Relationship of the Reflectivity Factor to other Meteorological Quantities

c

jj

V

D

Z

6

Precipitation content (W): The mass of condensed water substance (water or ice) present in the form of precipitation-sized particles (detectable with radar), per unit volume.

c

jj

c

jj

V

D

V

m

W

3

6 Where:

mj is the contribution to the total mass from each raindrop j

Precipitation Content

Basic units: kg/m3

Simple interpretation: Mass of water in a unit volume

Extreme values:0.1 gram/m3 in light drizzle10 gram/m3 in rain in hurricane eyewall

Example:A distribution of 1000 1-mm raindrops per cubic meter would have a precipitation content of about 0.5 grams/m3 .

j

jDW 3 j

jDZ 6

Problem: 2

36

jj

DD

Illustration of inequality

Consider two drops 1 mm and 2 mm2

36

jj

DD

81921

65212233

66

Therefore: There is no exactRelationship between precipitation content and radar reflectivity

Nevertheless, precipitation contents can be qualitatively related to the radar reflectivity factor, and radar scientists have sought empirical relationships of the type:

b

R W

WZZ

0

where ZR is the value of Zwhen W = W0

Relationship of the Reflectivity Factor to other Meteorological Quantities

c

jj

V

D

Z

6

Precipitation rate (R): The volume of precipitation passing downward through a horizontal surface, per unit area, per unit time.

c

jjj

c

jj

V

wD

V

r

R

3

6 Where:

rj is the contribution to the rainfall rate from each raindrop j

wj is the fall velocity of each drop j

Precipitation Rate

Basic units: m3/(m2sec) = m/sStandard units: mm/hr

Simple interpretation: Depth of accumulated rainfall on a runoff-free surface

Extreme values:0.1 mm/hr in light drizzle1000 mm/hr in a hurricane eyewall

Example:A distribution of 1000 1-mm raindrops per cubic meter, falling at their terminal fall speed of 4 m/s in

the absence of vertical motion, would give a precipitation rate of 2.1 10-6 m/s or

about 7.5 mm/hr.

c

jjj

V

wD

R

3

6

What is the fall velocity of a raindrop?

For drops with diameters between 0-2 mm (most drops) the fall velocity is proportional to diameter

Terminal velocity of raindropsIn still air (Foote and duTroit 1969)

j

DR 4

so what is the relationshipto the radar reflectivity?

Problem:

5.1

46

jj

DD

Illustration of inequality

Consider two drops 1 mm and 2 mm5.1

46

jj

DD

09.701721

65215.15.144

66

Therefore: There is no exactRelationship between rainfallRate and radar reflectivity

Nevertheless, rainfall rates are qualitatively related to the radarreflectivity factor, and radar scientists have sought empirical relationships of the type: b

R R

RZZ

0

where ZR is the value of Zwhen R = R0

Relationship of Z to Precipitation Rate

Methods of determining Z-R relationships

1. The direct method: Values of Z and R are measured by a radar and raingages. The data are compared using correlation statistics and a Z-R relationship is determined from a best fit.

Relationship of Z to Precipitation Rate

Methods of determining Z-R relationships

2. The indirect method: Values of Z and R are calculated from the same measured raindrop size distribution.

Methods to measure raindrop size distributions

Mechanical: stained filter paper: Uses water stains in filter paper to estimate raindrop sizes (used originally by Marshall and Palmer)

Impact disdrometer: Uses raindrop’s momentum when striking surface to estimate its size.

Ground Based Optical disdrometers

Airborne Optical disdrometers

Foil impactors

Determine drop sizes by shadows recorded on optical arrays

Foil impactors: determine drop sizes from impact craters

Example of raindrop images collected with an airborne optical array spectrometer in a shower in Hawaii with the largest raindropever recorded in nature (courtesy Ken Beard)

Typical measured raindrop size distributions

To estimate Z and R, exponential approximations to raindrop size distributions are often developed

The Marshall-Palmer Distribution

Developed from raindrop samples collected in Canada on powdered sugar filter paper in 1948 by radar pioneers Marshall and Palmer

DnDn exp0

The Marshall-Palmer Distribution

4640 10808.0 mcmn

c

R R

R

0

hrmmR /10 141 cmR

The Marshall-Palmer distribution stood as the standard for many decades although many subsequent studies showed that it was not universally applicable.

The exponential distribution has properties that make it useful because it is easy to relate the drop size distribution to rainfall rate, precipitation content, and radar reflectivity

General properties of an exponential size distribution

DnDn exp0

Total concentration of droplets

0

0, )(

ndDDNNT

Rainfall rate bt

bandDDNDwR

40

03 4

6)(

6

where the fall velocityb

t aDw

Precipitation content

40

03 4

6)(

6

ww n

dDDNDW

Radar reflectivity 7

0

06 7

)(

ndDDNDZ

Drop distributions do not extend to infinite size – the integrationmust be truncated at the maximum droplet diameter Dm

Effect of such a truncation:

D0 is mean diameter

0

0

)(

)(

dDDXn

dDDXn

F

mD

Calculation of Z from a measured drop size distribution

Note which dropscontribute most tothe radar reflectivity

Z = 1.7 105 mm6/m3

52.3 dBZ

General form of Z/R Relationships

b

R R

RZZ

0

b

Z

ZRR

1

00

Radar scientists have tried to determine Z-R relationships because of the potential usefulness of radar determined rainfall for

FLASH FLOOD NOWCASTING

WATER MANAGEMENT

AGRICULTURE(irrigation needs/drought impacts)

There have been hundreds of Z-R relationships published – here are just a few between 1947 and 1960 – there have been 4 more decades of new Z-R relationships to add to this table since!

Z-R relationships are dependent on the type of rainfall (convective, stratiform, mixed), the season (summer, winter), the location (tropics, continental, oceanic, mid-latitudes), cloud type etc.

For the NEXRAD radars , the NWS currently uses five different Z-R relationships and can switch between these depending upon the type of weather event expected.

        Default WSR-88D (Z= 300R1.4)        Rosenfeld tropical (Z=250R1.2)        Marshall/Palmer (Z=200R1.6)         East Cool Season (Z=200R2.0)        West Cool Season (Z=75R2.0)

The single largest problem in applying Z-R relationships has been accounting for effects of the radar bright band

The bright band: The melting level, where large snowflakes become water coated, but have not yet collapsed into small raindrops.

Wet snowflakes scatter energy very effectively back to the radar

D ista n c e (k m )

R ef le ct iv ity fa cto r (d B Z )

S tr a t i fo r m a r e a C o n v e c t io n

B BAltitude (km)

The bright band appears as a ring on PPI displays where the radar beam crosses the melting level

An extreme example of bright band contamination of precipitation estimation – radar estimates 6 inches of rain in a winter storm on January 31, 2002!

Other problems:

1. Estimating R from Z in regions of storms that are mixed phase(e.g., hail vs. rain)

2. Regions affected by ground clutter or blocking (particularly a problem for estimating rainfall during flash floods in mountainous regions)

SNOW

Few attempts have been made to develop Z-S relationships

1. Snow density varies significantly from storm to storm and within storms2. Scattering by ice is non-Rayleigh (not spheres) and so the

relationship between mass and Z is even less certain3. Radars calibrated for rain (Z determined from K for rain, not

ice, even in winter)

Measurements have been made of the size distributions of snowflakes and related to precipitation rates (melted equivalent), and Z-S relationships have been proposed but these relationships have largely been ignored in practice

Hail

Very few attempts have been made to quantity hailfall from thunderstorms. Most work focuses on trying to identify whether hail is reaching the surface. This work is now focused on studies using polarization radar technology, which we will examine later in the course.