Wavelet Analysis for Engineered Log Jams William Chen Eco-informatics Summer Institute 22 August

Goal To create an informed set of wavelet data that may be quickly analyzed by scientists working on Fish-ELJ data. We want to determine where fish like to reside near a log jam, but first we need to figure where the distribution of energy around a log jam. Wavelet analysis can help in this respect. Do fish like to stay in large eddies lower in the energy spectrum? 2

Note Wavelet theory is extremely complex, especially for undergraduate computer science majors, so only a high level view of wavelets has been applied in the creation of the set. 3

Background In general, wavelets are oscillating patterns with zero mean (Torrence et al); eddies are the swirling of fluid past large obstacles (Benitez-Nelson). Source: Wikimedia Commons 4

Background (con’t) For our purposes, we are looking at are oscillating patterns in a time series of velocities at a single location. Patterns in the time series can be interpreted as a transformation of any wavelet, the wavelet in question being the analyzing wavelet. By analyzing the quantitative properties of these patterns, we can gain insight into the energy properties of a different locations. 5

Background (con’t) cm/s s 6

Scales Source: Source: “Continuous Wavelet Transform” 7

Background (con’t) Scales are scalar quantities that represent factors of stretching and compression on a wavelet for “level”. Can be thought of as the “frequency” of wavelet. (“Continuous Wavelet Transform”) 8

Background (con’t) Large eddies tend to occur at low wave numbers, i.e. lower scales (Gibson). This makes sense because larger eddies would have more energy contained in them due to their size. 9

What We Need We want a data set to hold outputs of wavelet analysis. Why? High intellectual overhead Computational overhead In Matlab, we can do a continuous wavelet transform, from which Matlab will generate scales and their coefficients. (“Continuous Wavelet Transform”) The scales can be converted (or correlated) to frequencies/wave numbers The coefficients can be mapped to scales to generate an energy spectrum 10

Data Collection Data was collected around an engineered log jam at the Oregon Hatchery Research Center Velocity data is collected at 0.1 m spatial intervals over a grid The data in each point in space is saved as a single semicolon separated file 11

Data Processing Method Language used: Matlab Tested on Windows 8 We import the time series data and correlate them to a spatial location on the grid. The coordinates of each point, its energy information, the energy’s wavelet scales and coefficients are output to a CSV file. The data can be easily opened later and imported into Matlab or another programming/data analysis environment for study. 12

Example output 13

Data Processing Methods (con’t) Data is read into memory Using an Excel file, each file can be mapped to a point in space; the Excel file lists the file name and (x, y, z) coordinates in the same row WinADV data Memory File name in Excel Point in space File name of each WinADV data set 14

Data Processing Methods (con’t) The velocity data is used to calculate the kinetic energy of each point over time. Velocity data in each data file Vector of kinetic energy data at each point for each time step 15

Data Processing Methods (con’t) For each point in space (i.e. each data file), its kinetic energy vector is pushed through a continuous wavelet transform function. Vector of kinetic energy data at each point for each time step cwtft() scales scale coefficients 16

Implications Further processing of the generated data can help produce a visualization of the energy spectrum of the data. A visualization will help us reach a conclusion on the distribution of energy around log jams, and when mapped with fish count data, this can be used to determine where fish like to reside. 17

Future Data in relational database? 18

Acknowledgements Professor Desiree Tullos Sean McGregor William L’Hommedieu Cara Walter Professor Julia Jones Professor Tom Dietterich Amanda Reinholtz Peggy Lee 19

Questions? 20

Sources Benitez-Nelson, C. (n.d.). What are eddies? Retrieved from Gibson, M. (2011). Turbulence. Thermopedia, doi: /AtoZ.t.turbulence Torrence, C., Gilbert P. Compo (1998): A Practical Guide to Wavelet Analysis. Bull. Amer. Meteor. Soc., 79, 61–78. Continuous Wavelet Transform - MATLAB & Simulink.MathWorks. Retrieved from transform.html 21