Faculty Resource Network (FRN) Seminar

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

Background

The goal of this seminar is to introduce participants to the application of cross-correlation analysis to the measurement of fluid flows in the atmosphere and oceans. These flows carry heat around the planet and so dictate the state of our climate. It is possible to study aspects of the real-world flows by building physical laboratory analog models. Using particles that are artificially embedded in the laboratory flows it is possible to take successive images of the particle's positions and by cross correlation determine how far the particles have moved -- and by inference, since the time between images is controlled by the observer, basic statistical properties of the fluid flow can be determined.

Details

Schedule

Time (am) Activity
9:00 - 10:00 Lecture
10:00 - 10:30 Laboratory Visit
10:30 - 11:30 Computing Exercise
11:30 - 12:00 Participant Reports

The seminar is divided into four parts. First, a lecture will present an overview of basic elements of the earth's climate system and how atmospheric (e.g., the Jet Stream) and oceanic (e.g., the Gulf Stream) flows affect the climate we experience today and how that climate was radically different some 15,000 years ago (i.e., the last ice age). The basic mathematics of cross-correlation and the numerics and implementation of an efficient algorithm on a computer will be discussed. Third, the practical aspects of retrieving two sequential images of fluid flow with embedded particles will be covered, including details of image recovery (e.g., remote from artificial satellite or local by desktop camera), lighting (e.g., natural by sunlight or artificial by laser light sheet), particle seeding (e.g., natural by plankton or artificial by microspheres), and other pertinent aspects. The final part of the lecture will be the presentation of a computer code (in MATLAB) that implements the cross-correlation which receives as input sequential images of a fluid and produces as output the fluid's flow field. The lecture notes are available in PowerPoint format.

The second part of the seminar will be a visit to the nearby NYU Environmental Fluid Dynamics Laboratory (EFDL) where physical laboratory experiments of fluids on a turntable (mimicking the earth's rotation) are performed. A laser-based system which illuminates glass microspheres embedded in the fluid is used to measure the details of the fluid's flow. An underlying premise is that a laboratory experiment flow mimics certain aspects of the real flows that occur in the atmosphere and oceans.

A hands-on computational exercise using the MATLAB programming language will make up the third part of the seminar. A basic tutorial on using MATLAB is made available at the seminar web page. The application code used to compute fluid velocity by cross correlation is also found there. The participant exercises involve computing fluid flow under a variety of conditions, e.g. good lighting versus poor lighting, etc.

The final part of the seminar will consist of presentations by the participants of the results obtained in their computing exercises and summaries of their understanding of the basic principles underlying measuring fluid flow by cross-correlation analysis.

Flow-Feature-Tracking Technique

Oceanic flow fields can be determined by using an analysis technique known as feature tracking. The idea is to take two sequential images of the ocean surface and to employ an algorithm that can determine the spatial displacement of identifiable features embedded within the surface flow. For example, plant life in the ocean can be detected as phytoplankton pigment concentration by satellites. 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. An example of fluid flow computed for these waters using a feature-tracking algorithm is:

The red arrows show the surface layer drift as determined by a feature-tracking algorithm; the black arrows show surface drifter data (where available) for comparison. The ocean surface feature tracking images are from an article by Anthony K. Liu (EOS, 83(7), 12, February 2002, 61-64).

Particle-Image-Velocimetery Technique

A related technique, and one that is becoming quite popular in the study of fluid flows on small scales (such as in a physical laboratory), is particle image velocimetry (PIV). This technique and its application of cross-correlation analysis is the focus of this seminar. The technique can be used to produce a velocity map from two sequential images of "seeded" fluid flow. The seeds usually consist of small hollow glass spheres (typically 10 micron diameter and neutrally buoyant). The seeds are illuminated by a laser light sheet and their instantaneous spatial pattern is captured by a digital, cross-correlation camera. As an example of two sequential images we provide pictures base and cross. Using cross-correlation analysis, the PIV technique provides an estimate of the fluid velocity field (shown as red arrows):

Computing Practicals

Computer hardware and software resources will be provided during the seminar. For the former, we will use PCs; for the latter, MATLAB running under a Windows OS. Persons without MATLAB experience should browse an online tutorial prior to attending the seminar. For a hands-on demonstration of PIV analysis of the above sample images, participants will carry out the following actions:

PIV Code

The particular PIV software we are using is called URAPIV and it originates from a webpage maintained by Alex Liberzon. It is a collection of MATLAB routines (i.e., M-files) specifically developed for analysis of raw PIV data, and it comes complete with two sample images. The M-files and images are:

An alternative PIV package known as mpiv is available here, and one known as CIV available here. Additional PIV images are available from the PIV standards webpage.

PIV GUI Help

PIV Exercises

  1. Produce a velocity flow field using the MATLAB PIV software and the supplied sample images. Compare your result to that obtained during the lecture.
  2. Underlying the PIV technique is the ability to quickly perform a two-dimensional cross-correlation function c(x,y) of two, two-dimensional functions (i.e., the images, say f(x,y) and g(x,y)). Explain how, in general, a two-dimensional cross-correlation is achieved and how Fourier transforms of f(x,y) and g(x,y) are useful in this step of the PIV technique.
  3. The peak amplitude of the correlation function c(x,y) will be located at some position (x_o, y_o). This position tells us the relative vector displacement of the two images (in pixels). Explain.
  4. The displacement is in pixels, but to be useful it needs to be scaled to a practical unit, such as meters, representing the actual displacement of the real fluid. How is a pixel displacement in an image converted to a physical displacement of the real fluid? Explain why calibration of the images is necessary.
  5. Knowing the time between images, how is a physical, scaled velocity (e.g., m/s) finally determined?
  6. The correlations are actually carried out on contiguous subsets of the whole image. These subsets are know an as interrogation windows and generally there are of order 100 by 100 such windows used in a cross-correlation analysis. This implies an independent displacement estimate for each interrogation window. How does this lead to a spatial velocity field?

Background Readings

Other Web-Based Resources


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