garethgroberts/river_profile_spectra: Wavelet Power Spectra: River Profiles

Title: Wavelet power spectra of longitudinal river profiles Quick Summary: The code contained in this repository transforms longitudinal river profiles [one dimensional time series] from the spatial domain [i.e. elevation as a function distance, z(x)] into the distance-wavenumber [spatial frequency, k] domain [i.e. Z(x,k)]. Various techniques (e.g. generation of power spectra, filtering, inverse transforms, cross wavelet spectra, fitting of synthetic spectra, mirroring, testing noisy signals) are included in the source code and/or details can be found in Roberts et al. [2019, JGR, doi: 10.1029/2018JF004796] and Roberts [2019, GRL, doi:10.1029/2019GL082446]. Description: The software contained in this repository is designed to calculate power spectra of one-dimensional time series using wavelets. It calculates power [amplitude-squared] as a function of distance and wavelength [or time and frequency], and also distance-averaged power spectra, which is somewhat analogous to a Fourier transformation of the time series. A description of the methods used and examples are given in: Roberts, G. G., White, N., Lodhia, B., 2019. The Generation and Scaling of Longitudinal River Profiles, Journal of Geophysical Research - Earth Surface, doi:10.1029/2018JF004796. agupubs.onlinelibrary.wiley… And relatedly, a method to calculate similarities and disparities between time series [e.g. river profiles] across the scales of interest using a cross wavelet approach is given in: Roberts, G. G., 2019. Scales of Similarity and Disparity Between Drainage Networks, Geophysical Research Letters, doi: 10.1029/2019GL082446. agupubs.onlinelibrary.wiley… The code was designed to convert longitudinal river profiles [i.e. elevation as a function of distance, z(x)] into the distance-wavelength domain. The time series being transformed are monotonic functions with evenly sampled elevations [i.e. $\delta_x = C$]. Prior to transformation the time series are mirrored about the x and z axes in an attempt to minimise edge effects. Spectral bias was rectified using the approach described by Liu et al. [2007], in essence wavelet power spectra are normalised by scale. The code uses subroutines from the machine learning algorithms from Albanese et al. [2012], which can be found at mlpy.sourceforge.net, which are based on the methods described by Torrence and Compo [1998]. It also uses numpy and scipy routines to perform a basic Fourier transform and for other basic computational tasks. The code has been used to transform other (e.g. non-monotonic) time series successfully [see Roberts & Mannion, 2019, Scientific Reports, doi.org]. This repository includes [1] the source code to perform the wavelet transformation, it is written in python; [2] an example data file, which is the elevation of the Niger river extracted from the CGAIR SRTM digital elevation model down-sampled to 2 km [one.z, see Roberts et al., 2019; srtm.csi.cgiar.org]; [3] a plotting script to show results, this bash shell script uses routines from the Generic Mapping Tools [gmt; gmt.soest.hawaii.edu] toolkit to do the plotting. For completeness/benchmarking the plotting script is a pared down version of the one used to generate Figure 3 in Roberts et al. (2019). Note that the full resolution [~90 m] river profile is also included for completeness (obs_river), and for use in the plotting script. The code has been tested and benchmarked. However, I suggest that you regard it as developmental that you run your own tests to confirm veracity. If you have any issues with running the code and/or comments contact gareth.roberts@imperial.ac.uk. Software used to generate code: Python 2.7.13, GMT 5.1.1, mlpy 3.5.0. It was developed and run on OSX 10.12.6 (macOS Sierra), with DeveloperTools installed, and should be portable to most *nix systems/python environments. To run the code and plot output, assuming that you have already installed the dependencies (e.g. python, numpy, scipy, mlpy, gmt) try: python wavelets_rivers.py ./plot_wave_power_rivers.gmt If you want to run the code multiple times, say to test different mother wavelets, you can move output around using ./rename_output.sh. Examples of output for the DOG and Morlet mother wavelets are included in ./wavelet_tests. Note, however, that for compactness I've not uploaded the power spectral maps (i.e. file*.txt, or surf.grd). You will need to run the code first if you want to see that output (or email me). Some useful references: Albanese, D., Visintainer, R., Merler, S., Riccadonna, S., Jurman, G., Furlanello, C., 2012. mlpy: Machine Learning Python, arXiv:1202.6548. Liu, Y., Liang, X. S., Weinberg, R. H., 2007. Rectification of the Bias in the Wavelet Power Spectrum, Am. Met. Soc., doi: 10.1175/2007JTECHO511.1. Torrence, C., Compo, G. P., 1998. A Practical Guide to Wavelet Analysis, Bull. Am. Met. Soc., 79(1), 61-78. Gareth Roberts, 2017-2020, UK.