AQUARADAR -D

• The effects of inhomogeneous rain drop size

distributions (DSDs) in a radar volume

• Vertical structure of DSD profiles

• Spectral modes of DSDs

• The effects of inhomogeneous rain drop size

distributions (DSDs) in a radar volume

• Vertical structure of DSD profiles (Kowalewski & Peters,

2009)

• Spectral modes of DSDs (S. Melchionna, Met. Z. 2008)

1AQUARADAR - D, Bonn 2009

The effects of inhomogeneous rain drop size

distributions (DSDs) in a radar volume

Impact on Attenuation Correction Algorithms

Non-linear Z-R-relation

2AQUARADAR - D, Bonn 2009

Non-linear Z-R-relationbaRZ

S

dssRS

R 11

'ˆ

S

A

SA

AssR

AssR

/

/ 1

0

AsasZ

AssZb

1

0

/ b

b

b

S

a

dssZS

R

1

1

1

1

/

'ˆ

3AQUARADAR - D, Bonn 2009

Non-linear Z-R-relation

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Peters G., B. Fischer, Rain estimation from partially filled scattering volumes, ERAD 2006:

•Transition from small pixels (60 m ×160 m) to large pixels (960 m × 960 m) causes relative overestimation of rain rate up to 1.3 according to observations of convective and stratiform rain events. •Attempts to infer sub-scale variability from standard weather radar data were not successful.

Impact on Attenuation Correction Algorithms

5AQUARADAR - D, Bonn 2009

1. Iterative „robust“ algorithm by Hildebrand, 1978

2. Analytic „unstable“ algorithm by Hitschfeld and Bordan, 1954

Definitions:

Path Integrated Attenuation

Transmission

r

dxxr0

2 expPIA

/PIA1T

Iterative “robust” algorithm

6AQUARADAR - D, Bonn 2009

Hildebrand, 1978

rxrrrr

x

1

101 2 )()( expPIA

xrxrxr )()( PIA 101

rxrrrr

x

1

112 2 )()( expPIA

rxrrr

r

xii

1

112 expPIA )(

7AQUARADAR - D, Bonn 2009

r

Transmit Signal

)()( rZ 0

r

Scattered Signals

)()( 10 rZ

1r r

120 rrrr exp

rTrZrZ )()(0

rTrr )(0

Z propto

1 assume rr

rrr 222 0 exp

or

TTT lnln 0

or

TTT 0

0

0

T

TT

lnW

lnhttp://mathworld.wolfram.com/PowerTower.html

8AQUARADAR - D, Bonn 2009

Good News:

It can be shown that the iteration T(i) actually converges to

0

0

T

TT i lnW

ln

Bad News:

0TlnW is defined only for eT /exp 10 corresponding to -1.6 dB

9AQUARADAR - D, Bonn 2009

T_(0),min

0.7

1

0 0.2 0.4 0.6 0.8

T_(0

)

0.5

0.8

0.9

0 1

1

T

0.6

The inverse retrieval function TTT 0

10AQUARADAR - D, Bonn 2009

-6

Itera

ted

and

true

trans

mis

sion

(dB)

T

T_(0)

T_(1) T_(2)T_(3)

T_(4)T_(5)

T_(0), min

0-1-2-3-4-20

-15

-10

-5

0

-5T_(0) (dB)�

Retrieved transmission after various iteration steps

11AQUARADAR - D, Bonn 2009

0

5

10

15

20

25

30

0 20 40 60 80 100 120 140 160 180 200

PIA

(dB)

R (mm/h)

SIBO

PIA

SIBO

PIA

SIBOPIAHL

PIA

HL

PIA

HL

PIA

500 m10

00 m

100 m

PIASIBO and PIAHL at K-band as function of simulated Marshall-Palmer rain ratefor Δr = 100 m and 3 ranges.

12AQUARADAR - D, Bonn 2009

0

5

10

15

20

25

30

35

0 5 10 15 20 25 30 35

PIA

SIBO

in d

B

PIAiterative in dB

Achern 2007-10-17

13AQUARADAR - D, Bonn 2009

Analytic „unstable“ algorithm by Hitschfeld and Bordan, 1954

If can be observed, the solution assumes a particularly simple form: rrra /PIA

r

a dxx

r

0

21

1

PIA

14AQUARADAR - D, Bonn 2009

r

a dxx0

by

n

ia rir

1

Due to the finite range resolution we have to replace

Problem: If we assume homogeneity for κ in the scattering volume,it will be inhomogeneous for κa

r

rx

rk

r

a

2

212

1

0

expexp

Solve this equation for κ:

r

ra

2

21

ln

1 iiaip rrDNrDN PIA,,

0

dDrDNDr ipeip ,

rrrrrDNrDN ipipipi 221 /ln,,

dDrDNDr iei

0

,

rrrr iii 21 expPIAPIA

1ii

1. Set , set

2. Calculate

3. Calculate

4. Calculate

5. Calculate

6. Calculate

7. Set

8. Go to step 2

10 rPIA 1i

15AQUARADAR - D, Bonn 2009

Modification of the original retrieval

New!

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Finite Range Gate

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R-k-Relation

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Z-k-relation

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PIA

= 10

dB

-5

0

5

10

-3 -2 -1 0 1 2 3calibration error delta (dB)

tota

l rel

ativ

e er

ror (

dB)

PIA = 3 dB

PIA

= 6

dB

-10