# Correlating Extremes in Wind Divergence with Extremes in Rain over the Tropical Atlantic

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## Abstract

**:**

## 1. Introduction

## 2. Data and Study Area

#### 2.1. ASCAT Winds

#### 2.2. MSG Rain

#### 2.3. Study Area

#### 2.4. Collocations

## 3. Wind Divergence and Regridded Rain

#### 3.1. Wind Divergence Calculation

#### 3.2. PDFs and Classification

#### 3.2.1. PDF Characteristics

#### 3.2.2. Thresholds and Classification

#### 3.2.3. Are the Heavy Tails Geophysical?

#### 3.3. Rain Classification and Regridding

#### 3.3.1. Classification

#### 3.3.2. Regridding

#### 3.3.3. PDFs

## 4. Correlation Methods

#### 4.1. Contingency Tables

#### 4.2. Odds and Odd Ratios

- (i)
- Range: $[0,\infty ]$;
- (ii)
- $OR$ is symmetric: $OR(A,B)=OR(B,A)$;
- (iii)
- if $OR=1$, A and B are independent;
- (iv)
- if $OR>1$, then A and B are positively correlated (the presence of B raises the odds of A); and
- (v)
- if $OR<1$, A and B are negatively correlated (the presence of B reduces the odds of A).

## 5. Results and Discussion

#### 5.1. Local Analysis

#### 5.1.1. Spatial Distribution

#### 5.1.2. Scatter Plot and Conditional Probabilities

#### 5.1.3. Contingency Tables

#### 5.1.4. Correlating Extremes: The Two-By-Two Contingency Table and the Odds Ratio

#### 5.2. Structural Analysis

#### 5.2.1. Spatial Structure and Reclassification

#### 5.2.2. Temporal Evolution

## 6. Summary and Future Work

#### Future Work

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Wind Components

## Appendix B. Comparison of Central Difference Schemes

**Figure A1.**Effect of numerical method on probability distributions $P\left(\delta \right)$ calculated using standard central differences (CD 3-by-3, black), and difference-then-average (CDA 3-by-3, blue; CDA 2-by-2, red). Probability distributions are plotted in standardized coordinates, $z=(\delta -\overline{\delta})/{\sigma}_{\delta}$, where $\overline{\delta}$ and ${\sigma}_{\delta}$ are mean and standard deviation of $\delta $, respectively.

**Table A1.**Comparison of first four moments of probability distribution of wind divergence $P\left(\delta \right)$ obtained for standard central difference scheme (CD), and two central-difference-then-average (CDA) schemes: on a 3-by-3 block and a 2-by-2 block of WVCs.

Statistic | CD | CDA 3-by-3 | CDA 2-by-2 |
---|---|---|---|

Mean [${10}^{-5}$ s${}^{-1}$] | 0.043 | 0.044 | 0.071 |

Std. Dev. [${10}^{-5}$ s${}^{-1}$] | 5.1 | 4.4 | 6.1 |

Skewness | −0.55 | −0.41 | −0.88 |

Kurtosis | 26 | 19 | 38 |

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**Figure 1.**Conceptual model of a mesoscale convective system showing 2D hydrometeor trajectories through stratiform region of a squall line with trailing stratiform precipitation. (Modified from original ([5]) to show surface winds). Reprinted with permission from Ref. [5]. Copyright 1991 American Meteorological Society.

**Figure 2.**A satellite view of interaction of rain and ocean winds in Atlantic Inter-Tropical Convergence Zone in an area a few degrees north of equator to west of coast of Africa. Figure shows superposition of wind vectors from ASCAT-A (black) and ASCAT-B (red) with contours of Meteosat Second Generation (MSG) rain at time corresponding to pass of ASCAT-A. Colorbar shows rain rate in mmh${}^{-1}$. Note convective rain cells (>10 mmh${}^{-1}$) and nearby areas of strong wind convergence (C) and divergence (D).

**Figure 3.**Overlapped portions of ASCAT-A (red) and -B (black) swaths over Tropical Atlantic. Tropics are divided into three latitude bands. Main region of interest lies within rectangle, labeled ITCZ. Latitude band to its north is labeled Region I, and to its south, Region III. In the tropics, half of the time the right swath of ASCAT-A overlaps the left swath of ASCAT-B and vice versa. The left edge of the later scatterometer is shifted slightly eastward relative to the earlier scatterometer. To distinguish the time-of-arrival, the scatterometers are re-labeled SCAT-1 and SCAT-2. In the figure, the SCAT-2 swath overlies the SCAT-1 swath.

**Figure 4.**Number of collocations in boreal winter (

**left**) and boreal summer (

**right**) versus collocation time (time recorded for middle MSG frame in each collocation). Collocations in most eastern part of study area occur at 08:30 UTC, while those in most western part occur at 13:30 UTC (see Figure 3). Total number of collocations is 801 (425 in winter and 376 in summer).

**Figure 5.**Schematic for calculation of horizontal divergence and assignment of WVC rain rate. Wind divergence is assigned to half-grid point $(i+1/2,j+1/2)$. ${R}_{i,j}^{*}$ is maximum MSG rain rate in area spanned by WVC $(i,j)$.

**Figure 6.**PDFs of wind divergence for Regions I, ITCZ, and III (December 2012). Green curve is a Gaussian distribution with zero mean and standard deviation for all data (${\sigma}_{All}$).

**Figure 7.**Normal probability plot of ITCZ PDF. PDF follows a Gaussian distribution where it falls on or close to red line. Vertical lines ($\pm 2{\sigma}_{All}$) partition $\delta $ into extremes (${\delta}_{xc}$ and ${\delta}_{xd}$) and background (${\delta}_{bg}$). Note that figure does not show full range of $\delta $ values. Left and right tails extend much further out (to about $\pm 14\times {10}^{-4}$ s${}^{-1}$)—see Figure 6.

**Figure 8.**Probability distribution of rain, $P\left({R}_{\delta}^{*}\right)$, for Regions I, ITCZ, and III (December 2012). Rain classes are shown along bottom of plot.

**Figure 9.**Spatial patterns of extremes. (

**a**) Extremes of convergence (blue) and divergence (red); (

**b**) rain extremes (black) and transition/stratiform rain (green). Rain extremes are from field of view of ASCAT at time of the satellite pass. Times of five swaths are, from left-to-right, 09:45, 10:30, 11:15, 12:00, and 13:00 UTC. (

**c**,

**d**) Magnification of swath within the small rectangle in (

**a**,

**b**): (

**c**) coinciding extremes. (

**d**) Segmentation into subsets of co-occurring points defined by thresholds in Table 2 and Table 3.

**Figure 11.**Probability structure of ${R}_{\delta}^{*}$ and $\delta $ for ITCZ: (

**a**) $\delta $ conditioned on rain classes $\mathcal{R}=\{E,T,S,LNR\}$, and (

**b**) ${R}_{\delta}^{*}$ conditioned on divergence classes $\mathcal{D}=\{{\delta}_{xc},{\delta}_{bg},{\delta}_{xd}\}$.

**Figure 12.**Nearest neighbor distribution ${P}_{nn}$ (using all observations in December 2012). Solid line: distance from points $({\delta}_{bg},E)$ to the nearest ${\delta}_{xcd}$. Dashed line: distance from points $({\delta}_{xcd},LNR)$ to nearest rain extreme E. Here ${\delta}_{xcd}={\delta}_{xc}\cup {\delta}_{xd}$, and $LNR=L\cup NR$.

**Figure 13.**Odds ratios for convergence (black) and divergence (red) for a winter month and a summer month. Rain product lag is relative to SCAT-1 pass ($\tau =0$).

**Figure 14.**Sample result that compares information in SCAT-1 with that in SCAT-2 (50 min later). The top panels (

**a**,

**b**) show, respectively, statistical distributions of extreme wind convergence (${\delta}_{xc}$) and divergence (${\delta}_{xd}$), and bottom panels (

**c**,

**d**) show, for the same ocean areas, the change in SCAT-1 distributions after 50 min (SCAT-2 pass).

Region | Latitudes | $\overline{\mathit{\delta}}$ | $\mathit{\sigma}$ | Skewness | Kurtosis |
---|---|---|---|---|---|

(${10}^{-4}$ s${}^{-1}$) | |||||

All | 25${}^{\circ}$S–25${}^{\circ}$N | 0.01 | 0.58 | −0.32 | 18 |

I | 15${}^{\circ}$N–25${}^{\circ}$N | 0.03 | 0.49 | −0.18 | 12 |

II (ITCZ) | 5${}^{\circ}$S–15${}^{\circ}$N | −0.02 | 0.64 | −0.77 | 26 |

III | ${25}^{\circ}$S–5${}^{\circ}$S | 0.03 | 0.57 | 0.15 | 11 |

Gaussian | – | 0 | ${\sigma}_{All}$ | 0 | 3 |

**Table 2.**Wind divergence classes and thresholds. Thresholds, $\pm 2{\sigma}_{All}$, map $\delta $ to classes $\mathcal{D}=\{{\delta}_{xc},{\delta}_{bg},{\delta}_{xd}\}$, where ${\sigma}_{All}$ is standard deviation of all $\delta $ between 25${}^{\circ}$S and 25${}^{\circ}$N.

Class Label | Type | Range | Source | |
---|---|---|---|---|

${\mathcal{D}}_{1}$ | ${\delta}_{xc}$ | Extreme Convergence | $\delta <-2{\sigma}_{All}$ | strong updrafts and gust fronts |

${\mathcal{D}}_{2}$ | ${\delta}_{xd}$ | Extreme Divergence | $\delta >2{\sigma}_{All}$ | strong downdrafts |

${\mathcal{D}}_{3}$ | ${\delta}_{bg}$ | Background | $\left|\delta \right|\le 2{\sigma}_{All}$ | weak to moderate convergence/divergence |

Class Label | Type | Rain Rate | Precipitating Cloud | |
---|---|---|---|---|

mm h${}^{-1}$ | ||||

${\mathcal{R}}_{1}$ | E | Extreme | >10 | young, vigorous convection |

${\mathcal{R}}_{2}$ | T | Transition | $[5,10)$ | aging/decaying convection |

${\mathcal{R}}_{3}$ | S | Stratiform | $[1,5)$ | stratiform cloud |

${\mathcal{R}}_{4}$ | L | Light | $[0,1)$ | — |

${\mathcal{R}}_{5}$ | $NR$ | No Rain | 0 | — |

**Table 4.**$IJ$ contingency table. Number of joint (${X}_{i},{Y}_{j}$) events is denoted by ${n}_{ij}$. Subscript + denotes sum over index it replaces: ${n}_{i+}$ is number of ${X}_{i}$ observations, ${n}_{+j}$ is number of ${Y}_{j}$ observations, and ${n}_{++}$ is total number of observations (${n}_{tot}$).

${\mathit{Y}}_{1}$ | ${\mathit{Y}}_{2}$ | ⋯ | ${\mathit{Y}}_{\mathit{J}}$ | X sums | |
---|---|---|---|---|---|

${X}_{1}$ | ${n}_{11}$ | ${n}_{12}$ | ⋯ | ${n}_{1J}$ | ${n}_{1+}$ |

${X}_{2}$ | ${n}_{21}$ | ${n}_{22}$ | ⋯ | ${n}_{2J}$ | ${n}_{2+}$ |

⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ |

${X}_{I}$ | ${n}_{I1}$ | ${n}_{I2}$ | ⋯ | ${n}_{IJ}$ | ${n}_{I+}$ |

Y sums | ${n}_{+1}$ | ${n}_{+2}$ | ⋯ | ${n}_{+J}$ | ${n}_{tot}={n}_{++}$ |

B | A | |||

1 | 0 | sums | ||

A | 1 | ${n}_{11}$ | ${n}_{10}$ | ${n}_{1+}$ |

0 | ${n}_{01}$ | ${n}_{00}$ | ${n}_{0+}$ | |

B sums | ${n}_{+1}$ | ${n}_{+0}$ | ${n}_{++}$ |

**Table 6.**Two-way contingency table between categories of wind divergence ($\mathcal{D}$) and rain intensity ($\mathcal{R}$) and conditional probability structure for observations in December 2012.

(a) Counts of joint events. | |||||

$\mathcal{D}$ | $\mathcal{R}$ | ||||

${\delta}_{xc}$ | ${\delta}_{bg}$ | ${\delta}_{xd}$ | sums | ||

$\mathcal{R}$ | E | 4403 | 13,791 | 3440 | 21,634 |

T | 1276 | 6626 | 865 | 8767 | |

S | 3610 | 34,139 | 2501 | 40,250 | |

L | 3767 | 78,652 | 2576 | 84,995 | |

$NR$ | 5496 | 56,4752 | 4725 | 574,973 | |

$\mathcal{D}$ sums | 18,552 | 697,960 | 14,107 | 730,619 | |

(b) Probability of $\mathcal{D}$ given $\mathcal{R}$. | |||||

$\mathcal{D}$ | |||||

$P\left(\mathcal{D}\right|\mathcal{R})$ | ${\delta}_{xc}$ | ${\delta}_{bg}$ | ${\delta}_{xd}$ | ||

$\mathcal{R}$ | E | 0.204 | 0.637 | 0.159 | 1 |

T | 0.146 | 0.756 | 0.099 | 1 | |

S | 0.090 | 0.848 | 0.062 | 1 | |

L | 0.044 | 0.925 | 0.030 | 1 | |

$NR$ | 0.010 | 0.982 | 0.008 | 1 | |

(c) Probability of $\mathcal{R}$ given $\mathcal{D}$. | |||||

$\mathcal{D}$ | |||||

$P\left(\mathcal{R}\right|\mathcal{D})$ | ${\delta}_{xc}$ | ${\delta}_{bg}$ | ${\delta}_{xd}$ | ||

$\mathcal{R}$ | E | 0.237 | 0.020 | 0.244 | |

T | 0.069 | 0.009 | 0.061 | ||

S | 0.195 | 0.049 | 0.177 | ||

L | 0.203 | 0.113 | 0.183 | ||

$NR$ | 0.296 | 0.809 | 0.335 | ||

1 | 1 | 1 |

**Table 7.**Two-by-two contingency tables for coextremes of (

**a**) rain and convergence ($A=E$, $B={\delta}_{xc}$), and (

**b**) rain and divergence ($A=E$, $B={\delta}_{xd}$).

(a) | ||||

$B={\delta}_{xc}$ | A | |||

1 | 0 | sums | ||

$A=E$ | 1 | 4403 | 17,231 | 21,634 |

0 | 1449 | 694,836 | 708,985 | |

B sums | 18,552 | 712,067 | 730,619 | |

(b) | ||||

$B={\delta}_{xd}$ | A | |||

1 | 0 | sums | ||

$A=E$ | 1 | 3440 | 18,194 | 21,634 |

0 | 10,667 | 698,318 | 708,985 | |

B sums | 14,107 | 716,512 | 730,619 |

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## Share and Cite

**MDPI and ACS Style**

King, G.P.; Portabella, M.; Lin, W.; Stoffelen, A. Correlating Extremes in Wind Divergence with Extremes in Rain over the Tropical Atlantic. *Remote Sens.* **2022**, *14*, 1147.
https://doi.org/10.3390/rs14051147

**AMA Style**

King GP, Portabella M, Lin W, Stoffelen A. Correlating Extremes in Wind Divergence with Extremes in Rain over the Tropical Atlantic. *Remote Sensing*. 2022; 14(5):1147.
https://doi.org/10.3390/rs14051147

**Chicago/Turabian Style**

King, Gregory P., Marcos Portabella, Wenming Lin, and Ad Stoffelen. 2022. "Correlating Extremes in Wind Divergence with Extremes in Rain over the Tropical Atlantic" *Remote Sensing* 14, no. 5: 1147.
https://doi.org/10.3390/rs14051147