Berenguer Ferrer, Marc

Lee, G.

Sempere Torres, Daniel

Zawadzki, Isztar

English

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A methodology for attenuation correction based on the minimization of a cost function

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A METHODOLOGY FOR ATTENUATION CORRECTION BASED ON THE MINIMIZATION OF A COSTFUNCTIONMarc Berenguer*(1), GyuWon Lee(2), Daniel Sempere-Torres(1), Isztar Zawadzki(2). (1) Grup de Recerca Aplicada en Hidrometeorologia. Universitat Politecnica de Catalunya, Barcelona (Spain).(2) J. S. Marshall Radar Observatory, McGill University, Montreal (Quebec, Canada).1. INTRODUCTIONAttenuation of radar signal due to precipitation isone of the important limitations for quantitative uses ofC-band radars. However, most of the operationalradars (outside USA) are C-band and, therefore,significantly affected by attenuation.Although in 1954 Hitschfeld and Bordan derivedthe analytical expression to invert the attenuationproblem (supposing a potential Z-k relationship), theyalready showed that this equation is very unstablewith errors in radar measurements (in particular witherrors in the radar calibration) and in the estimation ofthe Z-k parameters.More recently, the use of attenuated-frequenciesfor air- or satellite-borne radar has promoted thedevelopment of different techniques to correct forattenuation (using single- or double-frequency, single-or double-beam and single- or double-polarizedradars), constraining the attenuation equation. In thecase of single-frequency methodologies, most of themare based on the Surface Reference Technique(Iguchi and Meneghini, 1994; Marzoug and Amayenc,1994; Iguchi et al., 2000) that consists on constrainingthe total Path Integrated Attenuation (PIA) with thesurface return. This idea was implemented by Delrieuet al. (1997) in ground-based radars, using mountainreturns as constraint.In this study, a methodology for attenuationcorrection based on the minimization of a costfunction is formulated. The aim of this methodology isto take into account as many sources of informationas possible to constrain the attenuation equation.In the presented formulation, this informationcomes from measurements at rain gages and theradar returns from mountains.2. ATTENUATION EQUATIONSAssuming a potential relationship between thespecific attenuation, k(r), and the (non-attenuated)reflectivity, k r( ) =α ⋅ Z r( )β , the attenuation equationcan be written as:Zm r( ) = Z r( ) ⋅δC ⋅ A r( )= Z r( ) ⋅δC ⋅ exp −0.46 ⋅ α ⋅ Z s( )β ds0r∫⎡ ⎣ ⎢ ⎤ ⎦ ⎥ (1)where Zm(r) is the attenuated reflectivity at range raffected by a calibration error, δC ; the term A(r)represents the attenuation reduction of reflectivity.Expressed in dB, A(r) becomes the Path IntegratedAttenuation (PIA),PIA r( ) = −10 ⋅ log A r( )[ ] = 2 ⋅ α ⋅ Z s( )β ds0r∫ (2a)  PIA r( ) = z r( ) − zm r( ) −10 ⋅ log δC( )( ) (2b)where Z(r) is in mm6m-3 and z(r) in dBZ.3. THE PROPOSED METHODThe aim of the proposed methodology is to satisfythe attenuation model, but imposing some constraintsto avoid the instability problems. The idea is thatthese constraints may come from different sources(ideally, calculable from radar measurements). In aprevious formulation of this methodology, rain gagemeasurements were incorporated to constrainHitschfeld and Bordan’s equation (Berenguer et al.,2002). In this case, we will also use measured groundecho returns.The proposed methodology consists on theminimization of the following cost function, whichimposes the corrected reflectivity field to be coherentwith the attenuation model but limited by theconstraints:J δC,α, β, Z( ) = w1 ⋅ z ri( ) − zm ri( ) + PIA ri( )( )[ ]2i=1nbins∑+w2 ⋅ z rg,j( ) −10 ⋅ log a ⋅ Rjb( )[ ]2j=1ngages∑+w3 ⋅ PIA rgc,k( ) − PIAgc rgc,k( )[ ]2k=1ngc∑(3)where, wi are the weights given to each term of thecost function; z(ri) is the corrected reflectivity field, thatis also used to calculate PIA(ri) from equation (2a);zm(ri) is the measured (attenuated) reflectivity to becorrected. a, b are the parameters of the Z - Rrelationship used to convert the rain-rate measured atthe gages, Rj, into reflectivity. Finally, PIA(rgc,k) andPIAgc(rgc,k) are the PIAs at ground clutter areas, on theone hand, calculated using equation (2a), and, on theother, estimated from the difference betweenmeasured ground echo returns and mean groundecho returns in dry weather conditions (using equation(2b)).P2C.1*Corresponding author address: Marc Berenguer,GRAHI-UPC, Jordi Girona, 1-3 D1, E08034-Barcelonae-mail: berengue@grahi.upc.es.Therefore, the philosophy of this methodology isquite similar to the idea of Serrar et al. (2000) in thesense that it estimates an “effective” calibration errorthat enables the attenuation correction.4. PRELIMINAR DISCUSSION OF THE METHODA methodology based on the minimization of acost function is proposed for the correction ofattenuation of radar signal due to precipitation. Theproposed function avoids the divergence of theanalytical solution and allows more constraints to beeasily incorporated.In this case, the formulation of the cost functionincludes a first term that imposes that the correctedreflectivity must satisfy the attenuation model and atthe same time, it is constrained by rain gageinformation and mountain returns (second and thirdterms, respectively).The use of rainfall rate from rain gages is notoptimal because it represents an external constraint(not directly derivable from radar).  010203040506018:01 February 2001             00:02      06: 12: 18: 00:03 06:Z(dBZ)Fig. 1. Evolution of the mean reflectivity for a givenmountain as measured by the Barcelona C-bandradar of the Spanish INM during a non-rainy period.It must be stated that using ground echoes asconstraint may have associated difficulties: (i) theirvariability in dry-weather conditions (see Fig. 1), (ii)the fact that mountain returns change from dry to wetconditions or (iii) that ground echoes are not uniformlydistributed in the radar scanning area to be corrected.Finally, an important point for future work is theinclusion of more constraints into the cost function,especially from radar measurements. Some of theseconstraints could come from the overlapping region oftwo or more radars or, imposing some kind of spatialcontinuity of the reflectivity field in contiguousazimuths (Vignal, 1998).Acknowledgements: This study has been carried outin the framework of the EC project VOLTAIRE (EVK2-CT-2002-00155) and of the Spanish CICYT projectREN2000-1755-C03-01. Thanks are also due to theINM for providing radar data.ReferencesBerenguer, M., G. W. Lee, D. Sempere-Torres, and I.Zawadzki, 2002: A variational method forattenuation correction of radar signal. 2 n dEuropean Conf. on Radar Meteor., 11-16.Delrieu, G., S. Caoudal, and J. D. Creutin, 1997:Feasibility of using mountain return for thecorrection of ground-based X-band weather radardata. J. Atmos. Oceanic Technol., 14, 368-385.Hitschfeld, W. F. and J. Bordan, 1954: Errors inherentin the radar measurement of rainfall at attenuatingwavelengths. J. Meteor., 11, 58-67.Iguchi, T. and R. Meneghini, 1994: Intercomparison ofsingle-frequency methods for retrieving a verticalrain profile from airborne or spaceborne radar data.J. Atmos. Ocean. Tech., 11, 1507-1516.Iguchi, T., T. Kozu, R. Meneghini, J. Awaka, and K.Okamoto, 2000: Rain-profiling algorithm for theTRMM precipitation radar. J. Appl. Meteor., 39,2038-2052.Marzoug, M. and P. Amayenc, 1994: A class ofsingle- and dual-frequency algorithms for rain-rateprofiling form a spaceborne radar. Part I: Principleand tests from numerical simulations. J. Atmos.Oceanic Technol., 11, 1480-1506.Serrar, S., G. Delrieu, J.-D. Creutin, and R. Uijlenhoet,2000: Mountain reference technique: Use ofmountain returns to calibrate weather radarsoperating at attenuating wavelengths. J. Geophys.Res., 105, 2281-2290.Vignal, B., 1998: Identificacion par methode inversedes profils verticaux de reflectivite et des profilsradiaux d'attenuation, LCPC. Ph.D. Thesis.

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A methodology for attenuation correction based on the minimization of a cost function

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