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Ocean and Sea Ice SAF

Scientific Report

OSI_AVS_15_02

Correlating extremes in wind and stress divergence with extremes in rain over the

Tropical Atlantic

Gregory P. King

Marcos Portabella

Wenming Lin

Ad Stoffelen

Version 1.0

13 November 2017

OSI_AVS_15_02 – Correlating extreme divergence with extreme rain

2

DOCUMENTATION CHANGE RECORD

Reference: OSI_AVS_15_02

Issue / Revision : Date : Change : Description :

Version 0.5 2017-10-27 Gregory King Draft version

Version 0.6 2017-10-28 Wenming Lin Revision

Version 0.7 2017-10-31 Marcos Portabella Revision

Version 1.0 2017-11-08 Ad Stoffelen Revision

OSI_AVS_15_02 – Correlating extreme divergence with extreme rain

3

Summary

Air-sea fluxes are greatly enhanced by the winds and wind structures generated by Mesoscale Convective Systems (MCSs). In contrast to global numerical weather prediction models, space-borne scatterometers are able to resolve the small-scale wind variability in and near MCSs. Heavy rain events (rain bursts) occurring in MCSs produce strong gusts and large divergence and curl in surface winds. In this report wind fields from the ASCAT-A and ASCAT-B tandem mission, collocated with Meteosat Second Generation rain fields, were used to develop a methodology capable of identifying and quantifying correlations between wind and rain. Categories of wind divergence, wind stress divergence, and rainfall intensity were defined and a spatial neighbourhood spanning a 25km-by-25km block of WVCs was searched to identify coincidences of extreme rain and extreme convergence/divergence. This showed that when there is extreme rain, there is extreme convergence/divergence in the vicinity. The coincidences were tabulated in 3-by-3 and 2-by-2 contingency tables from which cross-correlations were calculated for each time step in the collocation. The resulting response curves for extreme convergence and extreme divergence each had a well-defined peak. The time lag for the convergence peak was 30 minutes, implying that extreme rain generally appears after (lags) extreme convergence. The overall conclusion then is that the temporal scale of moist convection is determined by the slower updraft process. Results for wind divergence and wind stress were qualitatively similar, wind stress divergence showing the stronger response. This is probably due to its focus on high winds. The report also notes that extreme convergence/divergence are concentrated in spatial patches and recommends for further study to relate the spatial features with the statistics, and as a focus for studying the changes in the divergence fields between the ASCAT-A and ASCAT-B passes.

Correlating extremes in wind and stressdivergence with extremes in rain over the

Tropical Atlantic

Gregory P. King1

Marcos Portabella1, Wenming Lin1, and Ad Sto�elen2

November 13, 2017

1Institute of Marine Sciences, Barcelona, Spain2Royal Netherlands Meteorological Institute (KNMI), De Bilt, The Netherlands

Version 1.0

Abstract

Air-sea fluxes are greatly enhanced by the winds and wind structures generatedby Mesoscale Convective Systems (MCSs). In contrast to global numerical weatherprediction models, space-borne scatterometers are able to resolve the small-scalewind variability in and near MCSs. Heavy rain events (rain bursts) occurring inMCSs produce strong gusts and large divergence and curl in surface winds. In thisreport wind fields from the ASCAT-A and ASCAT-B tandem mission, collocatedwith Meteosat Second Generation rain fields, were used to develop a methodol-ogy capable of identifying and quantifying correlations between wind and rain.Categories of wind divergence, wind stress divergence, and rainfall intensity weredefined and a spatial neighbourhood spanning a 25km-by-25km block of WVCswas searched to identify coincidences of extreme rain and extreme convergence/di-vergence. This showed that when there is extreme rain, there is extreme conver-gence/divergence in the vicinity. The coincidences were tabulated in 3-by-3 and2-by-2 contingency tables from which cross-correlations were calculated for eachtime step in the collocation. The resulting response curves for extreme convergenceand extreme divergence each had a well-defined peak. The time lag for the con-vergence peak was 30 minutes, implying that extreme rain generally appears after(lags) extreme convergence. The overall conclusion then is that the temporal scaleof moist convection is determined by the slower updraft process. Results for winddivergence and wind stress were qualitatively similar, wind stress divergence show-ing the stronger response. This is probably due to its focus on high winds. Thereport also notes that extreme convergence/divergence are concentrated in spatialpatches and recommends for further study to relate the spatial features with thestatistics, and as a focus for studying the changes in the divergence fields betweenthe ASCAT-A and ASCAT-B passes.

1

OSI-AVS-15-02 Correlating extreme divergence with extreme rain

Contents1 Introduction 5

2 Data and Study Area 7

2.1 ASCAT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 MSG Rain Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3 Study area and period . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3 Methodology 11

3.1 Work flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.2 Collocations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.3 Derived scatterometer products . . . . . . . . . . . . . . . . . . . . . 12

3.3.1 Wind components . . . . . . . . . . . . . . . . . . . . . . . . 123.3.2 Wind stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.3.3 Divergence products . . . . . . . . . . . . . . . . . . . . . . . 13

3.4 The RR

max

product . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4 Results 17

4.1 Classification of divergence events . . . . . . . . . . . . . . . . . . . . 174.2 Classification of rainfall intensity . . . . . . . . . . . . . . . . . . . . 214.3 Are extremes in MSG rain correlated with extremes in ASCAT di-

vergence? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.3.1 Contingency tables . . . . . . . . . . . . . . . . . . . . . . . . 244.3.2 Neighbourhood size . . . . . . . . . . . . . . . . . . . . . . . 264.3.3 Time evolution of conditional probabilities . . . . . . . . . . . 27

5 Summary and future work 29

List of Tables1 Comparison of the mean, standard deviation, skewness and kurtosis

(flatness) of the probability distributions of wind/wind stress diver-gence using three numerical methods: standard central di�erences(CD), central-di�erence-then-average (CDA) on 3-by-3 and 2-by-2blocks of WVCs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2 Categories of divergence. The symmetrical thresholds for wind/windstress divergence are defined, respectively, as c

u

= 1.2 ◊ 10≠4 s≠1,and c

·

= 1.8 ◊ 10≠6 N m≠3. . . . . . . . . . . . . . . . . . . . . . . . 183 Thresholds for rainfall intensity classification used by (i) the United

Kingdom Meteorological O�ce (UKMO, 2007), and (ii) the presentwork. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4 A generic 3-by-3 contingency table. . . . . . . . . . . . . . . . . . . 245 Contingency table for rain rate and divergence, where the rain rate

is treated as the forecast variable. CNE means ’correct non-event’,XCD = XC fi XD (Table 2), and BGR = MR fi LR (Table 3). . . . 24

6 Joint probabilities P (DIV flRain) and marginal probabilities P (DIV )and P (Rain) of rain rate and wind/wind stress divergence (Decem-ber 2012 collocations); wind stress divergence probabilities are inparentheses under those for wind divergence. . . . . . . . . . . . . . 25

Version 1.0 (November 13, 2017) Page 2 of 32


List of Figures1 Study area. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 Study area and overlapped ASCAT-A and -B swaths. In the tropics,

half the time the right swath of ASCAT-A overlaps the left swath ofASCAT-B and vice versa. . . . . . . . . . . . . . . . . . . . . . . . . 9

3 Distribution of diurnal precipitation in the Atlantic ITCZ. . . . . . . 104 The number of collocations in Winter (left) and Summer (right) vs

collocation time (the time recorded for the middle MSG frame ineach collocation). The collocations in the most eastern part of thestudy area occur at 08:30 UTC, while those in the most western partoccur at 13:30 UTC (see Fig. 2). . . . . . . . . . . . . . . . . . . . . 11

5 The central-di�erence-then-average scheme for estimating numericalderivatives. Illustrated using 2-by-2 block of WVCs. . . . . . . . . . 13

6 E�ect of the numerical method on the probability distributions P (Ò·u) (left) and P (Ò · ·) (right) calculated using standard central di�er-ences (CD 3-by-3, black), and di�erence-then-average (CDA 3-by-3,blue; CDA 2-by-2, red). The probability distributions are plotted instandardized coordinates, Z = (x ≠ µ)/‡, where µ is the mean and‡ is the standard deviation. . . . . . . . . . . . . . . . . . . . . . . . 14

7 Comparison of ASCAT and ECMWF probability distributions P (Ò ·u) (left) and P (Ò · ·) (right). The ECMWF forecast winds arecollocated in space and time with the ASCAT winds. . . . . . . . . 15

8 Comparison of empirical and Gaussian probability distributions (PDs)for wind/wind stress divergence. The Gaussian PDs (green curves)have the same mean and variance as the corresponding empirical PD.The left panels compare distributions as a function of divergence andare plotted semilog to emphasize the tails. The right panels compareusing quantiles in the form of a normal probability plot. The part ofthe distribution that is well-approximated by a Gaussian falls on astraight line. The data begins to deviate from that line at about the2nd and 98th percentiles, and these percentiles are used to define thethresholds used to classify events as either background or extreme. 17

9 Wind divergence calculated for the test case (an ASCAT pass overthe tropical mid-Atlantic). The divergence is separated into back-ground convergence/divergence and extreme convergence/divergence.Background convergence/divergence is shaded blue/pink (left panel)and extreme convergence/divergence is blue/red (right panel). Theyellow shading in the left panel identify the areas containing extremeconvergence/divergence shown in the right panel. The bottom panelszoom in on the area with the largest divergence. . . . . . . . . . . . 19

10 As in Fig. 9, but for wind stress divergence. . . . . . . . . . . . . . . 2011 Histogram of rain rates for the area covered by SCAT1 passes at the

time of the pass (t = 7, blue) and 45 minutes later (t = 10, red) ac-cumulated for the month of December 2012. Vertical lines are drawnindicating the United Kingdom Meteorological O�ce thresholds forlight rain (below 2.5 mm h≠1), moderate rain (2.5 - 10 mm h≠1),heavy rain (10 - 50 mm h≠1), and torrential rain (greater than 50mm h≠1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21



12 Spatial distribution of maximum rain rate (RR

max

) classified as ei-ther background (BGR = LR + MR) or extreme (XR) as described inthe text. The BGR distribution is shown in the upper left panel andthe XR distribution in the upper right for test case 20121205_1115 ;the lower panels zoom in on the latitude band having the largest rainrates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

13 Scattergrams (left) and conditional probability distributions (right)of divergence and RR

max

. Top panels: wind divergence; Bottompanels: wind stress divergence. The scattergrams use only data fromthe test case using RR

max

observed at the time of the SCAT1 pass(lag time · = 0). The vertical lines are drawn at the thresholdsseparating the three divergence categories: XC (blue circles), BG(black dots), and XD (red circles). The horizontal lines are drawn atthe thresholds separating the three rain categories LR, MR, and XR.The panels on the right show probability distributions conditionedon the rain rate categories and use all samples from December 2012. 23

14 Zoomed in region of the test case swath with largest precipitationand divergence and showing WVCs that are hits (black filled circles),false alarms (red open circles) and misses (blue open squares): winddivergence (left); wind stress divergence (right). . . . . . . . . . . . . 26

15 Histograms of the shortest distance from an XR grid point to a XCDgrid point (top), and from a BGR grid point to an XCD grid point(bottom). Distance is measured in widths of a WVC. The last binin each histogram counts all distances greater than 10 WVCs. . . . . 27

16 Time evolution of the conditional probabilities for convergence P (XC|f)and divergence P (XD|f), where f denotes the rain categories: LR,MR, and XR. The calculations use all data from December 2012. . . 28

17 Extreme value cluster size distribution for wind divergence (left) andwind stress divergence (right). Probability distributions (in terms ofarea) are shown as quantiles (in terms of number of WVCs). Thevertical dashed lines in each panel indicate areas comprising of 10and 20 WVCs. The area of one WVC is approximately 156 km2.The probability distributions show that the size distribution followsan approximate power-law. . . . . . . . . . . . . . . . . . . . . . . . 29

18 Sample result that compares information in SCAT1 with that inSCAT2 (50 minutes later). The top panels show the statistical dis-tributions of extreme wind stress convergence and divergence andthe bottom panels show, for the same ocean areas, how the SCAT1distributions change after 50 minutes (SCAT2 pass). . . . . . . . . . 30



1 IntroductionRain-induced downbursts are extensive in the tropics and, with deep convection,result in near-surface turbulent winds that drive intense vertical exchanges of mo-mentum, heat and moisture. Moreover, gust fronts produced by downbursts inMesoscale Convective Systems (MCSs) promote cloud development, and triggeringand self-aggregation of convection (Feng et al., 2015). This has an important im-pact on convective organization, on ocean circulation, and a�ects the net Hadleycirculation.

Convection is largely unresolved in global Numerical Weather Prediction (NWP)forecasting models and Global Climate Models. As a result, important rain-induceddynamics are missed and must be parameterized. However, current parameteriza-tions are poor, which limits understanding of tropical circulation. Therefore, itis critically important for NWP and climate modelling that parameterizations areimproved (Bony et al., 2015).

Space-borne C-band scatterometers such as the Advanced Scatterometer (AS-CAT) have proven to be very e�ective in capturing small-scale dynamical featuresand are only slightly degraded under rainy conditions (Lin et al., 2015a). Thusthey can be used to calculate divergence and curl for both wind and wind stressdivergence at high resolution.

Moist convection is the process whereby moist air converges at the surface –in response to convective instability (updraft) – rises, expands, cools, and thencondenses into cloud and rain drops and freezes aloft when a critical tempera-ture is reached. On the way up, condensation and freezing releases heat into theair, increasing its buoyancy and uplift, while the aggregated precipitating particlesevaporate and melt on the way down cooling the surrounding air and forcing it todescend and diverge near the ocean surface (downdraft).

Under the right conditions, moist convection leads to the amalgamation of in-dividual cloud systems (thunderstorms) that self-organize upscale into a MesoscaleConvective System (MCS). MCSs have a very large upper cirriform cloud structure,rainfall covering large contiguous rain areas of about 100 km or more in at least onedirection (Houze, 2004), and normally persist for several hours or more. MCSs maybe round or linear in shape, and include systems such as tropical cyclones, squalllines, and Mesoscale Convective Complexes. MCSs are an important link betweenatmospheric convection and the larger-scale atmospheric circulation.

In an MCS, the collective outflow of cold air in downdrafts of individual convec-tive cells spreads out horizontally, displacing the warmer ambient boundary layerair, forming cold pools. The gust front that forms at the periphery of a cold poolpromotes triggering and self-aggregation of convection (Feng et al., 2015). Largegradients in the surface wind (divergence and curl) are created by the convectionand gust front. This has a large impact on turbulent air-sea fluxes, which in turnhave an important impact on convective organization and ocean circulation.

Surface wind stress curl is a direct and often dominant source term in the vor-ticity budget of the upper ocean, with implications for mesoscale and basin-scalecirculation. Surface wind stress divergence in conjunction with sea surface temper-ature (SST) provide indications of air-sea interaction and the related stability ofthe marine atmospheric boundary layer.

Wind stress divergence has no analog to the dynamical connection betweenbasin-scale circulation and zonal integrals of the wind stress curl. However, ex-tremes in averaged wind stress divergence highlight regions of important air-seainteractions where SST modifies the overlying wind field. Local feedbacks then



modify the upper-ocean thermodynamics (Chelton et al., 2004).Frenger et al. (2013) conclude from their study of Southern Ocean eddies that

transient mesoscale ocean structures can significantly a�ect much larger atmo-spheric low-pressure systems that swiftly pass by.

It has been less clear, however, how strongly the ocean and atmosphere interacton the mesoscale, especially in the extra-tropics. What is known is that mesoscalesea surface temperature (SST) anomalies are globally correlated with near-surfacewinds (Chelton et al., 2004). Bryan et al. (2010) find in an analysis of a suite ofcoupled atmosphere-ocean model output that characteristics of frontal scale ocean-atmosphere interaction, such as the positive correlation between SST and surfacewind stress, are realistically captured only when the ocean component is eddyresolving.

Concurrent modifications of winds and clouds were detected for the distinctGulf Stream rings (Park et al., 2006) and for large-scale fronts such as the semi-permanent Agulhas Return Current (O’Neill et al., 2005; Liu et al., 2007). Inaddition, an SST-related change of rain rate was observed for the Gulf Stream andthe Kuroshio (for example, Xu et al. (2011)).

Park et al. (2006) examine wind stress convergence/divergence and curl nearrings, and a relationship between observed surface wind convergence/divergenceand cloudiness is demonstrated. The pattern is that divergence (curl) is maximalwhere the mean wind is oriented perpendicular (parallel) to the frontal boundary ofthe ring and hence parallel (perpendicular) to the SST gradient. On the downwind(upwind) side of a ring the convergence (divergence) is expected to cause local uplift(subsidence) and hence enhance (decrease) cloudiness.

Large-scale fronts are outnumbered by mesoscale eddies, which dominate theoceans kinetic energy and typically feature an SST anomaly. Frenger et al. (2013)show on the basis of observations how oceanic mesoscale eddies impact the atmo-sphere by changing not only wind, but also clouds and rainfall.

In this project, data are used from the ASCAT-A and ASCAT-B tandem missioncollocated with Meteosat Second Generation rain rates retrieved from the SpinningEnhanced Visible and Infrared Imager (SEVIRI). The overall goal of this projectwas to use the datasets synergistically to provide information helpful to the under-standing of moist convection and facilitate improvements in convective parameter-izations in NWP and Climate models. In order to attain that goal a methodologywas created that could identify and quantify correlations between the winds andrain in the ASCAT and MSG datasets. That methodology was the project’s coretask and is fully described and validated in this report. Briefly, the methodologyborrows techniques from forecast verification. Based on key features easily observedin the statistical distributions of divergence and rain rate, thresholds were definedthat delineated the data into a small number of categories. After regridding therain rates to the ASCAT swath grid, contingency tables were constructed. In theterminology of contingency tables, hits, false alarms and misses were identified, aswell as information concerning their spatial configurations. Finally, the time evo-lution of the conditional probabilities of convergence/divergence given rain givesresponse curves that rise to a well-defined peak, showing that extreme rain lags ex-treme convergence by about 30 minutes, while extreme divergence is in synchronywith extreme rain. It is also shown that the response is stronger for wind stressdivergence than for wind divergence.

The report is structured as follows. Section 2 describes the data and study area,section 3 describes the methodology, and results are given in section 4. Finally,section 5 gives a brief summary and some ideas for future work.



2 Data and Study Area2.1 ASCATThe MetOp-A and MetOp-B satellites were launched in October 2006 and Septem-ber 2012, respectively, and are operated by the European Organisation for theExploitation of Meteorological Satellites (EUMETSAT). They are in quasi-sun-synchronous orbits with an inclination angle of � = 98.6¶, and both cross theequator at about 09:30 (descending pass) and 21:30 (ascending pass). MetOp-Aand MetOp-B fly in tandem, following one another 50 minutes apart with overlap-ping swaths. The scatterometers onboard the two satellites are named ASCAT-Aand ASCAT-B.

The ASCAT scatterometer uses a dual-swath fan-beam configuration with two550 km wide swaths separated by a nadir gap of about 700 km and transmits atC-band (5.3 GHz) (Figa-Saldaña et al., 2002). The fan-beam configuration has con-sistent measurement geometry and instrument perfomance. C-band scatterometerwinds have proved to be very e�ective in capturing small-scale dynamical featuresand are only slightly degraded under rainy conditions Lin et al. (2015a,b).

The radar backscatter detected by the scatterometers goes through two levels ofprocessing to produce wind speed and wind direction. Level 1 processing involvesaveraging individual backscatter measurements and produces them on a regularlyspaced grid. Level-2 takes the Level-1 data and applies quality control, an inversionstep, and an ambiguity removal step. The inversion step applies an empiricallyderived geophysical model function (GMF) to relate backscatter (as a function ofthe wind direction) with the equivalent neutral-stability vector wind at a heightof 10 meters. Due to the nature of radar backscatter from the ocean surface,this procedure usually provides multiple solutions referred to as ambiguities. Anambiguity removal algorithm is applied to produce the selected winds. Level-2processing is carried out at the Royal Netherlands Meteorological Institute (KNMI)using the ASCAT Wind Data Processor (AWDP). The GMF used in the AWDP isCMOD5.n and ambiguity removal is carried out using a two-dimensional variationalmethod (2DVAR) (Vogelzang et al., 2009).

The ASCAT product is collocated with NWP forecasts from the European Cen-tre for Medium range Weather Forecasting (ECMWF) model. The NWP forecastsare interpolated to the swath grid and packaged with the product.

Rain a�ects the radar backscatter measured by scatterometers: the higher theradar frequency, the larger the impact of rain attenuation and scattering. ASCATuses C-band radar frequency, which results in winds that are much less a�ectedthan winds derived by the higher Ku-band frequency scatterometers. ASCAT qual-ity control flags about 0.5% of the data – which correspond to the most variablewind conditions, including heavy precipitation (Portabella et al., 2012; Lin et al.,2015a,b).

2.2 MSG Rain RatesThe MeteoSat Second Generation (MSG) dataset we work with consists of rain rate(RR) and cloud top temperatures (CTT). These parameters are retrieved (alongwith other cloud/radiation properties) from the Spinning Enhanced Visible andInfrared Imager (SEVIRI) using the KNMI Cloud Physical Properties (CPP) al-gorithm (Roebeling et al., 2006). Cloud physical properties are inferred from solarbackscattered radiation, and hence rain rates are only available during daytime.



The MSG rain product is available for the Atlantic Ocean, Europe and Africabetween longitudes [≠50¶

, 50¶] and latitudes [≠80¶, 80¶] (Figure 1) and has an ac-

curacy of 1 mm h≠1. The MSG rain rate builds upon cloud optical thickness (COT)retrievals. As a result of the non-linear relation between reflectance and COT (re-flectance saturates for thick clouds), there is no limit to retrieved COT. In practicelimits are set to both COT and the derived precipitation, but these high values areassociated with very high uncertainty (Private communication, Jan Fokke Meirink,KNMI). The MSG rain rate does not discriminate between convective and strati-form, although potentially some indication of this can be obtained from cloud toptemperatures.

The URLs for the MSG Cloud Physical Properties wiki pages are:• Product Description: http://msgcpp.knmi.nl/mediawiki/index.php/MSGCPP_

product_description;• Data Viewer: http://msgcpp.knmi.nl/mediawiki/index.php/MSG_Cloud_

Physical_Properties_%28CPP%29.

2.3 Study area and periodThe study area is the Atlantic region between latitudes 25¶S and 25¶N and boundedlongitudinally by the coasts of Africa and South America (see Figure 2). Someoverlapped ASCAT-A and -B swaths are also shown in the figure. In the tropics,half the time the right swath of ASCAT-A overlaps the left swath of ASCAT-B andvice versa.

The study investigates the winter December 2012 - February 2013, and thesummer June - August 2013.

Some context for our study is provided by Figure 3 which shows the climatologyof diurnal variability of precipitation from 14 years of TRMM measurements overthe Atlantic Inter-Tropical Convergence Zone (ITCZ) (Liu et al., 2008). The leftpanel in the figure shows the overall mean activity, and the two columns on the rightshow the distribution during di�erent periods of the day. The distribution labelled"09-12" (row 2, column 2) best reflects the overall spatial pattern and intensity ofthe activity sampled in our study.1

1We thank Chuntau Liu for Fig. 3, generated from web tools at https://atmos.tamucc.edu/trmm/.


http://msgcpp.knmi.nl/mediawiki/index.php/MSGCPP_product_description

http://msgcpp.knmi.nl/mediawiki/index.php/MSGCPP_product_description

http://msgcpp.knmi.nl/mediawiki/index.php/MSG_Cloud_Physical_Properties_%28CPP%29

http://msgcpp.knmi.nl/mediawiki/index.php/MSG_Cloud_Physical_Properties_%28CPP%29

https://atmos.tamucc.edu/trmm/


Figure 1: MSG footprint.

Figure 2: Study area and overlapped ASCAT-A and -B swaths. In the tropics, half thetime the right swath of ASCAT-A overlaps the left swath of ASCAT-B and vice versa.



Diurnal Change

MCSs (Atlantic) from 14 years of TRMM

PR 00-03 03-06

06-09 09-12

12-15 15-18

18-21 21-00

Figure 3: Distribution of diurnal precipitation in the Atlantic ITCZ from 14 years ofTRMM measurements.



3 Methodology3.1 Work flowThe work proceeded as follows. First a collocation data set was compiled for thestudy area (section 3.2), where each file in the set consisted of data arrays fromASCAT-A, -B and MSG rain rate products. Since the order of arrival of ASCAT-Aand -B across the study area varied, the wind data for the earlier (later) pass wasidentified and referred to as SCAT1 (SCAT2).

As our interest is in identifying strong updrafts and downdrafts, empirical prob-ability distributions were constructed for wind divergence P (Ò · u) and wind stressdivergence P (Ò · ·). The distributions were found to be fat-tailed, meaning thatthere exist intermittent (extreme) values that cause the distribution to depart froma Gaussian shape. A normal probability plot was used to determine a thresholdthat was then used to distinguish extreme events from Gaussian (i.e., background)events.

The following steps were then carried out on the scatterometer data in eachcollocation and saved in arrays.

1. Wind speed and direction were transformed to wind components in the cross-track, along-track coordinate system (section 3.3.1).

2. Wind stress magnitude was calculated using a simplified model (section 3.3.2).3. Wind stress components were calculated in the cross-track, along-track coor-

dinate system (section 3.3.2).4. Wind divergence and wind stress divergence were calculated using the CDA

2-by-2 method (section 3.3.3).The 3-km MSG rain rates were binned to scatterometer WVCs to produce

products that could be easily correlated with the ASCAT data (section 3.4).

3.2 Collocations

7891011121314150

5

10

15

20

25

30

Collocation Time (UTC)

Num

ber o

f Col

loca

tions

Collocations Winter (DJF) 2012/3

7891011121314150

5

10

15

20

25

30

Collocation Time (UTC)

Num

ber o

f Col

loca

tions

Collocations Summer (JJA) 2013

Figure 4: The number of collocations in Winter (left) and Summer (right) vs collocationtime (the time recorded for the middle MSG frame in each collocation). The collocationsin the most eastern part of the study area occur at 08:30 UTC, while those in the mostwestern part occur at 13:30 UTC (see Fig. 2).



A collocation dataset was created comprising of (i) ASCAT-A and -B windsfrom the overlapped portion of the two swaths, and (ii) MSG rain rate and cloudtop temperatures. The first scatterometer in each pass is called SCAT 1 and crossesthe study area at time t1. The second scatterometer is called SCAT 2 and crossesat t2 = t1 +50 minutes. (Half of the time the crossing order is AB and the other halfis BA.) Since MSG CTT/RR data is only available during daytime, times t1 and t2are between 08h30 and 13h30 UTC. (It is for this reason that only ASCAT-A and-B descending orbits are considered in this study.) A 17 point segment of the MSGtime series (binned every 15 minutes) is extracted, for times between t1 ≠ 2 hoursand t2 + 2 hours. The collocated data were stored in a netCDF file on disk afterthey were trimmed longitudinally to a little more than the East-West extent of theASCAT tracks and latitudinally to the North-South boundaries of the study area.

All available collocations over the Atlantic Ocean during the two study periods(Winter: December 2012 - February 2013; Summer: June - August 2013) are used.The number of collocations at di�erent times of the day are shown in the middleand lower panels of Fig. 4. No significant di�erence is apparent in the distributionsfor Winter and Summer.

3.3 Derived scatterometer productsThe observation times for the ASCAT-A and ASCAT-B data were scanned to iden-tify the first scatterometer to pass over the study area. Thereafter data from thefirst scatterometer was renamed SCAT1 and the second SCAT2.

3.3.1 Wind componentsASCAT wind vectors come as the 10m stress equivalent wind speed U10s

(de Kloeet al., 2017) and wind direction „

m

(where „

m

is the direction in the meteorologicalconvention). These are transformed to wind components (u

c

, u

a

) in the cross-swath,along-swath directions via the following steps. First, (U10s

, „

m

) are transformed tozonal (u) and meridional (v) using:

„

u

= ≠„

m

≠ 90¶

u = U10s

cos „

u

,

v = U10s

sin „

u

,

and then to cross-swath, along-swath components using:

u

c

= u · c, (1)u

a

= u · a. (2)

where c, a are unit vectors related to the unit vectors x (zonal), y in the, respectively,zonal and meridional directions by

c = x cos ◊

p

+ y sin ◊

p

(3)a = ≠x sin ◊

p

+ y cos ◊

p

(4)

The angle ◊

p

is the angle that a makes with y, which is given by

◊

p

=I

◊

asc

= –, ascending pass◊

dsc

= fi ≠ –, descending pass(5)



where – > 0 and depends on latitude. The method used to estimate – is describedin Vogelzang and Verhoef (2014). 2

3.3.2 Wind stressThe wind stress magnitude · and wind stress vector · are given by

· = Èfla

ÍCDn

|U10s

|2 (6)· = Èfl

a

ÍCDn

|U10s

|u (7)

where Èfla

Í = 1.225 kg/m3, and C

Dn

is the drag coe�cient for the equivalent neutralwind approximated by de Kloe et al. (2017)

C

Dn

= aU10s

+ b (8)a = 7.94 ◊ 10≠5 (m≠1

s) (9)b = 6.12 ◊ 10≠4 (dimensionless) (10)

The values for a and b used here were taken from (de Kloe et al., 2017, fig 18). Theabove is a simplification based on the average relationship between U10s

and windstress in the ECMWF model, and what we use in this work. Note that di�erentsurface layer models (for example, Bourassa, 2006; Beljaars, 1998; Liu et al., 1979)and auxilliary information (i.e., air temperature, sea surface temperature, relativehumidity, surface pressure) imply di�erent drag coe�cients and thus stress (vector)values.

3.3.3 Divergence products

Central Difference then Average (CDA)

i+1 fi+1,j fi+1,j+1

i fi,j fi,j+1

j j+1

O

��f

�x

�

i+1,j+ 12

� fi+1,j+1 � fi+1,j

�x

��f

�x

�

i,j+ 12

� fi,j+1 � fi,j

�x

��f

�x

�

i+ 12 ,j+ 1

2

=1

2

��f

�x

�

i+1,j+ 12

+

��f

�x

�

i,j+ 12

�

��f

�y

�

i+ 12 ,j

� fi+1,j � fi,j

�y

��f

�y

�

i+ 12 ,j+1

� fi+1,j+1 � fi,j+1

�y

��f

�y

�

i+ 12 ,j+ 1

2

=1

2

��f

�y

�

i+ 12 ,j

+

��f

�y

�

12 ,j+1

�

Figure 5: The central-di�erence-then-average scheme for estimating numerical deriva-tives. Illustrated using 2-by-2 block of WVCs.

2Our calculations used fortran code extracted from the “convert” library in the ASCAT Wind DataProcessor (AWDP).



-60 -40 -20 0 20 40 60Zu

10-6

10-5

10-4

10-3

10-2

10-1

100

Probab

ility

∇ · u Atlantic (25S, 25N) 2012-12

CD3x32x2

-60 -40 -20 0 20 40 60Zτ

10-6

10-5

10-4

10-3

10-2

10-1

100

Probab

ility

∇ · τ Atlantic (25S, 25N) 2012-12

CDCDA 3x3CDA 2x2

Figure 6: E�ect of the numerical method on the probability distributions P (Ò · u) (left)and P (Ò · ·) (right) calculated using standard central di�erences (CD 3-by-3, black), anddi�erence-then-average (CDA 3-by-3, blue; CDA 2-by-2, red). The probability distribu-tions are plotted in standardized coordinates, Z = (x ≠ µ)/‡, where µ is the mean and ‡

is the standard deviation.

In order to calculate wind and wind stress divergence, a numerical di�erencingscheme must be selected. As taking di�erences adds noise, it is desirable to carry outsome averaging to reduce noise. After some consideration we decided to comparethree numerical schemes, all variations on central di�erences. The first is standardcentral di�erences (CD) applied to a 3-by-3 block of WVCs; in this scheme not allthe information in the block is used. The second is central-di�erence-then-average(CDA) scheme, applied to the same 3-by-3 block of WVCs: central di�erences areapplied to each row of the block and then averaged; this is repeated for the columns.The third scheme is the same as the previous one, but applied to a 2-by-2 block ofWVCs. Figure 5 illustrates CDA using a 2-by-2 block.

The three schemes are inter-compared after transforming the data produced byeach into standardized (dimensionless) coordinates, followed by a comparison oftheir probability distributions (see Figure 6). Standardized coordinates are calcu-lated as follows. Let x

u

= Ò · u and x

·

= Ò · · , then their means and standarddeviation are µ

u

= Èxu

Í, ‡

u

=

È(xu

≠ µ

u

)2Í, and similarly for µ

·

and ‡

·

, whereangle brackets denote an ensemble average. We then calculate for each scheme:

Z

u

= (xu

≠ µ

x

)/‡

u

(11)Z

·

= (x·

≠ µ

·

)/‡

·

(12)

In the new coordinates the mean and standard deviation are, respectively, ÈZÍ =0 and ÈZ2Í = 1. The higher (dimensionless) moments remain unaltered – e.g.,skewness (asymmetry)

skewness = ÈZ3Í, (13)

and kurtosis (measure of flatness or fatness of the tails)

kurtosis = ÈZ4Í. (14)

The mean, standard deviation, skewness and kurtosis for the three schemes aregiven in Table 1. Di�erences in the standard deviation indicate the a�ect of noiseand filtering. Comparing the values in the table indicates that the CDA 2-by-2 (3-by-3) scheme has more (less) noise than the CD scheme. If the data were gaussian



TABLE 1: Comparison of the mean, standard deviation, skewness and kurtosis (flat-ness) of the probability distributions of wind/wind stress divergence using three nu-merical methods: standard central di�erences (CD), central-di�erence-then-average(CDA) on 3-by-3 and 2-by-2 blocks of WVCs.

MethodStatistic CD CDA 3-by-3 CDA 2-by-2Mean [10≠5 s≠1] 0.043 0.044 0.071

Wind Std. Dev. [10≠5 s≠1] 5.1 4.4 6.1Divergence Skewness -0.55 -0.41 -0.88

Kurtosis 26 19 38Mean [10≠7 N m≠3] 0.045 0.045 0.110

Wind Stress Std. Dev.[10≠7 N m≠3] 8.3 7.3 9.2Divergence Skewness -0.58 -0.32 -1.64

Kurtosis 124 73 163

distributed, then they would only di�er in the mean and standard deviation, andafter centering and scaling produce identical PDFs. As Fig. 6 and Table 1 show,the PDFs are near-identical only in the middle portion of their range, negativelyskewed, and have fat but similar tails (large kurtosis). These non-gaussian charac-teristics can be explained as follows. As wind divergence (downbursts) and moistconvergence at the surface are governed by di�erent physical processes, skewness inthe distributions may be expected. Moreover, due to the self-organisation of moistconvection on the smallest scales, non-gaussian contributions to the PDFs are likelytoo, resulting in higher-order statistical contributions (i.e., kurtosis).

In summary, for the objectives of the present work, we anticipate that thedeciding factor for selecting the best numerical scheme is not the noise producedwhen taking derivatives. Instead, as we are carrying out a rain analysis study andrain processes are spatially highly variable, the scheme with the smallest spatialfootprint appears most appropriate. Therefore, in the remainder of this work onlyresults from the CDA 2-by-2 method recommended by KNMI will be shown.

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2x 10−3

10−6

10−5

10−4

10−3

10−2

10−1

100

Wind DIV (s−1)

Probability

Wind DIVECMWF DIV

−5 −4 −3 −2 −1 0 1 2 3 4 5x 10−5

10−6

10−5

10−4

10−3

10−2

10−1

100

Stress DIV (Nm−3)

Probability

Stress DIVECMWF Stress DIV

Figure 7: Comparison of ASCAT and ECMWF probability distributions P (Ò · u) (left)and P (Ò · ·) (right). The ECMWF forecast winds are collocated in space and time withthe ASCAT winds.



For completeness, we briefly compare the probability distributions of each diver-gence product with the global forecast ECMWF winds packaged with the ASCATdata. These are shown in Figure 7 and illustrate the well-known fact that ECMWFforecast winds miss out small-scale wind structures associated to moist convection(Portabella et al., 2012) – and hence miss out all extreme convection.

3.4 The RR

max

productThe KNMI MSG rain rates are presented on a 3-by-3 km grid. The original datais retained when copied to the collocations. In order to easily correlate divergencewith rain rates, both quantities are defined on the same grid. That is, the samespatial neighbourhood that is used to calculate velocity gradients at a grid pointis used to define a representative rain rate for that grid point. The maximum 3kmrain rate within a neighbourhood is the most influential one, and this is the oneused to create the RR

max

product. Specifically, the divergence calculation uses a2-by-2 block of WVCs centered on the half-grid point (i + 1

2 , j + 12). The RR

max

product is created by searching for the maximum 3km rain rate in the area spannedby that block of WVCs and assigned to the same half-grid point.



4 ResultsIn this section the rationale to identify and define thresholds to classify divergenceand rain events is explained and justified. Next, contingency tables – the mathe-matical framework used to organize the statistics – will be introduced and applied.Contingency tables enable calculation and consideration of conditional probabilitiesand these will be used to correlate extreme convergence/divergence with extremerain and study its time evolution.

The emphasis is on statistics, but examination of specific cases is important forverifying and interpreting those statistics. This report uses results from colloca-tion 20121205_1115 (hereafter, referred to as the test case) to help illustrate andinterpret the statistical results.

4.1 Classification of divergence events

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2x 10−3

10−6

10−5

10−4

10−3

10−2

10−1

100

Wind DIV (s−1)

Probability

Wind DIVNormal Dist

−5 −4 −3 −2 −1 0 1 2 3 4 5x 10−5

10−6

10−5

10−4

10−3

10−2

10−1

100

Stress DIV (Nm−3)

Probability

Stress DIVNormal Dist

Figure 8: Comparison of empirical and Gaussian probability distributions (PDs) forwind/wind stress divergence. The Gaussian PDs (green curves) have the same meanand variance as the corresponding empirical PD. The left panels compare distributionsas a function of divergence and are plotted semilog to emphasize the tails. The rightpanels compare using quantiles in the form of a normal probability plot. The part ofthe distribution that is well-approximated by a Gaussian falls on a straight line. Thedata begins to deviate from that line at about the 2nd and 98th percentiles, and thesepercentiles are used to define the thresholds used to classify events as either backgroundor extreme.

The classification of convergence and divergence into di�erent categories requiresdefining thresholds. In order to make the assignment less prone to subjective de-cisions, we use the following strategy. The first step is to consider the empirical



probability distribution (EPD) and compare it to a Gaussian distribution with thesame mean and variance. This step is carried out for both wind divergence andwind stress divergence EPDs calculated using all December 2012 collocations (Fig-ure 8). The panels on the left of Fig. 8 are plotted semilog in order to emphasizethe tails of the EPDs. The panels on the right of the figure compare the EPD andGaussian using quantiles in a normal probability plot. In such plots, where thedata falls on a straight line is well-described by a Gaussian. The figures show thatthe data strongly deviate from a line in the lower and upper 2%, and hence we saythe EPDs have non-Gaussian tails. Furthermore, lower and upper thresholds canbe objectively defined, respectively, as the value of the divergence at the 2nd and98th percentile. The use of symmetrical thresholds glosses over the fact that thedistributions are negatively skewed, but this is not felt to be an important problemfor the results reported here. Therefore, the divergence thresholds are defined tobe:

wind divergence: 1.2 ◊ 10≠4 s≠1 (15)wind stress divergence: 1.8 ◊ 10≠6 N m≠3 (16)

and are used to define three categories of divergence events (Table 2):(i) Extreme convergence (XC);(ii) Background convergence/divergence (BG);(iii) Extreme divergence (XD).Physically, XC and XD events are expected to correspond to large amplitude,intermittent updrafts and downdrafts in the near-surface ocean winds – that is,phenomena associated with Mesoscale Convective Systems.

TABLE 2: Categories of divergence. The symmetrical thresholds for wind/wind stressdivergence are defined, respectively, as c

u

= 1.2◊10≠4 s≠1, and c

·

= 1.8◊10≠6 N m≠3.

Category Wind Divergence Wind Stress DivergenceXC Extreme Convergence Ò · u Æ ≠c

u

Ò · · Æ ≠c

·

BG Background ≠c

u

< Ò · u < c

u

≠c

·

< Ò · · < c

·

XD Extreme Divergence Ò · u Ø c

u

Ò · · Ø c

·

Application of the thresholds to the test case is shown in Figure 9 for winddivergence and 10 for wind stress divergence. The left panel of each figure showsthe background convergence in light blue and background divergence in salmonpink; the yellow in the figures indicate the locations of extreme values. The rightpanels show only the extremes, where dark blue indicates extreme convergence anddark red extreme divergence. In these figures, and more generally, the extremes aremainly concentrated in equatorial latitudes – that is, within the band of latitudesdefined by the Inter-Tropical Convergence Zone.



-30 -20Longitude

-20

-15

-10

-5

0

5

10

15

20

Latitude

WindDivergenceBG

-30 -20Longitude

-20

-15

-10

-5

0

5

10

15

20

Latitude

WindDivergenceXC + XD

-30 -25Longitude

0

2

4

6

8

10

Latitude

WindDivergenceBG

-30 -25Longitude

0

2

4

6

8

10

Latitude

WindDivergenceXC + XD

Figure 9: Wind divergence calculated for the test case (an ASCAT pass over the tropi-cal mid-Atlantic). The divergence is separated into background convergence/divergenceand extreme convergence/divergence. Background convergence/divergence is shadedblue/pink (left panel) and extreme convergence/divergence is blue/red (right panel). Theyellow shading in the left panel identify the areas containing extreme convergence/diver-gence shown in the right panel. The bottom panels zoom in on the area with the largestdivergence.



-30 -20Longitude

-20

-15

-10

-5

0

5

10

15

20Latitude

Wind StressDivergenceBG

-30 -20Longitude

-20

-15

-10

-5

0

5

10

15

20

Latitude

Wind StressDivergenceXC + XD

-30 -25Longitude

0

2

4

6

8

10

Latitude

Wind StressDivergenceBG

-30 -25Longitude

0

2

4

6

8

10

Latitude

Wind StressDivergenceXC + XD

Figure 10: As in Fig. 9, but for wind stress divergence.



TABLE 3: Thresholds for rainfall intensity classification used by (i) the United King-dom Meteorological O�ce (UKMO, 2007), and (ii) the present work.

Rain Rate [mm h≠1] UKMO Present work< 2.5 Light LR BGR2.5 - 10 Moderate MR

10 - 50 Heavy XR XR> 50 Torrential

0 10 20 30 40 50 60 70 80Rain Rate [mmh−1]

100

101

102

103

104

105

106

107

108

Cou

nts

2012-12

t = 7t = 10

Figure 11: Histogram of rain rates for the area covered by SCAT1 passes at the time ofthe pass (t = 7, blue) and 45 minutes later (t = 10, red) accumulated for the month ofDecember 2012. Vertical lines are drawn indicating the United Kingdom MeteorologicalO�ce thresholds for light rain (below 2.5 mm h≠1), moderate rain (2.5 - 10 mm h≠1),heavy rain (10 - 50 mm h≠1), and torrential rain (greater than 50 mm h≠1).

4.2 Classification of rainfall intensityFigure 11 shows the empirical probability distribution of rainfall intensity con-structed from all 3-km MSG rain rates during December 2012 at the time of theSCAT1 pass (blue) and 45 minutes later (red). Only rain pixels from the area cov-ered by the SCAT1 pass were used. The vertical lines in the figure are thresholdsdefined by the United Kingdom Meteorological O�ce (UKMO): see Table 3 andUKMO (2007). As can be seen, the rain rate at which significant slope changesoccur in the MSG rain rate distribution are very close to the UKMO thresholds.Therefore, we define the rain rate categories as light (LR), moderate (MR), extreme(XR), using the thresholds given in Table 3. The XR category includes the UKMOheavy and torrential categories. The table also defines the background rain (BGR)category, the union of LR and MR.



-30 -20Longitude

-20

-10

0

10

20Latitude

RRmax

BGRt = 7

-30 -20Longitude

-20

-10

0

10

20

Latitude

RRmax

XRt = 7

-30 -25Longitude

0

5

10

Latitude

RRmax

BGR

-30 -25Longitude

0

5

10

Latitude

RRmax

XR

Figure 12: Spatial distribution of maximum rain rate (RR

max

) classified as either back-ground (BGR = LR + MR) or extreme (XR) as described in the text. The BGR dis-tribution is shown in the upper left panel and the XR distribution in the upper rightfor test case 20121205_1115 ; the lower panels zoom in on the latitude band having thelargest rain rates.



4.3 Are extremes in MSG rain correlated with extremesin ASCAT divergence?We now turn to making a quantitative correlation between rain and divergence.This task is carried out using the RR

max

product described in the previous section(section 3.4). In the left panels of Figure 13 data from the test case are used to con-struct scattergrams of RR

max

(at the time of the SCAT1 pass) versus wind/windstress divergence. The vertical lines in the figures are drawn at the thresholds sepa-rating the three categories of divergence, and the horizontal lines at the thresholdsseparating the three rain rate categories. The di�erent categories are indicatedalong the top and right-hand boundaries of each figure. A tendency can be seen forthe width of the scatter to increase with increasing RR

max

. In order to confirm thisobservation, probability distributions conditioned on the three rain categories arecalculated using all December 2012 collocations (right panels). These show that thetendency observed in the test case holds, with their width, skewness and flatnessall increasing with increasing rain rate. This demonstrates that extremes in rainfallintensity correlate with extremes in surface wind/wind stress divergence. A moredetailed look at the correlation is investigated next.

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2Wind Divergence [s−1] ×10-3

10-1

100

101

102

RR

max

[mmh−1]

τ = 0

XC BG XD

XR

MR

LR

-3 -2 -1 0 1 2 3Wind Divergence [s−1] ×10-3

10-6

10-5

10-4

10-3

10-2

10-1

100

Histogram

Conditioned onXR, MR and LR

LRMRXR

-3 -2 -1 0 1 2 3Wind Stress Divergence [Nm−3] ×10-5

10-1

100

101

102

RR

max

[mmh−1]

τ = 0

XC BG XD

XR

MR

LR

-5 -4 -3 -2 -1 0 1 2 3 4 5Wind Stress Divergence [Nm−3] ×10-5

10-6

10-5

10-4

10-3

10-2

10-1

100

Histogram

Conditioned onXR, MR and LR

LRMRXR

Figure 13: Scattergrams (left) and conditional probability distributions (right) of diver-gence and RR

max

. Top panels: wind divergence; Bottom panels: wind stress divergence.The scattergrams use only data from the test case using RR

max

observed at the time of theSCAT1 pass (lag time · = 0). The vertical lines are drawn at the thresholds separatingthe three divergence categories: XC (blue circles), BG (black dots), and XD (red circles).The horizontal lines are drawn at the thresholds separating the three rain categories LR,MR, and XR. The panels on the right show probability distributions conditioned on therain rate categories and use all samples from December 2012.



4.3.1 Contingency tables

TABLE 4: A generic 3-by-3 contingency table.

Forecast Observedo1 o2 o3 f sums

f1 r s t

qf1

f2 u v w

qf2

f3 x y z

qf3

o sums qo1

qo2

qo3 N

tot

TABLE 5: Contingency table for rain rate and divergence, where the rain rate istreated as the forecast variable. CNE means ’correct non-event’, XCD = XC fi XD(Table 2), and BGR = MR fi LR (Table 3).

Rain Event Divergence Event Rain sumsXCD BG

XR Hit False alarma + b

a b

BGR Miss CNEc + d

c d

DIV sums a + c b + d

N

tot

=a + b + c + d

The task is to quantify the relation between divergence and rain using thecategories identified above. The number of joint events are recorded in a contingencytable, as shown in Table 4. In forecast verification studies, the rows represent theforecast events (f) and the columns the observed events o. Thus if event f

i

isforecast and event o

j

is observed, a count is recorded in table position (i, j). Inthis way the contingency table is used to determine conditional probabilities. It isimportant to note that although contingency tables are useful in measuring forecastperformance, its principle use is to gain insight into what is right and what is wrongabout the forecasts (c.f., Doswell and Flueck, 1989; Doswell et al., 1990).

Returning to Table 4, of primary interest are the joint extreme events (XR,XC)and (XR,XD). As a result, LR and MR are merged to form event class BGR =LR fi MR, while XC and XD are merged to form event class XCD = XC fi XD.This reduces the 3-by-3 table to a 2-by-2 table. Before proceeding, we must decidewhich variable to identify as ”forecast” and which ”observed”. Since in fact bothvariables are observations, the identification is subjective and could be argued eitherway. However, because our objective is to relate ASCAT divergence to rain, thenatural choice is to choose rain rates as the forecast variable and divergence as theobservation. With these choices, the contingency table takes the form shown inTable 5. The meaning of the four outcomes in the table are:

(i) a is the number of hits (XCD is forecast and observed);



(ii) b is the number of false alarms (XCD is forecast but not observed);(iii) c is the number of misses (XCD is observed but not forecast); and(iv) d is the number of correct non-events or CNE (XCD is not forecast and not

observed).Counts are turned into probabilities by dividing all entries in the table by N

tot

.This yields the table of joint probabilities P (o fl f) and marginal probabilities P (f)and P (o) shown in Table 6. A perfect correlation would produce no false alarms(P (BG fl XR) = 0), that is, the scattergrams in Fig. 13 would have zero black dotsin the (XR,BG) rectangle. Similarly, a perfect correlation would produce no misses(P (XCD fl BGR) = 0), and hence no blue or red circles in the (LRfiMR,XCfiXD)rectangles. As the scattergrams show, this is clearly not the case. Indeed, Table 6shows that the probability of a hit is less than either of the probabilities of falsealarm or miss. Our next task is to determine how the number of false alarms andmisses might be reduced and the number of hits increased.

TABLE 6: Joint probabilities P (DIV fl Rain) and marginal probabilities P (DIV )and P (Rain) of rain rate and wind/wind stress divergence (December 2012 colloca-tions); wind stress divergence probabilities are in parentheses under those for winddivergence.

Rain Event Divergence Event P(Rain)XCD BG

XR 0.005 0.010 0.015(0.007) (0.009) (0.015)

BGR 0.036 0.945 0.985(0.031) (0.954) (0.985)

P(DIV) 0.041 0.959 1(0.037) (0.963) (1)



4.3.2 Neighbourhood sizeFigure 14 shows the locations of hits [(XR,XCD): black filled circles], false alarms[(XR,BG): red open circles], and misses [(BGR,XCD): blue open squares] for theregion of the test case swath with the greatest precipitation and divergence (seeFigs. 9 and 10). The figure suggests that false alarms and misses are always closeto a hit – typically within 2 or 3 WVCs. This suggests that the size of the spatialneighbourhood used to declare a hit was too small. If true, then the number of hitswould increase and false alarms and misses decrease if a neighbourhood larger thanthe 2-by-2 block of WVCs was used.

The question of neighbourhood size is investigated further using all data fromDecember 2012. Histograms compiled for the shortest distance from an XR gridpoint to an XCD grid point are shown in the top panels of Figure 15. The histogramsshow that 35% of all XR grid points are with a 2-by-2 area of WVCs, 80% within a3-by-3 area, and 90% within a 4-by-4 area. Thus a neighbourhood of about 4-by-4WVCs (50km-by-50km) would reduce the number of false alarms to near zero.

The bottom panels of Figure 15 show histograms of the shortest distance froma BGR grid point to an XCD grid point. These indicate that using a 4-by-4 neigh-bourhood would result in about 25% of BGR events to be reclassified as hits, whichwould greatly reduce the number of misses.

To summarize in physical terms, the histograms show that when there is extremerain, there is extreme convergence/divergence in the vicinity.

Due to the limited time available for this study, a test of increased neighbour-hood size is deferred to a future work.

-30 -29 -28 -27 -26 -25 -24 -23 -22Longitude

4

4.5

5

5.5

6

6.5

7

7.5

8

Latitude

WindDivergence

HitsFalse AlarmsMisses

-30 -29 -28 -27 -26 -25 -24 -23 -22Longitude

4

4.5

5

5.5

6

6.5

7

7.5

8

Latitude

Wind StressDivergence

HitsFalse AlarmsMisses

Figure 14: Zoomed in region of the test case swath with largest precipitation and di-vergence and showing WVCs that are hits (black filled circles), false alarms (red opencircles) and misses (blue open squares): wind divergence (left); wind stress divergence(right).



0 5 10Distance [WVC widths]

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Histogram

WindDivergenceXR → XCDτ = 0


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Wind StressDivergenceXR → XCDτ = 0


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Histogram

WindDivergenceBGR → XCDτ = 0


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Wind StressDivergenceBGR → XCDτ = 0

Figure 15: Histograms of the shortest distance from an XR grid point to a XCD grid point(top), and from a BGR grid point to an XCD grid point (bottom). Distance is measuredin widths of a WVC. The last bin in each histogram counts all distances greater than 10WVCs.

4.3.3 Time evolution of conditional probabilitiesConditional probabilities are easily calculated from the information in contingencytables. The probability of observation o given forecast f is defined to be (Papoulisand Pillai, 2001)

P (o|f) = P (o fl f)P (f) (17)

= N

of

N

f

(18)

where N

of

is the number of forecasts that were observed (hits), and N

f

is the totalnumber of forecasts made.

Earlier it was argued that because the rain rates were measured at severaltimes per ASCAT swath, it would be more appropriate to regard it as the forecastvariable. This can also be justified as follows. As the day progresses, air warms andcan hold more moisture. In consequence rain decreases with time, and hence thenumber of ’forecasts’ (N

f

) decrease with time. Eq. 18 shows that the calculationof P (o|f) properly takes into account the decreasing value of N

f

.Figure 16 shows the time evolution of P (XC(0)|f(·)) and P (XD(0)|f(·) for

each rain category f . Of most interest is the response to extreme rain [P (XC(0)|XR(·))and (P (XD(0)|XR(·))] – there are well-defined peaks at t(ASCAT) + 30 minutes(· = 30) for convergence, and at t(ASCAT) (· = 0) for divergence. The divergence



peak at · = 0 implies that the rain field at the time of the ASCAT pass shows moreextreme rain in the vicinity of extreme divergence than at any other time. This isbecause rain falls fast: the most vigorous heat exchange between precipitation andair occurs at the melting and liquid stage, accelerating the cooled air downwardswith the precipitation. At 20 m/s, it takes rain drops about 4 minutes to fall over5 km height. The convergence peak at · = 30 implies that the rain field 30 minutesafter the ASCAT pass shows more extreme rain in the vicinity of extreme conver-gence than rain fields at any other time. In other words, extreme rain generallyappears after (lags) extreme convergence. The overall conclusion then is that thetemporal scale of moist convection is determined by the slower updraft process.

Finally, note that wind stress divergence shows a stronger response than winddivergence. This is probably due to the latter’s focus on high winds (wind stress isa nonlinearly increasing function of wind speed).

-5 0 5 10τ/(15 min)

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

P(X

C|f)

Wind DIV201212

-5 0 5 10τ/(15 min)

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

P(X

C|f)

Wind Stress DIV201212

XR

MR

LR

XR

MR

LR

-5 0 5 10τ/(15 min)

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

P(X

D|f)

Wind DIV201212

-5 0 5 10τ/(15 min)

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

P(X

D|f)

Wind Stress DIV201212

XR

MR

LR

XR

MR

LR

Figure 16: Time evolution of the conditional probabilities for convergence P (XC|f)and divergence P (XD|f), where f denotes the rain categories: LR, MR, and XR. Thecalculations use all data from December 2012.



Figure 17: Extreme value cluster size distribution for wind divergence (left) and windstress divergence (right). Probability distributions (in terms of area) are shown as quan-tiles (in terms of number of WVCs). The vertical dashed lines in each panel indicate areascomprising of 10 and 20 WVCs. The area of one WVC is approximately 156 km2. Theprobability distributions show that the size distribution follows an approximate power-law.

5 Summary and future workIn this project, wind fields from the ASCAT-A and ASCAT-B tandem mission,collocated with Meteosat Second Generation rain fields, were used to develop amethodology capable of identifying and quantifying correlations between wind andrain. The methodology borrows techniques from forecast verification and was usedsuccessfully to calculate cross-correlations from 3-by-3 and 2-by-2 contingency ta-bles.

In the current work a spatial neighbourhood spanning a 25km-by-25km blockof WVCs was searched to identify coincidences of extreme rain and extreme diver-gence. This resulted in more false alarms and more misses than hits(see Table 6).This was found to be too restrictive (Fig. 14), and a spatial structure analysis(Fig. 15) suggested that a 50km-by-50km spatial neighbourhood would greatly re-duce the number of false alarms and misses. This should be investigated in futurework. In physical terms, the analysis shows that when there is extreme rain, thereis extreme convergence/divergence in the vicinity.

Figs. 9, 10, 12, and 14 all show evidence of spatial structuring of extreme events,that is, patches of extreme convergence/divergence and extreme rain – indicatinginteractions of Mesoscale Convective Systems and sea surface temperature gradi-ents. Future study of this structuring is important, as it relates coherent phenomenawith the point statistics. The structuring can be investigated using the clusteringtools (in MatLab, R, and python) that have been developed in the context of thisproject. Some work on clustering was carried out during the project and one resultis shown in Figure 17: quantiles of the distribution of cluster sizes. The verticaldashed lines in both panels divide the distribution into three size categories: small(less than 10 WVCs), medium (between 10 and 20 WVCs), and large (20 or moreWVCs).

The time evolution of conditional probabilities of convergence/divergence givenrain were calculated and revealed some interesting characteristics overall as well asin individual cases. The overall response curves were found to have a well-definedpeak (see Fig. 16). The response curve for convergence (P (XC|XR)) implies that



extreme rain lags extreme convergence by about 30 minutes, while that for extremedivergence (P (XCD|XR)) was found to be in synchrony with extreme rain. Wealso found that wind stress divergence demonstrated a stronger response than winddivergence, probably due to the latter’s focus on high winds.

We end this report with some additional recommendations. First, future workshould include an investigation of wind/wind stress curl that parallels and com-bines with the work reported here. Second, future work should investigate theSCAT1/SCAT2 temporal di�erences for which the clustering tools should provehelpful. After identifying individual clusters of extreme wind stress convergence/-divergence in the SCAT1 pass, the areas spanned by those clusters were identified inthe SCAT2 pass. Figure 18 shows the before (top) and after (bottom) of the statis-tical distributions of convergence/divergence. Something that could be interestingto do with such data is to calculate transition probabilities of, say, XC events to thedi�erent categories {XC,BG,XD} at the time of the SCAT2 pass. This informationcould be useful to modellers.

Finally, it is recommended that work be carried out in collaboration with mod-ellers who can help sharpen the results and suggest further refinements in theanalysis.

Figure 18: Sample result that compares information in SCAT1 with that in SCAT2 (50minutes later). The top panels show the statistical distributions of extreme wind stressconvergence and divergence and the bottom panels show, for the same ocean areas, howthe SCAT1 distributions change after 50 minutes (SCAT2 pass).



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