Radar Meteorology

Theoretical work (Mie scattering theory) in the late 1940s showed that “weather clutter” arose from the scattering of electromagnetic radiation by precipitation particles (resonant interaction between propagating EM wave and a dielectric such as water and ice).

Today modern radars can not only detect hydrometeors (both precipitation and cloud particles), but “clear air” targets such as insects and large aerosol particles, as well as changes in the index of refraction, the latter caused by turbulent motions in the atmosphere.

RADAR-Radio Detection and Ranging

Radar is the “art of detecting by means of radio echoes the presence of objects, determining their direction and range, recognizing their characteristics and employing the data thus obtained”.

“Object” refers to meteorological targets such as raindrops, hailstones, cloud ice and liquid particles and snowflakes. For the purpose of clear air detection, insects are considered the “objects”. Birds also are readily detected and hence are of interest.

Radar is based on the propagation of electromagnetic waves through the atmosphere, a non-vacuum. EM waves propagate at the speed of light in a vacuum, c = 2.998 x 108 m s-1.

Propagation speed in a non-vacuum determines the index of refraction, n = c/νwhere ν is the wave speed (Note : water and ice have different refractive index)

Electromagnetic Waves and Their Propagation Through the Atmosphere

Electromagnetic Waves are characterized by:

Wavelength, λ [m, cm, mm, m etc]

Frequency, ν [s-1, hertz (hz), megahertz (Mhz), gigahertz (Ghz)

where: c = λν

Polarization of electromagnetic waves

The polarization is specified by the orientation of the electromagnetic field.

The plane containing the electric field is called the plane of polarization.

ftEE xmx 2cosFor a monochromatic wave:

ftEE ymy 2cos

Electric field will oscillate in the x,y plane with z as the propagation direction

where f is the frequency and is the phase difference between Exm and Eym and the coordinate x is parallel to the horizon, y normal to x, and z in the direction of propagation.

If Eym = 0, Electric field oscillates in the x directionand wave is said to be “horizontally polarized”

If Exm = 0, Electric field oscillates in the y directionand wave is said to be “vertically polarized”

If Exm = Eym, and = or - electric field vector rotates in a circle and wave is circularly polarized

All other situations: E field rotates as an ellipse

How does radar scan ?

Ground/ship radar

VCP 31“clear air mode”

VCP 11“severe weather mode”

VCP 21Wide-spread precip

Scanning strategies for scanning radars must take into account the propagation path of the beam if certain operational or scientific objectives are to be addressed. Here, 3 common NWS NEXRAD Volume Coverage Patters (VCPs) are illustrated. NEXRADs have a 5-6 minute scan update requirement for severe weather detection, so they vary their VCPs and scan rates depending on the weather situation.

6 min update, slow scan rate

5 min update, fast scan rate

6 min update, slow scan rate

Airborne

Commercialairplanes

Airborne

Research airplanes

Space borne

What kind of electromagnetic pulse do we send?

PULSEElectric

FieldSidelobes

DuplexerKlystronAmplifier

Pulsemodulator

STALOMicrowaveOscillator

FrequencyMixer

COHOMicrowaveOscillator

Amplifier

PhaseDetector

DISPLAY

switch

Half-power beamwidth

TRANSMITTER

RECEIVER

ANTENNA

FrequencyMixer

Block Diagram of a Radar System

Transmitter106 W

Antenna

Receiver10-14 W

Display

T/R switch

Why is wavelength important?

• Longer wave length -> sensitive to larger objects -> larger penetration ability (long range), but require a larger antenna to obtain enough return signal

• Shorter wave length -> sensitive to smaller objects -> more scattering and more attenuation of signal, require a smaller antenna to obtain enough return signal

W and K band radars are “cloud radars”X, C, S and L band radars are “precipitation radars”

Also - Wind Profilers (UHF & VHF; ~50 to 900 MHz; ~6 to 0.3 m)

How does electromagnetic wave travel in the atmosphere?

Electromagnetic waves:

Interact with matter in four ways:

Reflection:

Refraction:

Scattering:

Diffraction:

Snell’s law:

n - n

n i

r

Vi

Vr

r

i

V

V

r

i

n

nn

sin

sin

Where: i is the angle of incidence r is the angle of refraction Vi is the velocity of light in medium n Vr is the velocity of light in medium n - n

In the atmosphere, n normally decreases continuously with height…

Therefore: due to refraction, electromagnetic rays propagating upward away from a radar will bend toward the

earth’s surface

Propagation of electromagnetic waves in the atmosphere

Speed of light in a vacuum: C

Speed of light in air: V

Refractive index: n=C/V

At sea level: n = 1.0003In space: n = 1.0000

c = 2.998 x 108 m s-1

The Refractive Index is related to:

1. Density of air (a function of dry air pressure (Pd), temperature (T), vapor pressure (e)

2. The polarization of molecules in the air

(molecules that produce their own electric field in the absence of external forces)

The water molecule consists of three atoms, one O and two H. Each H donates an electron to the O so that each H carries one positive charge and the O carries two negative charges, creating a polar molecule – one side of the molecule is negative and the other positive.

2121615 375.0106.51076.71T

embK

T

embK

T

PmbKn d

Earth curvature

Electromagnetic ray propagating away from the radar will rise above the earth’s surface due to the earth’s curvature.

Ray Path Geometry

Consider the geometry for a ray path in the Earth’s atmosphere. Here R is the radius of the Earth, h0 is the height of the transmitter above the surface, φ0 is the initial launch angle of the beam, φh is the angle relative to the local tangent at some point along the beam (at height h above the surface at great circle distance s from the transmitter).

Equation governing the path of a ray in the earth’s atmosphere:

01112

22

2

2

dh

dn

nhRR

hR

ds

dh

dh

dn

nhRds

hd

where R is the radius of the earth, h is the height of the beam above the earth’s surface, and s is distance along the earth’s surface.

To simplify this equation we will make three approximations

1. Large earth approximation RhR

2. Small angle approximation 1tan ds

dh

3. Refractive index ~ 1 in term: n

1

(1)

01112

22

2

2

dh

dn

nhRR

hR

ds

dh

dh

dn

nhRds

hd X1

XX1/R 1

dh

dn

Rds

hd

12

2

Approximate equation for the path of a ray at small angles relative to the earth’s surface:

dh

dn

Rds

d

1

Or, in terms of the elevation angle of the beam

(2)

Curvature of Ray Paths Relative to the Earth

θ

An additional equation of interest is the equation that provides the great circle distance s, from the radar, for the r, h pair (slant range, beam height), which is

s = keR sin-1[rcosθ/(keR + h)]

Here ke=4/3We can get even simpler and consider a the height of the beam at slant range R and elevation angle θ,

h (km) = R2/17000 + R sin θ

hR

Use standard atmosphere, solve Diff. Eq. describing ray path for height of beam above surface of earth (assumes dn/dh is small):

d2h/ds2 – (2/R + 1/n * dn/dh)(dh/ds)2 – (R/a)2(1/R + 1/n * dn/dh) = 0

Where:

a= earth radius; s= arc distance; h= height above earth surface

n= refractive index; R= h + a; r= slant range along beam

Physically: Via equation for refractivity, we expect the beam to bend toward the surface since dP,e/dz < 0 and < dT/dz. However, h increases with s due to 1/R (curvature of earth’s surface, which diverges from beam position). DEQ above expresses this relationship as it relates earth’s geometry and the assumed refraction of the standard atmosphere to beam height and arc distance.

Doviak and Zrnic (1993) Sec. 2.2 show how this can be reduced to two equations for h and s using the 4/3 Earth radius model (4/3 Earth radius - dn/dh assumed to be constant - of order 0.25/a)

So, let ae= 4/3 a; then for convenience of computation:

h=[r2 + (ae)2 + 2raesinΦ0]1/2 – ae

s=aesin-1(rcosΦ0/[ae+h])

STANDARD REFRACTION: What we expect the beam to do over the curved surface of the earth

Φ0h

s

r

4/3 Earth Radius Model for Beam Propagation

(Standard Refraction/Reference Atmosphere Assumed)

h=[r2 + (ae)2 + 2raesinθe]1/2 – ae

S=aesin-1(rcosθe/[ae+h])

Θe = elevation angle

To get h as a f(slant range:R), which is measured by the radar, use this simple formula:

h (km)= R2/17000 + R sinθe (with R in km)

Doviak and Zrnic (1993)

•

Non-Standard Refraction• Non-standard refraction typically occurs with the temperature distribution does not

follow the standard lapse rate (dn/dh ≠ -1/4 (R)). As a result, radar waves may deviate from their standard ray paths predicted by the previous model. This situation is known as abnormal or anomalous propagation (AP).

Abnormal downward bending ------- super-refraction (most common type of AP)Abnormal upward bending ----------- sub-refraction

• Super-refraction is associated most often with cold air at the surface, giving rise to a near surface elevated temperature inversion in which the T increases with height. Most commonly caused by radiational cooling at night, or a cold thunderstorm outflow.

• Since T increases with height, n decreases (rapidly) with height (dn/dh is strongly negative). Since n = c/v, v must increase with height, causing downward bending of the ray path.

Recall Snell’s Law:

n1sinθ1 = n2sin θ2

n1

n2

θ1

θ2

n1 n2>

v2/v1 = sinθ2/sinθ1 v2 > v1

Wave (beam) is bent downward (refracted) in the atmosphere

So relative to the refractivity, what’s important here? dN/dZ – change in refraction with height- this causes velocity differences across the beam.

4 cases of refraction (dN/dZ):

Standard: dN/dZ ~ 0 and -40 km-1

Super: dN/dZ < -79 km-1 and > -158 km-1

Sub: dN/dZ > 0

Ducting: dN/dZ < -158 km-1 (dn/dh = -1/R)

Non-Standard

Non-Standard Refraction

Super-Refraction (most common)

dN/dZ < -79 km-1 and > -158 km-1

Beam is bent downward more than standard

Situations:

1. Temperature inversions (warm over cold air; stable layers)

2. Sharp decrease in moisture with height

(1) And (2) can occur in nocturnal and trade inversions, warm air advection (dry), thunderstorm outflows, fronts etc.

Result:

1. Some increased clutter ranges (side lobes)

2. Overestimate of echo top heights (antenna has to be tilted higher to achieve same height as standard refracted beam)- see figure above

Most susceptible at low elevation angles (e.g., typically less than 1o)

Φ0Φ0

hhh’h’

Sub-Refraction (not as common)

dN/dZ > 0 km-1

Beam is bent upward more than standard

Situations:

1. Inverted-V sounding (typical of desert/intermountain west and lee-side of mountain ranges; microburst sounding; late afternoon and early evening; see figure)

Result:

1. Underestimate of echo top heights (beam intersects top at elevation angles lower than in standard refraction case)- see figure above

Most susceptible at low elevation angles (e.g., typically less than 1o)

DP T

Inverted-V sounding

Φ0Φ0

hh h’h’

Ducting or Trapping (common)

dN/dZ < -158 km-1

Beam is severely bent downward and may intersect the surface (especially at elevation angles less than 0.5o) or propagate long distances at relatively fixed heights in an elevated “duct”.

Situations:

1. Strong temperature inversions (surface or aloft)

2. Strong decreases in moisture with height

Result:

• Markedly Increased clutter ranges at low elevation angles

• Range increases to as much as 500% in rare instances (useful for tracking surface targets)

Most susceptible at low elevation angles (e.g., typically less than 1o)

Elevated ducts can be used as a strategic asset for military airborne surveillance and weapons control radars. E.g., if a hostile aircraft is flying in a ducting layer … it could be detected a long way away, while its radar cannot detect above or below the ducting layer. Conversely, friendly aircraft may not want to be located in the duct.

Doviak and Zrnic (1993)

Modeled with 100 m deep surface inversion with dN/dz=300 km-1 and standard thereafter.

Example of ray paths in surface ducting

One moral of the whole refraction story……..knowing the exact location of the beam can be problematic. Remember this when you have the opportunity to compare the measurements of two radars supposedly looking at the same storm volume!

Big implication of radar beam height increasing with range (under normal propagation conditions) combined with broadening of the radar beam: The radar cannot “see” the low level structures of storms, nor resolve their spatial structure as well as at close ranges. Thus, for purposes of radar applications such as rainfall estimation, the uncertainty of the measurements increases markedly with range.

Storm 1 Storm 2

4343

Beam Blockage in Complex Terrain• Beam propagation is a function of the vertical refractivity gradient

(dN/dz)– N = 77.6(p/T) - 5.6(e/T) + 3.75x105(e/T2)

• dN/dz is sensitive to p, T, e• Thus, changes in the vertical profiles of these quantities can change

the height of the ray path as it propagates away from the radar• This is especially important in complex terrain, because the amount

of beam blockage will change depending on the vertical refractivity gradient

dN/dZ = -40/kmdN/dZ = -80/km

)

False Data• Ground Clutter

– Portion of radar beam hits buildings, trees, hills

• Also can be due to dust, aerosols in the air near the radar

– Gives false indication that precip is present

– Radar location is in the black area surrounded by blue/green reflectivities

False Data• Anomalous propagation

(AP)– Occurs when

temperature inversions are present in low-levels

• Radar beam bent into ground, returning strong signal

– Common during early morning hours after a clear night

– Again, no precip really present

False Data

• Virga– Radar detects precip occurring at upper levels, but not making it to the ground

• Precip quickly evaporates in dry air below cloud

– Precipitation is thus overestimated

False Data

• Overshooting Beam– Some precip can form from clouds with minimal height

– Beam may overshoot a large portion of the cloud, underestimating the intensity of the precipitation

False Data

• Storm Interference– Storms closest to radar may absorb or reflect much of the radar energy

• Leaves reduced amount of energy available to detect distant storms

• Underestimates precipitation

False Data• Wind Shear

– Falling precip may be displaced by the wind as it falls

– Some regions may be experiencing precip where the radar indicates nothing, and vice versa

Height of Lowest Unobstructed Sampling Volume Radar Coverage Map

Mid-Atlantic River Forecast Center (MARFC)

Height of Lowest Unobstructed Sampling Volume Radar Coverage Map

West Gulf River Forecast Center (WGRFC)

6767

Southeast

PRECIPITATION MOSAIC RADAR COVERAGE MAP Northeast

Say you’d like to site a radar for a research experiment. In a perfect world…you’d like to be able to take a swim after work, but AP and beam blockagemay be a problem. Sidelobes may intersect the highly reflective ocean – creating “sea clutter”

Mountains can be a problem…

0.5°1.5°

Local effects can be a problem too – topographic maps and DEMs can help, butstill need to conduct a site survey to see trees, antennas, buildings, and overpasses.

Often times you end up in places like this…

Height of a ray due to earth’s curvature and standard atmospheric refraction

Assignment #1

Here we have reviewed the calculation of slant path of the radar beam. But I did not describe the equation for the beam width change with the distance. Here is the question: assuming you have a radar with beam width 1 degree. How big is the beam width at 100 km?

There is an airplane flying at 10 km altitude with a radar sending a 1 degree beam tangentially. At 200 km distance, a) what is the beam width? b) what are the heights of the top and bottom of the beam respect to the Earth surface? If there is a storm reaching 10 km at 200 km distance, can pilot see the storm on his screen? (assuming the standard atmospheric refraction)

ELECTRIC FIELD

An Electric field exists in the presence of a charged body

ELECTRIC FIELD INTENSITY (E)

A vector quantity: magnitude and direction (Volts/meter)

MAGNITUDE OF E: Proportional to the force acting on a unit positive charge at a point in the field

DIRECTION OF E: The direction that the force acts

The Electric Field (E) is represented by drawing the Electric Displacement Vector (D), which takes into account the characteristics of the medium within which the Electric Field exists.

EmcoulD 2

, the Electric Conductive Capacity or Permittivity, is related to the ability of a medium, such as air to store electrical potential energy.

112120 10850.8 mjoulecoulVacuum:

112121 10876.8 mjoulecoulAir:

Ratio: 003.10

1

The Electric Displacement Vector, D, is used to draw lines of force.

2mcoulUnits of D:

MAGNETIC FIELD

A Magnetic field exists in the presence of a current

MAGNETIC FIELD INTENSITY (H)

A vector quantity: magnitude and direction (amps/meter)

MAGNITUDE OF H: Proportional to the current

DIRECTION OF H: The direction that a compass needle points in a magnetic field

The Magnetic Field (H) is represented by drawing the Magnetic Induction Vector (B), which takes into account the characteristics of the medium within which the current flows.

HB , the Magnetic Inductive Capacity, or Permeability, is related to the ability of a medium, such as air, to store magnetic potential energy.

1260 10260.1 mampjouleVacuum:

Air:

Ratio: 000.10

1

1261 10260.1 mampjoule

Magnetic Fields:

Magnetic fields associated with moving charges (electric currents)

BIForce I: Current ampsorscoul 1

B: Magnetic Induction 21 mampjoule

Magnetic Field Lines are closed loops surrounding the currents that produce them

Maxwell’s Equations for time varying electric and magnetic fields in free space

0

E

t

BE

0 B

t

EIB

000

(where is the charge density)

Simple interpretation

Divergence of electric field is a functionof charge density

A closed loop of E field lines will exist whenthe magnetic field varies with time

Divergence of magnetic field =0(closed loops)

A closed loop of B field lines will exist inThe presence of a current and/or time varying electric field

Electromagnetic Waves: A solution to Maxwell’s Equations

Electric and Magnetic Force Fields

Propagate through a vacuum at the speed of light:

18103 smc

Electric and Magnetic Fields propagate as waves:

€

E(r,θ,φ, t) =A(θ,φ)

rexp i2πν t −

r

c ⎛ ⎝

⎞ ⎠+ iΨ

⎛ ⎝ ⎜

⎞ ⎠ ⎟

€

E(r,θ,φ, t) =A(θ,φ)

rcos 2πν t −

r

c ⎛ ⎝

⎞ ⎠+ Ψ

⎛ ⎝ ⎜

⎞ ⎠ ⎟

)sin()cos()exp( xixix where:

or:

ρ, θφare coordinates, A is an amplitude factor, ν is the frequency and Ψ is an arbitrary phase

All energy stored in

electric field

All energy stored in

magnetic field

wavelength

Time variations in charge, voltage and current in a simple Dipole Antenna

Energy is 1) stored in E, B fields, 2) radiated as EM waves, 3) Dissipated as heat in antenna

Near antenna: Energy stored in induction fields (E, B fields) >> energy radiated(near field)

More than a few λ from antenna: Energy radiated >> energy stored in induction fields (far field)

Pt. A

Pt. B

Spherically Stratified Atmosphere; Ray Path Equation

Integrating (2) yields,

(dh/ds)2 = 2∫ (1/R + dn/dh) dh + constant (3)

Since dh/ds ≈ φ for small φ, (3) can be written as,

1/2(φh2 - φ0

2) = (h - h0)/R + n - n0

= (h/R + n) - (h0/R + n0)

Letting M = [h/R + (n-1)] x 106, we have

= (M - M0)10-6

M is the so-called modified index of refraction. M has a value of approximately 300 at sea level.

Curvature of Ray Paths Relative to the Earth

• If the vertical profile of M is known (say through a sounding yielding p, T and q), φh can be calculated at any altitude h, that is, the angle relative to the local tangent.

• Lets now consider the ray paths relative to the Earth. For the case of no atmosphere, or if N is constant with height (dN/dh = 0), the ray paths would be straight lines relative to the curved Earth.

dφ/ds = 1/R + dn/dh 1/R for n constant with height(No atmosphere case?)(“Flat earth” case?)

• For n varying with height,

dφ/ds = 1/R + dn/dh < 1/R since dn/dh < 0

• For the special case where dn/dh = -1/R, dφ/ds = 0. Hence the ray travels around the Earth concentric with it, at fixed radius, R + h. This is the case of a trapped wave. “DUCTING”

Curvature of Ray Paths Relative to the Earth

For convenience, it is is easier to introduce a fictitious Earth radius, 1/R’ = 1/R + dn/dh

For typical conditions, dn/dh = -1/4 R m-1

Hence R’ = R/(1 - 1/4) = 4/3 RThis is the effective Earth radius model, to allow paths to be treated as straight lines.

Doviak and Zrnic (1993) provide a complete expression for h vs. r, where r is the slant range (distance along the ray).

h = {r2 + (keR)2 + 2rkeRsinθ}1/2 - keR

where h is beam height as slant range r, θ is the elevation angle of the antenna, and ke is 4/3 (R is the actual Earth radius).