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**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**

range active remote system target mean received power active remote sensing system constant range dependent measurement geometry target characteristics (backward-scattering) transmission term (attenuation)

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**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

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**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

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**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

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**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):

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**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

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**Range Resolution of a pulsed Active Remote Sensing Instrument**

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**Range Resolution of a pulsed Active Remote Sensing Instrument**

e.g. pulse duration 1 µs 300 m f0

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**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.

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**Radar Equation for Volume Targets**

2 r R θ/2 tan(α) ≈ α (rad) for small α volume reduction factor due to Gaussian antenna beam pattern

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**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

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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 << normalised radar cross section Resonance / Mie region: electrical size Optical region: a >> Figure: D. Pozar, “Microwave Engineering”, 2nd edition, Wiley.

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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: 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**

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

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**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.

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**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)

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**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

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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 approximated by 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 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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**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

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**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)

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**Lidar Equation for Volume Targets**

laser laser beam receiver receiver field of view telescope area

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**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

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**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.

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**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

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**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.

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**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

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**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

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**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

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**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

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**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!)

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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)

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**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

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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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**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.

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