata available at http://www.cesar-database.nl


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Content




• how to apply the active remote sensing equation for• calibration• system performance analysis


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Calibration of Active Remote Sensing Measurements

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, howeverfor 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

( )rP R R R C G B TC active remote sensing system constant



T(R) transmission term (attenuation)

( )rPR RR

B TC G


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Content




• how to apply the active remote sensing equation for• calibration• system performance analysis


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22

min 23 2

1024ln 2

mds

t t r

Bz R P

PGG c K

Radar performance

22

23 2

1024ln 2

r

t t r

Bz R P

PGG c K

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).

minMDS r

SP kTB

N

PMDS

kTBr

.. minimum detectable signal.. Boltzmann constant.. noise temperature.. receiver bandwidth

radar receiversignal-to-noise ratio

radar receiver noise expressed in terms of thermal noise using the Rayleigh-Jeans approximationwhich is valid at microwaves (not for lidar!)


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Radar performance

Result of radar performance calculation of an arbitrary weather radar:

22

min 23 2

1024ln 2

mds

t t r

Bz R P

PGG c K

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)


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IDRA reflectivity measurement of insects in summer

Why are there only insects close to the radar, because theradar 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


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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


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e-mail [email protected] 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.

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

Tobias Otto