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Understanding Atmospheric & Oceanic Flows:Understanding Atmospheric & Oceanic Flows: Laboratory Application of Cross-Correlation Laboratory Application of Cross-Correlation

David M. HollandDavid M. Holland

Courant Institute of Mathematical SciencesCourant Institute of Mathematical Sciences

New York UniversityNew York University

June 10June 10thth, 2003, 2003

Faculty Resource Network SeminarFaculty Resource Network Seminar

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Seminar Schedule Seminar Schedule

09:00 – 10:00 Lecture – Cross Correlation

10:00 – 10:30 Laboratory Visit

(Room 103, 251 Mercer St. WWH)

10:30 – 11:30 MATLAB Computing Exercises

11:30 – 12:00 Group Presentations

(Answers to MATLAB Exercises)

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Introduction to LectureIntroduction to Lecture

Atmospheric & Oceanic Flows

Planetary Scale Flows – Feature Tracking

Laboratory Scale Flows – Particle Image Velocimetry

Cross Correlation Analysis

MATLAB Implementation – Particle Image Velocimetry

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Atmospheric FlowsAtmospheric Flows

Jet Stream (Discovery Video)

Hurricane

Tornado

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Oceanic FlowsOceanic Flows

Great Conveyor Belt

Gulf Stream

Further Information: Read Chapter 1 of Handout:

The Oceans and Climate by Bigg

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Image “a” Image “b”

Here are two sequential images (a and b) of chlorophyll-a data collected over the US east coast on May 8, 2000 by two different satellites at time spacing of 67 minutes.

Planetary Scale Flows – Planetary Scale Flows – Feature TrackingFeature Tracking

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Flow Field Vectors - Derived by Feature Tracking Algorithm

Question: How are these flow arrows derived?

Planetary Scale Flows Planetary Scale Flows – Feature TrackingFeature Tracking

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Laboratory Analog of Planetary Scale Flows (Jet Stream)

PIV Principles

Further Information: Read Chapter 3 of Handout:

Particle Image Velocimetry by Raffel et al.

NYU Laboratory

Laboratory Scale Flows –Laboratory Scale Flows –Particle Image Velocimetry (PIV)Particle Image Velocimetry (PIV)

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Cross Correlation Analysis – Cross Correlation Analysis – Basic ConceptsBasic Concepts

One-Dimensional Example(Convolution, but similar to Cross Correlation)

( ) ( ) ( )N

k Ny n h k x n k

( )* ( ) ( ) ( )N

k Nh n x n h k x n k

Also use notation ‘*’ to indicate convolution

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Cross Correlation Analysis –Cross Correlation Analysis –Image Displacement Image Displacement

Demonstration of Cross Correlation to find (dis)placement of one image within another

(see MATLAB handout for details)

MATLAB • “demos”• Toolbox “Image Processing”• “Image Registration”• Set Path to “.”• Enter Commands

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Cross Correlation Analysis –Cross Correlation Analysis –Fast Fourier TransformFast Fourier Transform

One-Dimensional Example

0

1 1

2 2

( )

+ cos( )

+ cos(2 )

... + cos( )

o

o

N o N

x t a

a t

a t

a N t

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Cross Correlation Analysis – Cross Correlation Analysis – Convolution TheoremConvolution Theorem

One-Dimensional Example

(using functions f(k) and g(k))

( )* ( ) ( ) ( )N

k Nf n g n f k g n k

Convolution Theorem gives Convolution as Inverse Transform of Product of Fourier Transforms

( )* ( ) ( ) ( )N

k NInverse FourierTransform F v G v f k g n k

where F and G represent Fourier Transform of f and g.

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MATLAB Implementation –MATLAB Implementation –Particle Image Velocimetry (PIV) Particle Image Velocimetry (PIV)

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Concluding RemarksConcluding Remarks –– Cross Correlation

Atmospheric & Oceanic Flows are Complex – Laboratory Models Provide Insight

Particle Imaging Velocimetry – Non-Invasive Measurement

Cross Correlation Analysis – Plays Central Role

Future Research – Faster/Better Computer Algorithms

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Concluding RemarksConcluding Remarks –– Educational Applications

MATLAB is a powerful teaching tool Various Demo Modules for most all

aspects of Mathematics

Interesting Applications of Statistics and Probability in the Geosciences (e.g., Fluid Flow Measurement)

This Seminar Web Site available http://fish.cims.nyu.edu/educational_pages/frn_2003/syllabus.html (see Handout)

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Seminar Schedule – Seminar Schedule – Remainder of MorningRemainder of Morning

10:00 – 10:30 Physical Laboratory Visit (Room 103, 251 Mercer St.)

(see NYU Map Handout for details)

10:30 – 11:30 MATLAB Computing Exercises (Break into Groups of Two) (Room 305, 197 Mercer St.)

11:30 – 12:00 Group Presentations (Answers to MATLAB Exercises)

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