ANALYSIS OF THE EFFECT OF RAIN ON THE ENVISATALTIMETER: DEFINITION OF A RAIN FLAG

Jean Tournadre, IFREMER, DRO-OS

Graham Quartly, SOC, JRD

Meric Srokosz, SOC, JRD

Technical Report Ifremer DRO-OS N˚00/01

2

1. Introduction

Past experiences with ERS and Topex/Poseidon altimeter data have shown that rain can signifi-cantly alter the quality of altimeter measurements (dynamic heights, significant wave heights, windspeed) (Guymer et al, 1995, Tournadre and Morland, 1997, Quartly et al, 1996, Quartly, 1997).

Among all the different atmospheric phenomena which can affect the altimeter measurement, rainis one of the less well understood and at present no reliable correction can be made for the wholerange of geophysical parameters. To avoid any contamination by rain, the data which might possi-bly be affected are simply discarded using a flag set by using concurrent passive microwave radiom-eter measurements. These passive microwave data are also used to calculate an atmospheric watervapour correction to the height and can also give an estimate of atmospheric liquid water (Ruf et al,1995).

Recent studies (Quartly et al., 1996, Tournadre and Morland, 1997) have analysed the effects ofrain on the dual frequency (Ku and C band) Topex altimeter (NRA) data. They showed that the fre-quency dependence of the attenuation of electromagnetic signal by rain can be used to define a rainflag based on the detection of departures from the normal relationship between the two frequenciesbackscatter measurements. They also showed that this kind of rain flag based on the altimeter meas-urements itself performs better than the one based on coincident passive microwave data.

The future Envisat altimeter will have a dual frequency capability, and will operate in Ku and Sband. Compared with the ERS altimeters, this dual frequency capability can be used to defined asimple rain flag. The aim of the present study is firstly to analyse in some details the influence ofrain on the Envisat altimeter waveforms and secondly to propose a rain flag for operational use.

The possibility of defining a rain rate measurement will also be described.In section 2, the altimeter waveform model used in the study (Brown, 1977) is presented. The

modelling of the effect of rain on the waveform is then described in section 3, with a particular at-tention to the integration methods. The impact of rain on Envisat data is analysed in section 4. Sec-tion 5 discusses the definition of a robust rain flag, with section 6 showinghow the data from themicrowave radiometer may be added to give further details on the size and strength of rain cells<The recommendations are provided in section 7.

3

2. Altimeter waveform model

The model of altimeter echo during rain events presented here is based on the work of Brown(1978), Barrick (1978) and Barrick and Lipa (1985). The backscatter coefficient is expressed as adouble convolution product of the radar point target response, the flat sea surface response and thejoint probability density function of slope and elevation of the sea surface (Brown, 1977).

Figure 1: The altimeter coordinate system.

The radar cross section for backscatter as a function of time for an altimeter pulse and for a Gaus-sian random distribution of rough-surface specular points, can be expressed in the coordinate sys-tem defined in Fig.1 as (Barrick, 1972):

( 1)

where is the sea surface elevation above the mean local surface, is the angle at the antennafrom nadir to a point of the surface, is the angle at the earth centre from the satellite to ,

is the two-way antenna normalized gain pattern, is the normalized effective pulseshape as a function of spatial propagation distance, , is the earth’s radius, is theFresnel reflection coefficient of the sea surface at normal incidence, is the angle between the lo-cal normal to the surface at and the satellite, and is the joint height-slope distribution prob-ability density function at elevation and wave slopes corresponding to specular angle .

As , and are small, the following relations are verified at the second order:

( 2)

a

H

Satellite

ct/2

ζ

ψ

θ

φ

σ t( ) 2π2a2 R 0( ) 2 g ψ( ) θ )sec( 4 φ P ξ ζ–( )p j ξ( ) ξd∞–

∞

∫sin φd0

∞

∫=

ζ ψζ φ ζ

g ψ( ) P x )(x ct( ) 2⁄= a R 0( )

θζ pi ζ( )

ζ θφ ψ θ

ψ= aφ H⁄

θ aH''φ=⎝⎛

H' H 1 H a⁄+( )=

H'' H 1 Ha----+⎝ ⎠

⎛ ⎞⁄=⎝⎜⎜⎛

4

and, ( 3)

where, is the satellite altitude. Using (2) and (3), (1) becomes:

( 4)

Furthermore,

( 5)

Using the small angle trigonometric approximation (5) becomes:

( 6)

where,Using again the small-angle trigonometric approximations (as and are small), and the fact

that is small compared to and the satellite height, the normalized radar cross section can beexpressed as:

( 7)

The new antenna gain factor is defined as .

We assume the Gaussian shape of half-power width for the compressed altimeter pulse, i.e.,

( 8)

where .

We also assume a Gaussian shape for the antenna beam pattern near its boresight. From the two-way half-power antenna beamwidth is , we define . Thus,

( 9)

or, ( 10)

where, .

The joint probability of slope and elevation is assumed Gaussian and is defined by: ( 11)

φsin φHa----ψ= =

θsec 1=

H

σ t( ) 2π2H2 R 0( ) 2 g ψ( ) P ξ ζ–( )p j ξ( ) ξd∞–

∞

∫ ψd0

∞

∫=

ζ a+( )2 H ct 2⁄+( )2 a H+( )2 2 H ct 2⁄+( ) a H+( ) ψcos–+=

ζ ct2---- H′ψ2

2-------------+– x– u+= =

uH′ψ2

2-------------= x

ct2----=

ψ φζ a H

σ t )( 2π2H'' R 0( ) 2 G u( ) P ξ x u–+( )p j ξ( ) ξd∞–

∞

∫ ud0

∞

∫=

G u( ) g 2u( ) H'⁄( )=

τ

P x( ) e

x2

2στ2

---------–

=

στcτ

2 8 2ln-------------------=

ψH ψb ψH 8 2ln( )⁄=

g ψ( ) e

ψ2

2ψb2

---------–

=

G u( ) e

uub

-----–

=

ub

H'ψb2

2------------=

5

( 12)

where, is the rms wave height (related to the significant wave height by ), and are the rms wave slopes along two orthogonal directions of a plane tangent to the mean sphere,

and is the correlation coefficient of the wave slopes along these two axes.Under these assumptions, the cross section is given by:

( 13)

By integration over , and by defining :

( 14)

and by integration over ,

( 15)

For TOPEX or ENVISAT class altimeter, and , thus (14) can be simplified as:

( 16)

where, , is the average normalized backscattering cross section per unitarea at normal incidence.

p j ξ( ) p ξ 0 0, ,( )1

2π( )3 2⁄ hsxsy 1 ρxy2–

----------------------------------------------------e

ξ2

2h2--------–

= =

h Hs Hs 4h= sx

sy

ρxy

σ2xc

------⎝ ⎠⎛ ⎞ 2π2H′′ R 0( ) 2

2π( )3 2⁄ hsxsy 1 ρxy2–

---------------------------------------------------- e

uub

-----–

e

ξ2

2h2--------–

e

ξ x u–+( )2

2στ2

----------------------------–

ξd∞–

∞

∫ ud0

∞

∫=

ξ σp h2 στ2+=

σ2xc

------⎝ ⎠⎛ ⎞ πH′′ R 0( ) 2στ

σpsxsy 1 ρxy2–

------------------------------------ e

uub

-----–

ex u–( )2

2σp2

------------------–

ud0

∞

∫=

u

σ2xc

------⎝ ⎠⎛ ⎞ 2π( )3 2/ H′′ R 0( ) 2στ

4sxsy 1 ρxy2–

------------------------------------------------- 1 erfx

2σp

-------------σp

2

2ub

------------–+ e

2ub x– σp2+

2ub2

-----------------------------

=

σp ub« xσp

2

2ub--------»

σ2xc

------⎝ ⎠⎛ ⎞ 2π( )3 2/ H′′ R 0( ) 2στ

4sxsy 1 ρxy2–

------------------------------------------------- 1 erfx

2σp

-------------+ e

xub

-----–

=

σ0 1 2sxsy 1 ρxy2–( )⁄=

6

3. Waveform modelling in presence of rain

The presence of rain within the altimeter footprint changes the computation of the altimeterwaveform by introducing for every point of the altimeter footprint a perturbation in the form of anattenuation of the radar signal . Under this condition, relation (14) becomes:

( 17)

The attenuation by rain of the radar signal depends on the signal frequency and the rain rate. The rain rate field can be either analytically defined using a rain cell model or numerically

set using rain rate measurements from in situ data (meteorological radar for example). As thenumber of available in situ measurements over the ocean is small, analytical models of rain cellshave been used in this study.

Figure 2: Rain cell geometry (left) Coordinate system used for the computation of altimeter waveforms in presence ofrain. As the problem presents a circular symmetry, is always assumed nil (right).

3.1- Rain cell models

Several studies based on the analysis of meteorological radar have been published on the size ofrain cells and on the distributions of rain rate within the cell. Among them Capsoni et al (1987) andGoldhirsh and Musiani (1992) have been used in this study. These studies showed that in general,the rain rate fall-off observed within rain cells were best represented by exponential fall-off and thathigher rain rates were associated with smaller rain cells though the size dispersion is large.

As the purpose of our study is the analysis of the effects of rain on Ku and S band altimeter data,a wide range of rain cells have been considered.

The programmes that have been developed to compute the altimeter waveforms include differentfall-off scheme (constant, exponential, gaussian) and different cell shapes (circular, bands).

A u θ,( )

σ2xc

------⎝ ⎠⎛ ⎞ πH′′ R 0( ) 2στ

σpsxsy 1 ρxy2–

------------------------------------ e

uub

-----–

ex u–( )2

2σp2

------------------– 12π------ A u θ,( ) θd

0

2π

∫⎝ ⎠⎛ ⎞ ud

0

∞

∫=

R u θ,( )

Hc

x0

ALTIMETER

RAIN CELL

d

x

y

Hψ

(x0,0)θ

(x,y)

y0

7

For any rain cell, the rain rate field can easily been defined in the local terrestrial coordi-nate system. As the altimeter presents a circular symmetry, the circular cell can always been consid-ered centred on the x axis in and the band cell parallel to the y axis and centred in .

For practical computation we need to change from the local terrestrial coordinates to the local al-timeter coordinates using the following transform:

( 18)

Figure 3: Rain rate field for a 5 km, 5 mm/hr exponential rain cell centred 1 km of nadir in the local terrestrial coordi-nate system (left) and the local altimeter coordinate system (right).

R x y,( )

x0 0,( ) x0 0,( )

u θ,( )

x Hψ θcos H2uH′------ θcos 2uH′′ θcos= = =

y Hψ θsin H2uH′------ θsin 2uH′′ θsin= = =

0.2 0.4 0.6 0.8

u m

θ de

g

Expo. cell d=5km x0=1km

0 50 100 150

0

50

100

150

200

250

300

350

1 2 3 4 5

−10 −5 0 5 10−10

−5

0

5

10

Expo. cell d=5km x0=1km

x km

y km

8

3.2- Computation of the rain attenuation

The estimation of the rain attenuation of radar signals have been widely studied since the 1940’sand the literature on the subject is plentiful. The one way signal attenuation (dB.km-1), , is relat-ed to the rain rate by the Marshall-Palmer relation (Marshall and Palmer, 1948):

( 19)

where and are two frequency dependent coefficients.The two-way attenuation depends on the rain thickness and on :

( 20)

Thus, as a function of the rain rate field,

( 21)

The and coefficients for Ku and S band are given in Table 2.

Table 1: Rain cells models.For gaussian rain fields, where is the 3dB width, for the constant rate cases,

, for exponential cases,

circular

const

exp

gaus

Band

const

exp

gaus

r d 2 2log( )⁄= dr d 2⁄= r d 2 2log( )⁄=

R x y,( ) R u θ,( )

0 if x x0–( )2 y2+( ) r>

R0 if x x0–( )2 y2+( ) r≤⎩⎪⎨⎪⎧

0 if 2H′′u x02 2 2uH′′x0 θcos+ + r>

R0 if 2H′′u x02 2 2uH′′x0 θcos+ + r≤

⎩⎪⎨⎪⎧

R0e

x x0–( )2 y2+( )r

-----------------------------------------–

R0e

2H ′ ′u x02 2 2uH ′ ′x0 θcos+ +

r------------------------------------------------------------------------------–

R0e

x x0–( )2 y2+( )

r2------------------------------------–

R0e

2H ′ ′u x02 2 2uH ′ ′x0 θcos+ +

r2-------------------------------------------------------------------------–

0 si x x0– r>

R0 si x x0– r≤⎩⎨⎧ 0 si u θcos( )2 r x0+( )2>

R0 si u θcos( )2 r x0+( )2<⎩⎨⎧

R0e

x x0–r

----------------–

R0e

2uH ′ ′ θcos x0–r

---------------------------------------------–

R0e

x x0–( )2

r2--------------------–

R0e

2H ′ ′u θcos( )2 x02 2 2uH ′ ′x0 θcos+ +

2r2---------------------------------------------------------------------------------------------–

k p

k p αRβ=

α βHc k p

AR 10

2Hck p

10---------------–

=

A u θ,( ) 10

2HCαR u θ,( )β

10------------------------------------–

=

α β

9

1: after Ulaby et al (1981)2: after Golhirsh and Rowland (1982)

Note: for , , i.e. no signal attenuation, for , , i.e. com-plete signal attenuation.

The non-linearity of the Marshall-Palmer relation precludes the analytical integration of the atten-uation term of relation (17). However, Barrick and Lipa (1985) and Tournadre (1998) showed thatif this non-linearity is neglected, relation (17) greatly simplifies for gaussian and constant rain cells.The method decomposes as follows:

Neglecting the non-linearity of , the integral of the attenuation over can be expressedby:

( 22)

However, under this form, it is necessary to consider the fraction of the signal absorbed by therain by using:

( 23)

Indeed, if one considers only , then for zero rain, i.e., , there

should be nor attenuation, i.e., , but tends towards 1 and , thus

. If the rain rate tends towards infinity, , then . It is thus simpler to

consider the transmission than the attenuation.Relation (16) is thus expressed as

( 24)

or

( 25)

Table 2: Coefficients of the Marshall-Palmer relation for different radar frequencies

Frequency

S (3 GHz)1 4.6 10-4 0.954

Ku (13.5 GHz)2 3.4610-2 1.109

α β

R 0= A u θ,( ) 1= R ∞→ A u θ,( ) 0=

A u θ,( ) θ

A u( ) Ar1

2π------ 1

R0-----R u θ,( ) θd

0

2π

∫=

Ar 10

2HcαR0β

10--------------------–

1–=

Ar 10

2HcαR0β

10--------------------–

= R0 R u θ,( ) 0= =

A u( ) 1= Ar R u θ,( ) θd0

2π

∫ 0=

A u( ) 0→ R0 ∞→ A u( ) 0→

σ2xc

------⎝ ⎠⎛ ⎞ πH′′ R 0( ) 2στ

σpsxsy 1 ρxy2–

------------------------------------ e

uub

-----–

ex u–( )2

2σp2

------------------–

1 A u( )+( ) ud0

∞

∫=

σ x( )2π( )3 2/ H′′ R 0( ) 2στ

4sxsy 1 ρxy2–

------------------------------------------------- e

uub

-----–

ex u–( )2σp

2----------------

2

–ud

0

∞

∫ e

uub

-----–

ex u–( )2σp

2----------------

2

–A u( ) ud

0

∞

∫+⎝ ⎠⎜ ⎟⎛ ⎞

=

10

If the non-linearity of the Marshall-Palmer relation is not neglected then the computation of thewaveform requires the numerical integration of the attenuation term and then the numerical integra-tion of relation (17).

3.3- Barrick and Lipa integration

Circular gaussian cells

For a gaussian cell, the integration of (25) gives:

( 26)

where is the zero order modified Bessel function of the first kind. Using the relationship be-tween and , we obtain:

( 27)

is defined by:

( 28)

Circular constant cellFor a rain cell of diameter d and constant rain rate R, by simple geometric consideration we ob-

tain:

( 29)

where is defined by:

( 30)

or:

( 31)

Thus, the radar cross section can be expressed as:

( 32)

A u( ) ARe

2H ′ ′u

r2---------------–

e

x02

r2----–

e

2x0 2uH ′ ′

r2---------------------------- θcos–

θd0

2π

∫ ARe

2H ′ ′u

r2---------------–

e

x02

r2----–

I0

2x0

r2-------- 2H′′u⎝ ⎠⎛ ⎞= =

I0

u ψ

σN2xc

------⎝ ⎠⎛ ⎞ σ0στ

2π----------- 1 erf

x

2σp

-------------⎝ ⎠⎛ ⎞+ e

xub

-----– σ0στ

πσp----------- ARe

x02

r2----–

e

x u–( )2

2σp2

------------------– uu'b------–

Io

2x0

r2-------- 2H′′u⎝ ⎠⎛ ⎞ ud

0

∞

∫+=

u'b

u'bub 1 H a⁄+( )

1 2ubH( ) r2⁄+------------------------------------=

A u( )1

2π------ θd

θ– 1

θ1

∫=

θ1

2H′′u x02 2 2uH′′x0 θ1cos+ + r2=

θ12uH′′2x0

------------------x0

2 r2–

2x0 2uH′′---------------------------+

⎝ ⎠⎜ ⎟⎛ ⎞

acos=

σN2xc

------⎝ ⎠⎛ ⎞ σ0στ

2π----------- 1 erf

x

2σp

-------------⎝ ⎠⎛ ⎞+ e

xub

-----–

+=

σ0στ

πσp----------- AR e

x u–( )2

2σp2

------------------– uu'b------–

2uH′′2x0

------------------x0

2 r2–

2x0 2uH′′---------------------------+

⎝ ⎠⎜ ⎟⎛ ⎞

acos ud

0

∞

∫

11

Circular exponential cellNo analytical integration

Bands

No analytical integration for gaussian and exponential band.For a rain band of constant rain rate R, of width l, centred on x0, by simple geometric considera-

tion we obtain by a calculus similar to the one used for circular constant cell:

( 33)

User defined rain fields

In order to allow the possibility of analysing any rain field , a programme has been devel-oped in which the rain field can be defined in the local terrestrial coordinate system . I t isthen transformed into the altimeter system , to obtain . This field can be used tocompute the attenuation term (20). Relation (17) can be integrated using both integration methoddescribed earlier.

3.4- Comparison of integration methods

Two integration methods have been considered to compute the altimeter waveforms in presenceof rain, one neglecting the non-linearity of the Marshall-Palmer relation (called thereafter Barrickand Lipa method), and a numerical method (called Numerical Integration or NI). It should be notedthat for constant rate cell the two computations are equivalent. Fig. 4. presents a comparison of thetwo different methods for an exponential rain cell of 10 mm/hr and 5 km. The difference of attenu-ation reaches almost 0.5 dB and 7% of the leading edge slope. This difference is particularly notice-able near the rain cell centre.

The waveform deformation by rain is noticeable near the leading edge as well as within the pla-teau of the waveforms (see Fig. 5.) The approximate integration method underestimates the defor-mation in all part of the waveforms.

σN2xc

------⎝ ⎠⎛ ⎞ σ0στ

2π----------- 1 erf

x

2σp

-------------⎝ ⎠⎛ ⎞+ e

xub

-----–

+=

σ0στ

πσp----------- AR e

x u–( )2

2σp2

------------------– uu'b------– x0 r–

2uH′′--------------⎝ ⎠⎛ ⎞acos

x0 r+2uH-------------⎝ ⎠⎛ ⎞acos–⎝ ⎠

⎛ ⎞ ud

∞

∫

R x y,( )O x y, ,( )

O u θ, ,( ) R u θ,( )

12

Figure 4: Comparison of integration methods for an exponential rain cell defined and . Evo-lution of the waveform as the satellite fly over the rain cell, (a) Barrick and Lipa integration, (b) numerical integra-tion. (c) Variation of the attenuation, BL integration (red line), NI (blue line). (d) Variation of the leading edge slope.

0

0.2

0.4

0.6

0.8

−10 −5 0 5 10

50

100

150

200

250

300

350

distance from cell center (km)

T (

ns)

(b)

0

0.2

0.4

0.6

0.8

−10 −5 0 5 10

50

100

150

200

250

300

350

distance from cell center (km)

T (

ns)

(a)

−10 −5 0 5 10−2.5

−2

−1.5

−1

−0.5

0

distance from cell center (km)

Atte

nuat

ion

(dB

)

(c)

−10 −5 0 5 10

0.6

0.7

0.8

0.9

1(d)

distance from cell center (km)

lead

ing

edge

slo

pe

R 10mm/hr= d 5km=

13

Figure 5: Comparison of altimeter waveforms as the satellite fies over an exponential rain cell definedand . (a) Barrick and Lipa integration, (b) numerical integration (c) difference between the two methods.The waveforms are given every 1.3km, the most attenuated corresponding to zero distance between the satellite na-dir and the rain cell centre.

Joint influence of integration method and significant wave heightGiven the form of relation (17) in particular of the term

( 34)

there is an interaction between the method used to estimated the attenuation term and the distribu-tion of the sea surface elevations. The higher the significant wave height the more important the im-pact of the attenuation term on the waveform will be.

To better understand this influence of the attenuation term one can analyse a simple case. Let usconsider an exponential rain cell centred on the satellite nadir, the rain rate field R is defined by

( 35)

The attenuation term for the NI method is given by:

0 100 200 3000

0.2

0.4

0.6

0.8

1

T (ns)

(a)

0 100 200 3000

0.2

0.4

0.6

0.8

1

T (ns)

(b)

0 100 200 300−0.06

−0.04

−0.02

0

T (ns)

(c)

R 10mm/hr=d 5km=

T 1 ex u–( )2

2σp2

------------------– 12π------ A u θ,( ) θd

0

2π

∫⎝ ⎠⎛ ⎞=

R R0e2uH ′ ′

r-------------------–

=

14

( 36)

And for the BL method is given by:

( 37)

The difference between the two attenuation terms is evident in Fig. 7.

Figure 6: Atmospheric transmission as a function of u. for 10 km exponential rain cell of 10 and 25 mm/hr, NI: numer-ical integration, BL, Barrick and Lipa integration.

A u( )1

2π------ 10

0

2π

∫2HCα R0e

2uH ′ ′r

-------------------–

⎝ ⎠⎛ ⎞

β

10----------------------------------------------–

dθ 10

2HCαR0e2uH ′ ′

r-------------------–

β

10-----------------------------------------–

= =

A u( ) 1 10

2HCαR0β

10-----------------------–

1–⎝ ⎠⎜ ⎟⎛ ⎞

– e2uH ′ ′

r-------------------–

=

0 50 100 1500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

u

Atte

nuat

ion

(line

ar)

25 mm.hr−1 NI25 mm.hr−1 BL10 mm.hr−1 NI10 mm.hr−1 BL

15

4. Waveforms in presence of rain

A first estimate of the radar signal attenuation by rain cells can be obtained using the Marshall-Palmer relation, assuming the altimeter footprint is entirely filled by constant rain. The two-way at-tenuation is then given by:

( 38)

The rain thickness is assumed to be 5 km which corresponds to the mean altitude of the freez-ing level (Goldhirsh and Katz, 1979).

Fig. 7 presents the attenuation coefficient and the total two way attenuation computed from theMarshall-Palmer relation. The attenuation can be considered as the upper limit of the effect of rainon the backscatter coefficient

Figure 7: Marshall-Palmer relation (a) Attenuation by rain (dB/km) as a function of rain rate for Ku and S band. (b)Two-way total attenuation of the signal assuming the altimeter footprint is entirely filled by rain.

The non-uniformity of the rain within the altimeter footprint can strongly distort the altimeterwaveform and thus alter the estimation of the altimetric parameters. This kind of perturbation canbe observed for quite low rain rate. It is important when analysing the effect of rain on the altimeterwaveforms to quantify these impacts as well as possible. The best way to conduct such a studywould be to use the ocean waveform retracking as defined in the ENVISAT radar altimeter process-ing. This is however too computer intensive. In practice, we choose to estimate a number of param-eters which give an idea of firstly the attenuation of the signal, secondly the deformation of thewaveform leading edge, and thirdly the deformation of the waveform plateau.

A 10

2Hck p

10---------------–

=

Hc

10−1

100

101

102

10−4

10−3

10−2

10−1

100

Rain rate (mm/hr)

Atte

nuat

ion

(dB

.km−

1 )

S Ku

0 5 10 15 200

1

2

3

4

5

6

7

8

9

10

Rain rate (mm/hr)

Tot

al tw

o−w

ay A

ttenu

atio

n (d

B)

KuS

(a) (b)

16

4.1- Estimation of the waveform distortions

4.1.1- Attenuation

The attenuation of the signal is computed using the method defined to estimate the Automaticgain control of the TOPEX altimeter (Marth et al, 1993)

( 39)

where and are the average of samples 17 to 48 (assuming the waveform track point is 32.5)of respectively rain affected and rain free waveforms.

4.1.2- Leading edge slope

A very simple form has been adopted for the estimation of the leading edge slope variation. It isdefined as follow:

( 40)

where and are the rain affected and rain free waveforms.

4.1.3- Deformation of the plateau, off-nadir angle

As it has been shown in section 3.4, the plateau of the altimeter waveform can be strongly modi-fied by the presence of rain cells. The slope of the plateau is used in the altimeter waveformprocessing to estimate the off-nadir angle of the satellite. We used the estimate of the off-nadir an-gle as an indicator of the waveform distortion. The off-nadir angle is defined by (SSALTO, 1999)

( 41)

where and .

The term represents the slope of the plateau region. In practice, thisslope is determined by linear regression of the logarithm of for .

4.2- Practical computation

4.2.1- Envisat Radar altimeter configurationAltitude, : 800 km

■Ku band

Frequency: 13.575 GHz

Bandwidth: 320, 80, 20 MHz

3dB antenna beamwidth: 1.33˚

A 10σi

σio

-----⎝ ⎠⎜ ⎟⎛ ⎞

log=

σi σi0

slopeσ39 σ25–

σ390 σ25

0–---------------------=

σ σ0

ξ

ξ1

2 1 2γ---+⎝ ⎠

⎛ ⎞--------------------- 1

σilog σ jlog–α∆t

--------------------------------+⎝ ⎠⎛ ⎞=

γ4

2 2log--------------

ψb

2------sin⎝ ⎠

⎛ ⎞2

= α 4c

γH′--------=

σilog σ jlog–( ) α∆t( )⁄σi i 40...80=

H

17

compressed pulse duration: 3.125 ns (vertical resolution of 0.47 m) ■S band

Frequency: 3.2 GHz

Bandwidth: 160 MHz

3dB antenna beamwidth: 5.25˚

compressed pulse duration: 6.25ns (vertical resolution of 0.94 m)

4.2.2- Set of geophysical parameters

As the numerical integration method and exponential rain cell appear better suited for the analysisof the effect of rain on altimeter measurements they have been used to compute the rain affectedwaveforms. The rain thickness has been set for all computation to 5 km (corresponding to the meanaltitude of the freezing level). The distance nadir/ centre of the rain cell has been set to zero whichcorresponds to the maximum attenuation. It should be noted that for small intense cellsmight not represent the maximum distortion of the waveform plateau (see fig. 5).

The influence of the following parameters have been considered ■rain rate ■rain cell diameter ■significant wave-height .

The following values have been used for the 3 parameters ■ 1, 2, 4, 6, 8, 10, 14, 18, 22, 26, 30, 34, 38, 42, 46, 50 mm/hr ■d 1, 2, 3,4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20km ■ 0.5, 1, 2, 3, 4, 5, 6, 7, 8, 9,10 m

4.3- Ku band data

Fig. 8 presents several Ku band waveforms computed for different rain cell characteristics, andFigs. 9 to 11 presents the attenuation, the leading edge slope and off-nadir as a function of the dif-ferent parameters (R, , d).

From these figures, it can first be seen that the influence of significant wave height is only margin-al. It is only noticeable for very high sea states and light rain. In a first order approximation, the ef-fect of can be neglected (with less than 3% error) in the definition of a rain flag.

The attenuation, the variation of the leading edge slope and the off-nadir angle increase rapidlywith the rain rate and rain cell diameter. The attenuation exceeds 0.5 dB for rain rates over 3 mm/hrand 1dB for rates over 6 mm/hr. The distortion of the leading edge exceed 10% for rates over 2 mm/hr. The off-nadir angle is even more sensitive to rain and exceeds 0.1˚ for rain cells over 2 mm/hrand 3 km diameter.

It should be noted that the effect of the cell diameter is important for small cells and tend to satu-rate when the cell area is comparable to the altimeter footprint area.

These results are similar to the ones presented by Tournadre (1998) and Quartly el al. (1999a).

x0 0=

Rd

Hs

R

Hs

Hs

Hs

18

Figure 8: (Ku band Envisat waveforms in presence (a) of a 5 km exponential rain cell for different rain rate R 1, 2, 4, 6,8, 10, 14, 18, 22, 26, 30, 34, 38, 42, 46, 50 mm/hr., (b) of a 20 mm/hr exponential rain cell for different rain cell di-ameter d 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 14, 15, 16, 17, 18, 19, 20 km, (c) of a 10 mm/hr 10 km exponentialrain cell for 0.5, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 m. The dashed lines represents the rain free waveform (for (c) the redones for =0.5 m and the black one for =10m.

0 50 100 150 200 250 300 350 4000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

T (ns)0 50 100 150 200 250 300 350 400

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

T (ns)

0 50 100 150 200 250 300 350 4000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

T (ns)

(a) (b)

(c)

HsHs Hs

19

Figure 9: Attenuation as a function of rain rate , significant waveheight and cell diameter , for 5 km thick ex-ponential rain cells.

−0.2 −0.1 0

5 10 15 20

2

4

6

8

10

d (km)

Hs (

m)

R0 = 1 mm/hr

−2 −1.5 −1 −0.5

5 10 15 20

2

4

6

8

10

d (km)

Hs (

m)

R0 = 6 mm/hr

−8 −6 −4 −2 0

5 10 15 20

2

4

6

8

10

d (km)

Hs (

m)

R0 = 22 mm/hr

−4 −2 0

10 20 30 40 50

2

4

6

8

10

R0 (mm/hr)

Hs (

m)

d = 2 km

−15 −10 −5 0

10 20 30 40 50

2

4

6

8

10

R0 (mm/hr)

Hs (

m)

d = 8 km

−20 −10 0

10 20 30 40 50

2

4

6

8

10

R0 (mm/hr)

Hs (

m)

d = 20 km

−20 −10 0

10 20 30 40 50

5

10

15

20

R0 (mm/hr)

d (k

m)

Hs = 1 m

−20 −10 0

10 20 30 40 50

5

10

15

20

R0 (mm/hr)

d (k

m)

Hs = 2 m

−20 −10 0

10 20 30 40 50

5

10

15

20

Hs = 10 m

R0 (mm/hr)

d (k

m)

R0 Hs d

20

Figure 10: Leading edge slope as a function of rain rate , significant waveheight and cell diameter , for 5 kmthick exponential rain cells

0.9 0.95 1

5 10 15 20

2

4

6

8

10

d (km)

Hs (

m)

R0 = 1 mm/hr

0.4 0.6 0.8 1

5 10 15 20

2

4

6

8

10

d (km)

Hs (

m)

R0 = 6 mm/hr

0.2 0.4 0.6 0.8

5 10 15 20

2

4

6

8

10

d (km)

Hs (

m)

R0 = 22 mm/hr

0.2 0.4 0.6 0.8

10 20 30 40 50

2

4

6

8

10

R0 (mm/hr)

Hs (

m)

d = 2 km

0 0.5 1

10 20 30 40 50

2

4

6

8

10

R0 (mm/hr)

Hs (

m)

d = 8 km

0 0.5 1

10 20 30 40 50

2

4

6

8

10

R0 (mm/hr)

Hs (

m)

d = 20 km

0 0.5 1

10 20 30 40 50

5

10

15

20

R0 (mm/hr)

d (k

m)

Hs = 1 m

0 0.5 1

10 20 30 40 50

5

10

15

20

R0 (mm/hr)

d (k

m)

Hs = 2 m

0 0.5 1

10 20 30 40 50

5

10

15

20

R0 (mm/hr)

d (k

m)

Hs = 10 m

R0 Hs d

21

Figure 11: Off-nadir angle (˚) as a function of rain rate , significant waveheight and cell diameter , for 5 kmthick exponential rain cells

0.040.060.080.10.120.14

5 10 15 20

2

4

6

8

10

d (km)

Hs (m

)

R0 = 1 mm/hr

0.1 0.15 0.2 0.25

5 10 15 20

2

4

6

8

10

d (km)

Hs (m

)

R0 = 6 mm/hr

0.2 0.3 0.4 0.5

5 10 15 20

2

4

6

8

10

d (km)

Hs (m

)

R0 = 22 mm/hr

0.1 0.2 0.3 0.4 0.5

10 20 30 40 50

2

4

6

8

10

R0 (mm/hr)

Hs (m

)

d = 2 km

0.2 0.4 0.6 0.8

10 20 30 40 50

2

4

6

8

10

R0 (mm/hr)

Hs (m

)

d = 8 km

0.2 0.4 0.6

10 20 30 40 50

2

4

6

8

10

R0 (mm/hr)

Hs (m

)

d = 20 km

0.2 0.4 0.6 0.8

10 20 30 40 50

5

10

15

20

R0 (mm/hr)

d (k

m)

Hs = 1 m

0.2 0.4 0.6 0.8

10 20 30 40 50

5

10

15

20

R0 (mm/hr)

d (k

m)

Hs = 2 m

0.2 0.4 0.6 0.8

10 20 30 40 50

5

10

15

20

R0 (mm/hr)

d (k

m)

Hs = 10 m

R0 Hs d

22

Figure 12: Attenuation, slope of the leading edge and off-nadir angle (˚) as a function of rain rate , and cell diameter, for 5 km thick exponential rain cells and = 1.5m. Detail of figure 9 to 11. The solid lines represent the 1dB

attenuation isoline, the 0.9 slope isoline and the 0.1˚ isoline. The dashed lines represent the 0.5dB isoline and the0.95 slope isoline.

4.4- S Band data

The effect of rain on S band data is almost negligible as it can be seen in figure 13. Compared tothe rain free waveforms the attenuation is always less than 0.05 dB even for 50 mm/hr rain.

−4 −3 −2 −1

2 4 6 8 10

5

10

15

20

rain rate (mm/hr)

Dia

met

er (

km)

Attenuation

0.5 0.6 0.7 0.8 0.9

2 4 6 8 10

5

10

15

20

rain rate (mm/hr)

Dia

met

er (

km)

Slope

0 0.1 0.2 0.3

2 4 6 8 10

5

10

15

20

rain rate (mm/hr)

Dia

met

er (

km)

Offnadir angle

R0d Hs

23

Figure 13: S band altimeter waveform in presence (a) of a 10 km exponential rain cell for different rain rate R 1, 2, 4, 6,8, 10, 14, 18, 22, 26, 30, 34, 38, 42, 46, 50 mm/hr., (b) of a 50 mm/hr exponential rain cell for different rain cell di-ameter d 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 14, 15, 16, 17, 18, 19, 20 km.

100 200 300 4000

0.2

0.4

0.6

0.8

1

T (ns)

(a)

100 200 300 4000

0.2

0.4

0.6

0.8

1

T (ns)

(b)

24

5. Definition of a rain flag

The results of the Envisat waveforms modelling in presence of rain show that the S band data areunaffected by rain and that the Ku band data are strongly attenuated by rain rate over 3 mm/hr anddiameter over 5km.

The strong attenuation of the Ku band signal by rain and the differential effect between the twochannels can be used, as in the case of the TOPEX dual frequency altimeter, to define a rain flag.The method is identical to the one presented by Quartly et al, 1996 and Tournadre and Morland,1997 for Topex.

As the S band is unaffected by rain, the flag will look for occurrences where is significantlyattenuated as compared to .

( 42)

where is the Ku/S band relationship, A is an attenuation threshold (which can be fixed or equal to1.9 ,standard deviation of the relation. can be considered as the expectedfrom the measurement.Remark: The relation varies also with significant wave height (Elfouhaily et al., 1998). This ef-fect is only significant for wave heights less than 3m , and then the variation is only of order 0.1dB(Quartly et al., 1999b). An analysis of Topex data should be conducted to test this effect on rainflagging.

Fig. 14 presents the Ku/S band attenuation which can be expected for different rain rate andcell diameters, as well as the leading edge slope and off-nadir angle. For rain rates over 3 mm/hrand 5 km diameter the attenuation is over 0.7dB, the variation of the leading slope over 10% and theoff-nadir angle over 0.14˚. The retrieved altimeter parameters can therefore be considered as erro-neous. Going back to relation (42), considering fig. 14, a safe flag should detect any attenuationover 0.3-0.5dB to avoid any problem.

Analysis of Topex data using the rain flag defined by relation (42) showed that problems can ariseunder low wind speed conditions. To ensure the presence of cloud a liquid water criterion was alsointroduced (Tournadre and Morland, 1997) in the form of

where is the microwave radiometer cloud liquid water estimate and a threshold fixed to be0.2mm.

For each rain cell used in the study, the cloud liquid water content was estimated using first theMarshall-Palmer drop size distribution (Ulaby et al.,1981),

( 43)

where is the number of drops of diameter d (in m) per unit volume, and b (m-1)is related to the rainfall rate R (mm/hr) by .

For a given rain rate the liquid water content is given by

( 44)

σ0Ku

σ0S

∆σ0 f σ0S( ) σ0

Ku– A>=

frms σ0

S( ) f f σ0S( ) σ0

Ku

σ0S

f

∆σ0

Lz Lz0>Lz Lz0

p d( ) N0e bd–=

p d( ) N0 8 6×10=b 4100R 0.21–=

mv

mvπ6---106 p r( )r3 rd

0

∞

∫8 12×10

41004----------------πR0.84= =

25

Figure 14: Attenuation (a), leading edge slope (b) and off-nadir angle (c) as a function of rain rate for rain cell diameterfrom 1 to 20 km. The heavy solid lines indicate limit of validity for altimeter parameter retrieval.

For a circular rain exponential rain cell, the total water content is

( 45)

where (in g) is total mass of water within the rain cell, the rain thickness and r the rain celldiameter. The equivalent cloud liquid water (in mm) measured by the microwave radiometer of 20km resolution is defined as:

( 46)

Fig. 15 presents the Ku band attenuation as function of for different rain rates and cell diame-ters. It shows the large dispersion of the relation and points out the fact that a small heavy rain cellmight not be detected by the microwave radiometer. Except for very small cells (<4 km) a 0.2 mmliquid water threshold will ensure the presence of either intense small cells or large light rain cells.

Fig. 15 also presents the Ku band attenuation as a function of the atmospheric correction (dB)(SSALTO, 1999) computed from the cloud liquid water content by

5 10 15 20−5

−4

−3

−2

−1

0

Rain rate (mm/hr)

Atte

nuat

ion

(dB

)

1 2 3 4 5 6 7 8 9 1011121314151617181920

5 10 15 200.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Rain rate (mm/hr)

Lead

ing

edge

slo

pe

5 10 15 200

0.1

0.2

0.3

0.4

0.5

Rain rate (mm/hr)

Off−

nadi

r an

gle

(o )

Rain cell diameter

L8 12×10

41004----------------πHcR0

0.84 xr--–⎝ ⎠

⎛ ⎞exp0.84

x xd( ) θd

0

∞

∫0

2π

∫ 2π 0.126051 R00.84r2Hc⋅ ⋅= =

L Hc

LzL

2π108---------------10 3–=

Lz

26

( 47)

It can be seen that as already shown from Topex (Quartly et al., 1996, Tournadre and Morland,1997) and ERS (Guymer et al., 1995) data this atmospheric correction greatly underestimates theattenuation of Ku band signal by rain.

Figure 15: Attenuation as a function of equivalent liquid water content (top) and atmospheric correctiuon.The rain rates given by the colour coding.

atmcor 2 0.042 0.145Lz+( )=

5

10

15

20

0 0.5 1 1.50

1

2

3

4

5

Lz (mm)

Atte

nuat

ion

(dB

)

5

10

15

20

0 0.2 0.4 0.6 0.8 10

1

2

3

4

5

Atmospheric correction (dB)

Atte

nuat

ion

(dB

)

27

6. Estimation of rain rate and rain cell diameter

From the different modelling of waveforms in the presence of a rain cell and from the estimates ofmicrowave radiometer cloud liquid water it is possible to derive a relation between attenuation/cloud liquid water and rain rate/cell diameter. This relation is presented in fig. 16. The figure clearlyshows that even if further studies are needed to test the reliability and precision of the rain rate/cellretrieval, a possibility exists to obtain much needed information on the oceanic precipitation.

Some preliminary studies could be done using the Topex/Poseidon archive. It should also be not-ed that an extra information on the rain cell diameter can be obtain by analysis of the along-trackvariation of attenuation and/or off-nadir angle.

Figure 16: Relationship between cloud liquid water attenuation and (left) rain rate and (right) cell diameter.

0 5 10 15

0 0.5 1 1.50

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Lz (mm)

Atte

nuat

ion

(dB

)

Diameter (km)

0 5 10 15 20 25

0 0.5 1 1.50

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Lz (mm)

Atte

nuat

ion

(dB

)

Rain rate (mm/hr)

28

7. Conclusion and recommendations

The analysis of the Envisat modelled waveforms in presence of rain shows that as in the case ofthe Topex/Poseidon altimeter a rain flag can be defined using the altimeter dual frequency capabili-ties. As this rain flag is mainly based on the measurements of the altimeter itself, it should performbetter than one based on coincident passive radiometer measurements.

An operational flag could be defined by relations (42) and (43), where the , relationshipcould be defined during the validation phase.

As different studies (Quartly et al, 1999b, Ge et al., 1997) showed, dual frequency altimeter datacan be used to study the climatology of oceanic precipitation. Both studies used Ku band attenua-tion to estimate rain rate.

It would be of primary importance to provide in the Envisat GDR the attenuation (or en-hancement) of the Ku band versus the S band .

This parameter could be used for climatological studies of rain, verification of rain flag (no gener-al reprocessing needed in case of a change of threshold for example (past experience with thechange of liquid water threshold for the Topex operational flag proved to be annoying), possibleanalysis of blooms or wave damping by rain (Ge et al, 1998), estimation of a corrected windspeed in presence of rain,..

σ0Ku σ0

S

σ0 σ0

σ0

29

8. References

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