Nat. Hazards Earth Syst. Sci., 10, 149–158, 2010 www.nat-hazards-earth-syst-sci.net/10/149/2010/ © Author(s) 2010. This work is distributed under the Creative Commons Attribution 3.0 License.
Natural Hazards and Earth System Sciences
Time-dependent Z-R relationships for estimating rainfall fields from radar measurements L. Alfieri1,2 , P. Claps1 , and F. Laio1 1 Dipartimento 2 Institute
di Idraulica, Trasporti ed Infrastrutture Civili (DITIC), Politecnico di Torino, Torino, Italy for Environment and Sustainability (IES), Joint Research Centre, EC, Ispra, Italy
Received: 26 March 2008 – Revised: 29 October 2009 – Accepted: 22 December 2009 – Published: 26 January 2010
Abstract. The operational use of weather radars has become a widespread and useful tool for estimating rainfall fields. The radar-gauge adjustment is a commonly adopted technique which allows one to reduce bias and dispersion between radar rainfall estimates and the corresponding ground measurements provided by rain gauges. This paper investigates a new methodology for estimating radar-based rainfall fields by recalibrating at each time step the reflectivity-rainfall rate (Z-R) relationship on the basis of ground measurements provided by a rain gauge network. The power-law equation for converting reflectivity measurements into rainfall rates is readjusted at each time step, by calibrating its parameters using hourly Z-R pairs collected in the proximity of the considered time step. Calibration windows with duration between 1 and 24 h are used for estimating the parameters of the Z-R relationship. A case study pertaining to 19 rainfall events occurred in the north-western Italy is considered, in an area located within 25 km from the radar site, with available measurements of rainfall rate at the ground and radar reflectivity aloft. Results obtained with the proposed method are compared to those of three other literature methods. Applications are described for a posteriori evaluation of rainfall fields and for real-time estimation. Results suggest that the use of a calibration window of 2–5 h yields the best performances, with improvements that reach the 28% of the standard error obtained by using the most accurate fixed (climatological) Z-R relationship.
1
Introduction
Advances achieved in the recent past in radar technology and in the methods for processing data are leading to an increasing confidence toward the use of radar-based rainfall Correspondence to: L. Alfieri (
[email protected])
estimates into hydrologic analyses and simulations. An accurate knowledge of the spatial characteristics and the amount of rainfall falling on a catchment area is of crucial importance in flood forecasting and warning systems (Smith, 1993; Claps and Siccardi, 1999; Arnaud et al., 2002; Brath et al., 2004), and can substantially improve the allocation of water resources for agricultural uses as well as for hydroelectric production (Alfieri et al., 2006). Further, an accurate quantitative precipitation estimation through radar measurements can provide an aid to the understanding of relations between point and areal rainfall (Bacchi and Ranzi, 1996) and for defining the probability distribution of annual rainfall intensity for large return periods (i.e., high rainfall rates), due to the large amount of data that the radar collects at each scan (e.g., Koistinen et al., 2006). The procedure for evaluating radar-based rainfall fields requires (i) to define in the best possible manner the reflectivity field (Z) induced by precipitation falling toward the Earth surface, and (ii) to link Z to an estimate of the rainfall rate (R), eventually with the aid of actual precipitation measurements. Ciach and Krajewski (1999) stressed the importance to distinguish the search for a physical dependence between rainfall reflectivity and rainfall intensity in a specific precipitation system from the goal of producing the most accurate radar-based predictions of the rainfall field at the ground level. This second objective, which is the main focus of this paper, is often pursued by pairing radar measurements (aloft) with ground data provided by rain gauge networks. The adoption of a unique climatological relationship to link Z and R is a widespread practice for estimating rainfall fields, due to its simplicity of use and its ability to provide, on average, low-biased estimates. The literature concerning this topic reports a large number of different Z-R relationships between radar reflectivity and the corresponding rainfall rate in the form of power-laws, such as those listed by Battan (1973) and Doviak and Zrnic (1984), which include the widely adopted formulations proposed by Marshall and Palmer (1948), Joss and Waldvogel (1970),
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Woodley et al. (1975), among others. However, because of the considerable variability that the raindrop size distribution shows during a rainfall event, the actual Z-R relation changes continuously in space and time. Therefore, the major drawback of using a unique relationship is that it cannot account for the broad variability of the true Z-R relation in the presence of different types of precipitation (e.g., convective or stratiform), as well as the variations that occur within each rainfall event (e.g., see Richards and Crozier, 1983; Smith and Krajewski, 1993; Lee and Zawadzki, 2005). Several approaches for improving the radar estimates of rainfall based on radar-gauge comparisons have been proposed in the past years (e.g., Brandes, 1975; Zawadzki, 1975; Collier et al., 1983; Austin, 1987; Ulbrich and Lee, 1999). These methods often try to reduce the estimation uncertainty by introducing additional information such as the precipitation type (e.g., convective or stratiform), the distance of the gauge from the radar, the elevation of the radar beam, and so forth (see for example Joss and Lee, 1995; Anagnostou and Krajewski, 1999a, b; Gabella and Amitai, 2000). Recent applications of real-time procedures are reported in Seo and Breidenbach (2002), Chumchean et al. (2006), Germann et al. (2006), Chiang et al. (2007). Legates (2000) carried out a real-time calibration of the Z-R power-law relationship by considering the radar-gauge pairs of the previous month characterized by non-zero rainfall, in order to account for seasonal fluctuations due to different types of precipitation and for possible drifts of the hardware calibration. It is known that the actual Z-R relation is subject to variations on much shorter time scales, but it is still not clear whether an optimal calibration period for providing the best rainfall estimates is definable. In the present paper, we propose a simple methodology for producing accurate radar-based estimates of rainfall intensity, by readjusting the coefficients of the Z-R relationship continuously in time considering short calibration windows. The procedure is tested on 19 rainfall events, with no recorded snow on the ground, that occurred in the northwestern Italy between 2003 and 2006. We use reflectivity measurements from a weather radar and rainfall data from a network of 20 rain gauges located within a range of 25 km from the radar. The next section describes the procedure devised for estimating rainfall fields from radar-gauge pairs. Section 3 gives some information on the case study and the data adopted; then it shows the results obtained with the two proposed techniques and a comparison with those of some literature methods. Some conclusions are reported in the final section. 2
Methods
The basic idea behind this work is that, for each time step t, one can estimate as many Z-R relationships as the number of the available data pairs in the Z-R plane. When a two-para-
Nat. Hazards Earth Syst. Sci., 10, 149–158, 2010
meters power-law of the form Z = a · Rb
(1)
is adopted, the coefficients a and b can be obtained by fitting the relation to match at least two points identified by non-zero concurrent measurements of rainfall rate and radar reflectivity. The Z-R pairs can be taken either at the same spatial coordinates (i.e., the same rain gauge j ) at different times (i.e., {Z1 = Z(ti ,j ),R1 = R(ti ,j )} and {Z2 = Z(ti+1 ,j ),R2 = R(ti+1 ,j )}), or for the same time step ti , considering two neighboring rain gauges (i.e., {Z1 = Z(ti ,j ),R1 = R(ti ,j )} and {Z2 = Z(ti ,j + 1),R2 = R(ti ,j + 1)}), where it is conventionally assumed that j + 1 is the rain gauge closest to j . If the measurements were not affected by errors, for (ti+1 − ti ) approaching zero (or for the distance between j and j + 1 approaching zero) the relation obtained by using only two Z-R pairs would be the most suitable to convert reflectivity measurements in rainfall rates. Of course, the relation would change when moving to a different place or considering another instant in time. The above described procedure is not practicable, due to a number of limitations which affect the problem. First, the radar reflectivity and the ground-rainfall measurements are subject to several sources of error (e.g., see Steiner et al., 1999). As a consequence, the adoption of only two Z-R pairs for estimating the coefficients a and b would provide scarcely-reliable and highly fickle relationships. A further major source of uncertainty is due to the non-homogeneity of the volume sampled by the two instruments and to a possible temporal lag between the two measured variables (e.g., Zawadzki, 1975). The radar carries out instantaneous measurements of volume aloft, whose size varies with the distance from the radar. The rain gauges that we considered measure cumulative rainfall depths at the ground level, by sampling an area of 200 cm2 for a duration as long as the sampling time (10 min for this study). Because of the difficulty to define Z-R pairs which refer to the same volume of atmosphere sampled, rainfall measurements are usually aggregated over longer durations. Likewise, radar data are averaged over time and space, by considering a number of pixels from the reflectivity maps nearby the position of the rain gauges and a duration corresponding to the same one adopted for the rainfall data aggregation. In this work, the choice of the aggregation period is one hour in time and nine radar pixels in space (i.e. a 1.5×1.5 km2 area) centered on the pixel which contains the considered rain gauge. In order to obtain more stable and reliable Z-R relationships it is necessary to increase the number of Z-R pairs to be considered for estimating the a and b coefficients. We propose to estimate a different power-law relationship for each hourly time step ti , where the coefficients a and b are obtained by considering all the available Z-R pairs with nonzero rainfall rate recorded in a calibration window of duration www.nat-hazards-earth-syst-sci.net/10/149/2010/
L. Alfieri et al.: Continuous-time radar rainfall estimation d, taken in the proximity of the selected instant ti . As a consequence, each power-law regression is carried out on a number of Z-R pairs with an upper bound NRG ·d, given by the number of rain gauges NRG times the duration d (in hours) of the calibration window. Different calibration windows of durations between 1 and 24 h are tested, and results are shown and discussed in Sect. 3.2 and 3.3. Parameters of each powerlaw relationship are calculated by minimizing the squared differences between the observed and the estimated rainfall values. The latter are obtained through ∗ Z log aˆ R = 10 ∧ − , (2) 10bˆ bˆ where aˆ and bˆ are the coefficients to estimate. Equation (2) is obtained from Eq. (1), by considering that the reflectivity data are provided in the form Z ∗ =10 log Z [dBZ]. Note that, despite Eq. (2) represents a straight line in the bilogarithmic plane log Z-log R, the regression procedure is carried out in the Z-R plane by means of non-linear techniques. The adopted optimization algorithm is a subspace trust region method and is based on the interior-reflective Newton method. The optimized coefficients aˆ and bˆ are estimated iteratively by taking those obtained from the linear regression method (see Eq. 3) as first attempt values. The non-linear optimization causes a substantial heteroscedasticity of the estimation residuals, but it is necessary for obtaining regressions producing small errors also at high rainfall intensities, which is important when dealing with extreme precipitation events. Further, this approach has the advantage of producing almost unchanged Z-R relationships when a threshold for the minimum considered value of reflectivity is set. This is better clarified at the end of this section. In order to assess the performances of the mentioned nonlinear regression method we carried out a comparison of the results obtained with this method and with the commonly adopted method based on the equation logZ = log aˆ 0 + bˆ 0 logR,
(3)
derived from Eq. (1), where aˆ 0 and bˆ 0 are the estimated values of a and b. Results of such comparison are discussed in Sect. 3.2. The calibration procedure was then used both for (i) a posteriori evaluation of the rainfall field, hereafter referred to as “continuous-time” (CT) readjustment, and (ii) for real-time (RT) estimation. In the first case, for each time step ti , the Z-R relationship is estimated by considering the Z-R pairs for a time window of duration d centered on ti , i.e., taking all the available pairs between ti − d/2 and ti + d/2. In real-time monitoring the aim is to estimate the current rainfall field from reflectivity measurements at the same instant ti , where each relation is estimated from the Z-R pairs of the preceding hours. In this case the relation to use at the time ti was inferred from the Z-R pairs between ti−1 − d and ti−1 . Figure 1 shows a scheme of the calibration windows to assume www.nat-hazards-earth-syst-sci.net/10/149/2010/
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Fig. 1. Calibration windows considered for estimating the Z-R relationship at a given time step ti . Both windows for real-time (RT) estimation and continuous-time (CT) readjustment are shown.
(both for CT readjustment and RT estimations) for a certain duration d and time step ti , where t and d are expressed in hours. Some further details are needed to clarify the operational use of the method. For some time windows no valid Z-R relationship was found, either because few Z-R pairs were available for the corresponding calibration window (i.e., rainfall was zero for most of the rain gauges) or because the coefficients of the power-law did not comply with the imposed constraints, aˆ >1 and bˆ >1. We rejected the estimated Z-R relationships characterized by coefficients aˆ
