Errors in surface rainfall rates retrieved from radar due to wind-drift

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1 ATMOSPHERIC SCIENCE LETTERS Atmos. Sci. Let. 6: (2) Published online in Wiley InterScience ( DOI:.2/asl.96 Errors in surface rainfall rates retrieved from radar due to wind-drift Steven A. Lack* and Neil I. Fox Department of Soil, Environmental and Atmospheric Sciences, University of Missouri Columbia, Columbia, USA *Correspondence to: Steven A. Lack, Department of Soil, Environmental and Atmospheric Sciences, 32 ABNR Bldg., University of Missouri Columbia, Columbia, MO 6211, USA. Abstract This article describes a series of experiments based on real data wherein the advection of the precipitation below the radar-sampled volume is estimated using Doppler radar determined wind fields. The experiments show that even at standard resolutions of 2 km the error can be extensive, and at higher resolutions and greater ranges (higher beam elevations) the errors become very large. Errors are assessed using different Z R relationships and resolutions as high as. km. Copyright 2 Royal Meteorological Society Received: 13 May 24 Revised: 24 January 2 Accepted: 2 January 2 Keywords: wind drift; drop sorting; errors in surface rainfall estimates 1. Introduction Wind-drift has been recognized as an issue since early in the days of attempts to use radar reflectivity in estimating rainfall totals at the surface, but this has been relatively ignored in today s applications of weather radar. Wind-drift has been identified as a problem, but it is never addressed as something that leads to large errors in estimating surface rainfall fields or as an error that could be corrected. In hydrological applications of weather radar, incorporating surface rainfall fields accurately, not only intensity but also spatial accuracy, is of key importance. Ignoring the effects of wind-drift on falling precipitation could cause errors in predicted streamflow, and other variables important to hydrological studies, especially on finer scales such as urban hydrological applications. The first publication that identifies the problem of wind-drift is found in Gunn and Marshall (19). They identify the parabolic trajectory of raindrops in a constant wind shear environment, and allude to the possibility that the distances along the ground could be quite large from the original location of the drop. A more modern reference to wind-drift and urban hydrology comes from Collier (1999); however, like earlier studies in wind-drift, no real quantification of the possible error is accomplished. Various other articles allude to the problem of wind-drift when incorporating estimated rainfall fields into hydrological models, but it is secondary to advanced methods in estimating variables inside the radar volume scan aloft. Even with dual-polarization and other techniques for accurately assessing drop shapes and drop size distribution in the target radar volume aloft, this will not enhance the spatial accuracy or surface rainfall fields without examining wind-drift. This article is an exploration in two methods for attempting to account for the wind-drift of falling precipitation to give a more accurate representation of the surface rainfall field that could be incorporated into various hydrological models. The first method is a bulk-advection scheme, involving a single representative fall speed for each grid square based upon that cell s reflectivity measurement. The second method involves a drop-sort method, which involves using drop size bins based upon the reflectivity for each grid square. Both of the above methods use adjoint wind fields with u- and v-components to advect the entire representative reflectivity of the grid square (method 1) or each bin within one grid square (method 2). In the following sections, the data sets used will be discussed followed by an explanation of the methodology for the two aforementioned methods in correcting surface rainfall rates by examining the effect of winddrift on precipitation. Case studies of both a stratiform and convective case will be examined, and a final discussion of the correction schemes, given different circumstances, will be addressed along with future directions for expanding this research. 2. Data sets The data sets used and referenced throughout this article come from the C-POL radar, which was located outside of Sydney, Australia used during the 2 Summer Olympics (Keenan et al., 23). Two cases were selected to run the wind-drift schemes, a severe convective case from 3 November 2 and a stratiform case from 18 November 2. The data used as input in the correction scheme include reflectivity, u-, v-, and w-components of the wind. The reflectivity Copyright 2 Royal Meteorological Society

2 72 S.A.LackandN.I.Fox was smoothed for a maximum possible return of 3 dbz and values less than dbz were filtered out of the data set. The u- and v-components were obtained using the adjoint wind retrieval scheme of Sun and Crook (1994). The w-components, to roughly simulate contributions of vertical velocity from updrafts and downdrafts, were obtained from a consideration of the continuity equation using the horizontal winds at a number of levels. For the first runs of the adjustment scheme, the CAPPI height for the four input variables was m with grid spacing of 2. km. For later considerations of topography within this study, high-resolution topography provided by the Bureau of Meteorology was reduced to a similar resolution to the Cartesian grid of reflectivity and wind. 3. Methodology The background problem to this theoretical advection of precipitation is to calculate the contribution from individual gridded data squares (pixels) to other nearby squares. This is accomplished by taking a Cartesian grid of reflectivity and overlaying u- and v-components of the wind, given by the Doppler radar, and applying some simplified expressions relating these variables. From the given data, calculations of drop fall speed can be made within an individual grid square leading to the wind-drift in the x- and y-directions of the drops, which results in a given pixel s contribution to another grid location. Finally, each square in a new grid is summed on the basis of the contributions from the other nearby squares in the system yielding a new distribution of reflectivity, or rainfall rate, at the surface. The correction scheme changes slightly when using a bulk-advection scheme or the drop-sorting scheme. A further explanation of the difference in the two schemes will follow. In either scheme, the first step in the process to generate a new field of surface rainfall rate by taking into account wind-drift is to calculate a fall time for the drops within one pixel. In the bulk-advection case, this is accomplished by assuming that all drops within one pixel have one average drop diameter. By making this assumption, one can apply any Z R relationship to convert reflectivity to a rainfall rate that is assumed uniform throughout the entire pixel. An equation (1) relating rainfall rate (R) in mm h 1 to fall speed (V f ) in m s 1, derived from Lacy (1977), can be incorporated. V f = 4.R (1/9) (1) Once fall speed is calculated, the time it takes for the entire grid square represented by the average drop to fall a certain distance can be found. This is accomplished by simply dividing the height of the CAPPI by the fall speed modified by the w-component of the wind to simulate the effects of updrafts or downdrafts. In the drop-sorting case, the estimation of fall speed comes from a distribution of drops divided into 2 drop-size bins. The fall speed of the drops is found by applying the relationships given by Rogers and Yau (1989) based on the average drop radius in each bin (2a and 2b). V f = k 2 r, 4 µm < r< 6 µm (k 2 = 8 3 s 1 ) (2a) V f = k 3 r., r > 6 µm (k 3 = (ρ a /ρ a ). cm. s 1 ) (2b Depending on the CAPPI height, the fall speed of the drops within a given bin is dependent on the density of the air (ρ a ) at that altitude. In the case of drop sorting, the time it takes to fall to the surface can be calculated for each bin. This allows for more dispersion of the intensity given by one grid square to other nearby grid squares based upon the number of drops in each bin. The number of drops in any given bin is given by integrating a modified Marshall Palmer distribution function for the given stratiform and convective cases in this study. The time it takes for each bin of drops to impact the surface is calculated by the same method as the bulk-advection case above. The individual grid square or size bin contribution to other grid squares is determined by simply multiplying the wind speed in the x- and y-directions by the fall time. The wind speed is determined by making an assumption on the shape of the wind profile from the surface to the elevation of the radar beam or CAPPI, which in the following examples will be a constant shear profile. Therefore, it is assumed that friction will take the velocity down to zero at the surface. Given the dimension of a single pixel, a critical radius of influence can be determined by finding the magnitude of the combined u- and v-components. If the critical radius is greater than the dimension of the individual square, the contribution to the original grid box is zero. This means that all of the precipitation from that square is being advected to a different location and most likely contributes to more than one other grid square within the given area. Once the critical radius is found for each grid square, it must be determined which original grid squares aloft contribute to each individual surface grid square. In most cases, multiple grid boxes will contribute to a single pixel, unless the wind is calm in a given column, making the critical radius zero resulting in no wind-drift effect. Given the distance each area of precipitation travels in the x- and y-directions, the fraction of overlap onto other grid squares is calculated using simple geometry, and these overlapping areas are represented as fractions of the original square s reflectivity to a new grid square (Figure 1). All the fractions over one grid square are summed yielding a new power, or reflectivity at that grid square. Once Copyright 2 Royal Meteorological Society Atmos. Sci. 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3 Errors in surface rainfall rates 73 Figure 1. A simplified illustration of the drop-sorting scheme showing the spread of reflectivity of the 2 drop-size bins leading to contributions to multiple grid squares at the surface due to the different fall speeds for each bin after a north-westerly wind is applied. Only four size bins are shown in this example, but, in reality, there should be 2 different squares of the same dimensions each grid square or size bin in the given grid is accounted for, the new field can be compared with the original field of reflectivity, or rainfall rate, and the errors associated with wind-drift can be calculated for both a severe convective case and a stratiform case. 4. Initial data results Before examining the cases using real data from the Sydney 2 games, a look at the fundamental difference between the drop-sorting and bulk-advection scheme for one grid box can be shown. In this highly simplified example, the CAPPI height is set to m and the u- and v-components applied to the grid box with a given reflectivity are 6 m s 1 and 6 ms 1 respectively. The wind profile decays parabolically with height to zero at the surface. Setting up a small Cartesian grid, with grid spacing of 2. km, the dispersion of 4 dbz reflectivity in the upper-left corner of the grid for both the drop-sorting and bulk-advection schemes for convective and stratiform cases is shown (Table I). When evaluating the schemes from the actual precipitation data in Sydney, a comparison of adjusted rainfall totals, adjusted reflectivity and accumulation differences between the non-corrected (raw) data and each correction scheme are examined. The first comparison was done in the data s original framework, m CAPPI height with 2.-km grid spacing. For this abbreviated extended abstract, only the accumulation errors will be discussed, as they are of greatest importance. Illustrations of both the convective and stratiform cases are easily viewed once the correction schemes are run in the framework desired, giving useful data on under- and over-estimation by the raw reflectivity. In the convective case, the differences are much higher than the stratiform cases (see Figure 1 and Figure 2), which is to be expected, given the nature of the intensity, wind speed differences and areal coverage of the two events. However, differences between the bulk-advection and drop-sorting scheme are difficult to visualize on the below figures. In the convective case (Figure 2), the coupling of under- and over-estimation is to be expected, given the pattern of precipitation and veering wind profile. The accumulation differences at their highest absolute value are around 2 mm (Table II), which is significant. Even more significant is that an area of under- or overestimation can span distances of around km. When looking at the stratiform case (Figure 3), the differences are less obvious and follow no set pattern. The area and intensity differences are much less significant to the point where an adjustment scheme may not be needed when looking at only 3-h intervals. Both the drop-sorting and bulk-advection schemes give similar results graphically. It is noticeable that the errors resulting from the use of the CAPPI interpolation are as observable as those from the winddrift. Although the differences in the above cases illustrate the need for taking into account the wind-drift of precipitation for accurate spatial rainfall rates and totals especially for convective events; the errors due to wind-drift must be examined on smaller scales using data that simulates high-resolution radar data for incorporation into high-resolution hydrological models. In this study, there was no high-resolution data available, so the raw data from the Sydney games was run on a scale that simply reduces both the grid Table I. A simple advection example is shown for the different schemes used in the experiment. The drop-sorting and bulk-advection schemes are shown in dbz Convective drop-sorting Convective bulk-advection Stratiform drop-sorting Stratiform bulk-advection Copyright 2 Royal Meteorological Society Atmos. Sci. Let. 6: (2)

74 S.A.LackandN.I.Fox 2 2 2 2 1 1 1 1 2 2 (a) 2 (b) 2 Figure 2. Animations of the accumulation differences from the convective case (3 November 2) for the 3-h period with grid spacing of 2.

Some statistics for the 3-h accumulations (in mm) for the different wind-drift schemes Case Scheme Height Res Max over Max under Max abs val Max R Max R corr Difference Conv DS 1. km 2. km 8.64 14.

4 74 S.A.LackandN.I.Fox (a) 2 (b) 2 Figure 2. Animations of the accumulation differences from the convective case (3 November 2) for the 3-h period with grid spacing of 2. km; bulk-advection (left) and drop-sorting (right). The scale used is ±2 mm. Table II. Some statistics for the 3-h accumulations (in mm) for the different wind-drift schemes Case Scheme Height Res Max over Max under Max abs val Max R Max R corr Difference Conv DS 1. km 2. km Conv BA 1. km 2. km Conv BA 1. km 1. km Conv BA 1. km. km Strat DS 1. km 2. km Strat BA 1. km 2. km Strat BA 1. km 1. km Strat BA 1. km. km Conv BA.7 km 2. km Conv BA.7 km. km Strat BA.7 km 2. km Strat BA.7 km. km spacing and height for the different iterations of the correction schemes.. Increasing resolution results When increasing the resolution from 2. km to 1. km to m, the differences become spread over a greater number of pixels within the domain and the magnitudes of the differences increases significantly (Figure 4). In the convective case, there are regions where the absolute value of the differences exceeds the range (2 mm) of the scale on the image. The differences for the stratiform case also approach the maximum on its scale ( mm) in some locations; in these regions, the higher deviations from the raw data are either highlighting differences from the original rainfall field and the correction scheme or pointing to areas where there could be errors in the raw data from anomalous propagation. 6. Incorporating topography The images in Figure illustrate the importance of topography when correcting for wind-drift. The drops in each bin or within the entire grid square are falling toward the surface at what is assumed to be a constant terminal velocity. If elevation changes are accounted for, some of the drops reach the ground sooner over higher terrain than over radar level, causing the effect of wind-drift to be reduced. In Figure, the difference after including topography can be seen over the higher elevations in the north-western region when compared with Figure Discussion Dispersion in the drop-sorting case can easily be shown in the single cell examination, or in the convective case s reflectivity field. In some areas, especially toward the north-east corner, there is noticeably more dispersion of reflectivity than in the bulk-advection scheme, which is to be expected. However, since the Copyright 2 Royal Meteorological Society Atmos. Sci. Let. 6: (2)

Errors in surface rainfall rates 7 2 2 2 2 1 1 1 1 2 2 (a) 2 (b) 2 Figure 3. Animationsoftheaccumulationdifferencesfromthestratiformcase(18Nov.2)forthe3-hperiodwithgridspacing of 2.

Animations of the accumulation differences from the convective case (3 November 2) (left) and from the stratiform case (18 November 2) (right) for the 3-h period using the bulk-advection scheme with

contribution of the smallest drops most likely contributes to the higher dispersion in the drop-sorting scheme, including the effects of evaporation would eliminate the smaller drop sizes in some

In this brief exploratory article, the bulk-advection scheme was used when increasing resolution in order to save computational time.

5 Errors in surface rainfall rates (a) 2 (b) 2 Figure 3. Animationsoftheaccumulationdifferencesfromthestratiformcase(18Nov.2)forthe3-hperiodwithgridspacing of 2. km; bulk-advection (left) and drop-sorting (right). The scale used is ± mm (a) 2 (b) 2 Figure 4. Animations of the accumulation differences from the convective case (3 November 2) (left) and from the stratiform case (18 November 2) (right) for the 3-h period using the bulk-advection scheme with -m grid spacing. contribution of the smallest drops most likely contributes to the higher dispersion in the drop-sorting scheme, including the effects of evaporation would eliminate the smaller drop sizes in some areas where the air is less close to saturation beneath cloud base. In this brief exploratory article, the bulk-advection scheme was used when increasing resolution in order to save computational time. The results are quite similar with the drop-sorting events except for slightly higher dispersion in the drop-sorting case, which can be neglected if an assumption on evaporation or coalescence of the smallest drop sizes is made. Some of the numerical errors are shown in Table II. Again, the bulk-advection scheme is used mainly for time efficiency, but similar numerical values should be expected for the drop-sort scheme as shown in the comparison of the DS and BA schemes at the lowest resolution (1. km height and 2. km horizontal). Most of the errors are to be expected, in that, as the resolution is increased the magnitude of the errors increases, and as the CAPPI height is decreased, the errors also decrease. In attempting to use highresolution radar data in hydrological models, the ideal data source to use is the lowest scan and the lowest grid spacing. The terrain of the area of interest governs the lowest scan available and the technology of the radar limits the horizontal resolution. Table II shows that fairly high error results in reducing the lowest scan to 7 m with m horizontal resolution. The errors are of similar order of magnitude that would represent a spatial error of an entire grid square. This could be quite significant over small catchments and applications of urban hydrology where a fine horizontal resolution is necessary. If using the lowest Copyright 2 Royal Meteorological Society Atmos. Sci. Let. 6: (2)

76 S.A.LackandN.I.Fox C-POL centered Sydney topography 8 2 6 2 1 2 2 3 3 4 2 2 4 1 4 6 4 1 2 2 3 3 4 4 8 (m) 1 2 (a) (b) 2 Figure.

km; however, in this case, -m spacing is used to exaggerate the error, the scale used is again ±2 mm.

6 76 S.A.LackandN.I.Fox C-POL centered Sydney topography (m) 1 2 (a) (b) 2 Figure. C-POL centered topography (left) applied to the bulk-advection scheme for the convective case (3 Nov. 2) (right, animation). The actual grid spacing is 2. km; however, in this case, -m spacing is used to exaggerate the error, the scale used is again ±2 mm. beam elevation, errors will result at higher ranges from the radar; however, locations near the radar will benefit from smaller errors due to wind-drift. Also of importance is the final column, the difference between the total rainfall of the correction and the raw data; this highlights that the wind-drift schemes implicitly handle enhanced precipitation due to convergence and reduced precipitation due to divergence. Although there are some differences between the maximum and minimum estimations, one needs to be aware that the coding allows precipitation to be advected out of the entire region. Thus, in some cases, convergence and divergence are handled well within the grid; however, a net overall divergence often occurs since, inherently, precipitation will be advected out of the domain at any given time step without new echoes replacing what is lost. Further conclusions, and additional tables and figures can de found at the web address listed in the next section. 8. Conclusion and future work Although this method of adjustment is solely based on winds at one level that are estimated from Doppler velocities, surface and model data (Sun and Crook 1994), it illustrated the need to account for the wind-drift of precipitation even at high resolutions for hydrological modelling. As the horizontal resolution is increases, the errors become more significant within the domain. Reducing the height of the lowest reflectivity scan or the elevation of the beam reduces the error, but the error resulting from changes in the horizontal resolution still dominates. When incorporating topography into the wind-drift scheme, errors are reduced in regions of higher elevations, but increase in areas that are below the radar height. The lowest scan used is a function of the height of the topography, so that beam blockage and anomalous propagation is reduced. The dropsorting scheme is more computationally intensive, and therefore takes longer to run as it has to account for 2 drop-size bins. However, even in real time application, this is not significant. For more information, visit: research.html. Future work in the area of enhancing the corrections to surface rainfall fields derived from radar reflectivity will involve the addition of an evaporation scheme. To be successful, the scheme will have to incorporate below cloud relative humidity information and apply it to the falling drop sizes with multiple level horizontal wind fields. Different fall speed calculations will also be explored in future tests of the different correction schemes. The evaporation scheme and different experiments with the Sydney 2 data will continue in the hope of making the data as physically realistic as possible. The current schemes and other advances in the schemes will be used on cases from the United States in the near future and will be compared against a field of rainfall ground truth. Eventually, the corrected rainfall fields will be incorporated into a hydrological model so that direct comparisons of streamflow and the resultant errors can be explored. Acknowledgements The authors would like to thank Michael Sleigh and Andrew Crook for their help and advice concerning the processing of the radar data. Copyright 2 Royal Meteorological Society Atmos. Sci. Let. 6: (2)

7 Errors in surface rainfall rates 77 References Collier CG The impact of wind-drift on the utility of very high spatial resolution radar data over urban areas. Physics and Chemistry of the Earth (B) 24: Gunn KLS, Marshall JS. 19. The effect of wind shear on falling precipitation. Journal of Meteorology 12: Keenan T, Joe P, Wilson J, Collier C, Golding B, Burgess D, May P, Pierce C, Bally J, Crook A, Seed A, Sills D, Berry L, Potts R, Bell I, Fox N, Ebert E, Eilts M, O Loughlin K, Webb R, Carbone R, Browning K, Roberts R, Mueller C. 23. The Sydney 2 world weather research programme forecast demonstration project: overview and current status. Bulletin of the American Meteorological Society 84: Lacy R Climate and Building in Britain. Building Research Establishment Report, Department of the Environment. HMSO: London. Rogers RR, Yau MK A short course in cloud physics, 3rd edn. Butterworth-Heinemann: Woburn, MA; p 29. Sun J, Crook NA Wind and thermodynamic retrieval from single-doppler measurements of a gust front observed during PHOENIX II. Monthly Weather Review 122: Copyright 2 Royal Meteorological Society Atmos. Sci. Let. 6: (2)

Lei Feng Ben Jong-Dao Jou * T. D. Keenan

P2.4 CONSIDER THE WIND DRIFT EFFECTS IN THE RADAR-RAINGAUGE COMPARISONS Lei Feng Ben Jong-Dao Jou * T. D. Keenan National Science and Technology Center for Disaster Reduction, Taipei County, R.O.C National