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I have a time series of tidal data for one month and success with identifying tidal characteristic such as

  • semidiurnal or diurnal
  • spring and neap position
  • power spectrum

However, I don't have success in obtaining tidal properties such as amplitude and phase in diurnal (d1), semidiurnal (d2) and fortnightly (d4).

The tidal data are as follows:

{{{-0.92}, {-1.38}, {-1.45}, {-1.04}, {-0.38}, {0.23}, {0.76}, \
{1.32}, {1.65}, {1.51}, {1.1}, {0.6}, {0.11}, {-0.3}, {-0.6}, \
{-0.49}, {0.01}, {0.58}, {1.05}, {1.39}, {1.53}, {1.3}, {0.72}, \
{0.18}, {-0.4}, {-0.91}, {-1.24}, {-1.08}, {-0.57}, {0.19}, {0.86}, \
{1.41}, {1.83}, {1.95}, {1.55}, {0.97}, {0.44}, {-0.05}, {-0.57}, \
{-0.77}, {-0.47}, {0.11}, {0.68}, {1.1}, {1.31}, {1.34}, {0.92}, \
{0.28}, {-0.28}, {-0.94}, {-1.46}, {-1.63}, {-1.22}, {-0.49}, {0.22}, \
{0.84}, {1.42}, {1.83}, {1.78}, {1.34}, {0.73}, {0.18}, {-0.28}, \
{-0.67}, {-0.69}, {-0.18}, {0.43}, {0.95}, {1.28}, {1.49}, {1.3}, \
{0.66}, {0.06}, {-0.57}, {-1.2}, {-1.64}, {-1.57}, {-0.99}, {-0.21}, \
{0.49}, {1.09}, {1.66}, {1.91}, {1.64}, {1.11}, {0.52}, {0.01}, \
{-0.47}, {-0.73}, {-0.49}, {0.06}, {0.6}, {1.07}, {1.37}, {1.45}, \
{1.08}, {0.43}, {-0.19}, {-0.8}, {-1.37}, {-1.63}, {-1.35}, {-0.67}, \
{0.11}, {0.74}, {1.35}, {1.85}, {1.91}, {1.53}, {1.}, {0.4}, {-0.06}, \
{-0.49}, {-0.59}, {-0.24}, {0.36}, {0.91}, {1.28}, {1.52}, {1.42}, \
{0.91}, {0.31}, {-0.25}, {-0.87}, {-1.36}, {-1.42}, {-0.94}, {-0.24}, \
{0.47}, {1.08}, {1.63}, {1.98}, {1.81}, {1.4}, {0.83}, {0.34}, \
{-0.12}, {-0.47}, {-0.45}, {0.01}, {0.53}, {0.98}, {1.27}, {1.44}, \
{1.25}, {0.7}, {0.15}, {-0.37}, {-0.94}, {-1.31}, {-1.18}, {-0.63}, \
{-0.01}, {0.59}, {1.13}, {1.64}, {1.85}, {1.63}, {1.13}, {0.6}, \
{0.19}, {-0.23}, {-0.47}, {-0.35}, {0.08}, {0.51}, {0.91}, {1.18}, \
{1.29}, {1.03}, {0.49}, {-0.02}, {-0.47}, {-0.91}, {-1.18}, {-1.01}, \
{-0.54}, {0.07}, {0.59}, {1.08}, {1.47}, {1.57}, {1.29}, {0.87}, \
{0.37}, {-0.05}, {-0.45}, {-0.64}, {-0.5}, {-0.12}, {0.28}, {0.64}, \
{0.9}, {1.01}, {0.81}, {0.36}, {-0.05}, {-0.46}, {-0.88}, {-1.08}, \
{-0.86}, {-0.42}, {0.08}, {0.56}, {1.02}, {1.39}, {1.42}, {1.15}, \
{0.75}, {0.28}, {-0.1}, {-0.5}, {-0.72}, {-0.62}, {-0.3}, {0.12}, \
{0.49}, {0.71}, {0.85}, {0.74}, {0.4}, {0.04}, {-0.33}, {-0.67}, \
{-0.87}, {-0.72}, {-0.33}, {0.11}, {0.53}, {0.91}, {1.23}, {1.31}, \
{1.04}, {0.67}, {0.26}, {-0.13}, {-0.59}, {-0.8}, {-0.74}, {-0.45}, \
{-0.07}, {0.34}, {0.62}, {0.84}, {0.92}, {0.66}, {0.29}, {-0.04}, \
{-0.33}, {-0.59}, {-0.58}, {-0.27}, {0.12}, {0.49}, {0.85}, {1.23}, \
{1.33}, {1.1}, {0.76}, {0.35}, {-0.08}, {-0.57}, {-0.88}, {-0.88}, \
{-0.58}, {-0.13}, {0.32}, {0.69}, {1.01}, {1.17}, {1.}, {0.69}, \
{0.33}, {-0.04}, {-0.36}, {-0.47}, {-0.25}, {0.1}, {0.51}, {0.92}, \
{1.22}, {1.36}, {1.17}, {0.74}, {0.28}, {-0.22}, {-0.76}, {-1.09}, \
{-1.1}, {-0.74}, {-0.23}, {0.3}, {0.73}, {1.18}, {1.41}, {1.28}, \
{0.94}, {0.47}, {0.08}, {-0.23}, {-0.49}, {-0.35}, {0.06}, {0.52}, \
{0.95}, {1.26}, {1.43}, {1.25}, {0.74}, {0.22}, {-0.34}, {-0.9}, \
{-1.27}, {-1.29}, {-0.85}, {-0.22}, {0.4}, {0.92}, {1.42}, {1.69}, \
{1.54}, {1.11}, {0.59}, {0.14}, {-0.24}, {-0.49}, {-0.39}, {0.11}, \
{0.64}, {1.07}, {1.38}, {1.49}, {1.25}, {0.67}, {0.07}, {-0.58}, \
{-1.17}, {-1.55}, {-1.43}, {-0.89}, {-0.15}, {0.53}, {1.11}, {1.64}, \
{1.84}, {1.62}, {1.13}, {0.56}, {0.08}, {-0.31}, {-0.54}, {-0.32}, \
{0.25}, {0.76}, {1.22}, {1.54}, {1.6}, {1.18}, {0.56}, {-0.04}, \
{-0.71}, {-1.31}, {-1.62}, {-1.4}, {-0.74}, {0.02}, {0.71}, {1.31}, \
{1.82}, {1.93}, {1.59}, {1.03}, {0.44}, {-0.03}, {-0.49}, {-0.63}, \
{-0.28}, {0.29}, {0.83}, {1.25}, {1.54}, {1.49}, {0.95}, {0.28}, \
{-0.35}, {-1.03}, {-1.59}, {-1.79}, {-1.37}, {-0.59}, {0.19}, {0.84}, \
{1.47}, {1.92}, {1.86}, {1.43}, {0.83}, {0.24}, {-0.24}, {-0.64}, \
{-0.61}, {-0.13}, {0.45}, {1.01}, {1.4}, {1.6}, {1.37}, {0.72}, \
{0.07}, {-0.6}, {-1.25}, {-1.75}, {-1.71}, {-1.11}, {-0.32}, {0.44}, \
{1.08}, {1.67}, {1.93}, {1.68}, {1.2}, {0.56}, {0.03}, {-0.46}, \
{-0.7}, {-0.53}, {0.03}, {0.61}, {1.11}, {1.4}, {1.49}, {1.09}, \
{0.47}, {-0.14}, {-0.81}, {-1.46}, {-1.76}, {-1.5}, {-0.79}, {-0.04}, \
{0.65}, {1.24}, {1.74}, {1.82}, {1.43}, {0.87}, {0.28}, {-0.2}, \
{-0.62}, {-0.75}, {-0.39}, {0.16}, {0.68}, {1.05}, {1.33}, {1.26}, \
{0.76}, {0.13}, {-0.45}, {-1.04}, {-1.54}, {-1.65}, {-1.22}, {-0.55}, \
{0.15}, {0.76}, {1.33}, {1.68}, {1.57}, {1.09}, {0.52}, {0.02}, \
{-0.38}, {-0.7}, {-0.69}, {-0.21}, {0.29}, {0.74}, {1.06}, {1.26}, \
{1.05}, {0.5}, {-0.04}, {-0.55}, {-1.08}, {-1.42}, {-1.33}, {-0.79}, \
{-0.14}, {0.45}, {0.97}, {1.45}, {1.63}, {1.33}, {0.86}, {0.35}, \
{-0.04}, {-0.43}, {-0.64}, {-0.42}, {0.02}, {0.44}, {0.79}, {1.08}, \
{1.16}, {0.85}, {0.35}, {-0.08}, {-0.53}, {-0.97}, {-1.14}, {-0.86}, \
{-0.33}, {0.23}, {0.74}, {1.2}, {1.51}, {1.44}, {1.06}, {0.66}, \
{0.18}, {-0.21}, {-0.55}, {-0.62}, {-0.34}, {0.02}, {0.37}, {0.62}, \
{0.82}, {0.81}, {0.52}, {0.11}, {-0.24}, {-0.62}, {-0.91}, {-0.91}, \
{-0.56}, {-0.14}, {0.31}, {0.73}, {1.15}, {1.32}, {1.14}, {0.79}, \
{0.4}, {0.03}, {-0.34}, {-0.61}, {-0.55}, {-0.3}, {-0.01}, {0.32}, \
{0.56}, {0.7}, {0.68}, {0.45}, {0.14}, {-0.17}, {-0.46}, {-0.66}, \
{-0.58}, {-0.3}, {0.06}, {0.43}, {0.77}, {1.06}, {1.14}, {0.94}, \
{0.65}, {0.29}, {-0.02}, {-0.36}, {-0.57}, {-0.52}, {-0.29}, {0.}, \
{0.28}, {0.51}, {0.69}, {0.73}, {0.59}, {0.36}, {0.12}, {-0.11}, \
{-0.27}, {-0.26}, {-0.02}, {0.3}, {0.57}, {0.81}, {1.02}, {1.12}, \
{0.94}, {0.66}, {0.37}, {0.05}, {-0.3}, {-0.53}, {-0.51}, {-0.33}, \
{-0.08}, {0.19}, {0.42}, {0.68}, {0.81}, {0.73}, {0.56}, {0.33}, \
{0.12}, {-0.06}, {-0.12}, {0.02}, {0.24}, {0.45}, {0.69}, {0.91}, \
{0.97}, {0.83}, {0.58}, {0.27}, {-0.08}, {-0.4}, {-0.64}, {-0.69}, \
{-0.51}, {-0.22}, {0.1}, {0.41}, {0.74}, {0.94}, {0.94}, {0.75}, \
{0.49}, {0.24}, {0.04}, {-0.14}, {-0.1}, {0.14}, {0.36}, {0.61}, \
{0.82}, {0.92}, {0.8}, {0.49}, {0.18}, {-0.15}, {-0.55}, {-0.85}, \
{-0.86}, {-0.61}, {-0.29}, {0.12}, {0.52}, {0.93}, {1.2}, {1.19}, \
{0.97}, {0.61}, {0.26}, {0.04}, {-0.16}, {-0.19}, {0.07}, {0.38}, \
{0.74}, {0.97}, {1.09}, {0.98}, {0.6}, {0.18}, {-0.25}, {-0.69}}}

After plotting the data and using ContinuousWaveletData as well as Periodogram:

So, there are three questions related to wavelet that I didn't understand:

  • how to draw Cone of Influence
  • how to draw Global Wavelet Spectrum
  • how to calculate/obtain amplitude and phase of diurnal (d1), semidiurnal (d2), and fortnightly (d4) using Morlet wavelet.
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ListPlot[Flatten[data]]

ListPlot

Periodogram[Flatten[data]]

Periodogram

cwd = ContinuousWaveletTransform[Flatten[data]]

ContinuousWaveletTransform

ListLinePlot[cwd[All, "Values"], PlotRange -> {{1, 710}, {-10, 10}}]

ListLinePlot cwd

WaveletScalogram[cwd]

WaveletScalogram

ListPlot3D[Re@cwt[All, "Values"], ColorFunction -> "DarkRainbow", 
 AxesLabel -> {"time", "scale"}, Mesh -> None, PlotRange -> All]

enter image description here

{ws, oct, voc} = cwd[{"WaveletScale", "Octaves", "Voices"}]

({0.251646, 9, 4})

cwd["SampleRate"]

(1)

{Precision[cwd], Accuracy[cwd]}

({MachinePrecision, 13.8445})

Identify features:

cwt = ContinuousWaveletTransform[data, 
  DGaussianWavelet[4], {Automatic, 12}, Padding -> "Fixed"]
WaveletScalogram[cwt]

identify features

cwt = ContinuousWaveletTransform[data, GaborWavelet6, {Automatic, 16}] WaveletScalogram[cwt, ColorFunction -> "BlueGreenYellow"] Amplitude GaborWavelet

WaveletScalogram[cwt, Automatic, Arg, 
 ColorFunction -> "BlueGreenYellow"]

Phase

cwt = ContinuousWaveletTransform[data, GaborWavelet[6], {10, 16}, 
   Padding -> 0.0, SampleRate -> 2047, WaveletScale -> Automatic];
WaveletScalogram[cwt, ColorFunction -> "SolarColors"]

enter image description here

Filter the cosine with frequency 140[Pi]: vec = ConstantArray[1, Length[data]]; vec[[1 ;; 1000]] := 0.0 cwtThresh = WaveletMapIndexed[#1 vec &, cwt, {{4 | 5, }, {6, u /; u < 8}}]; WaveletScalogram[cwtThresh, ColorFunction -> "SolarColors"] ![140[Pi]]11

This does suffer only slightly from the problem that lower frequencies are detected at higher ones:

cwd1 = ContinuousWaveletTransform[data, MorletWavelet[], Automatic]
ListLinePlot[{InverseContinuousWaveletTransform[cwd1], data}]

enter image description here

Scalogram:

cwd = ContinuousWaveletTransform[data, GaborWavelet[6], {4, 12}, 
   WaveletScale -> 100];
{WaveletScalogram[cwd, All, Re, ColorFunction -> "CherryTones"], 
 WaveletScalogram[cwd, All, Abs, ColorFunction -> "CherryTones"]}

Scalogram

I recovered a definition from evaluating-hough-functions-by-using-ndeigensystem-on-the-laplace-tidal-equation.

So there is no real information in the raw data about the form of the diurnal wave all of the above tests are searches for reconstruction in terms of well known diurnal waves. Whether there are semidiurnal wase involved is hidden too. But the test did found all and that might hold in composition.

semidiurnal or diurnal corresponds to the form of the wavelets analyzed! spring and neap position correspond to the dynamics of the wavelets: amplitudes and phases, precision, and reconstructability. power spectrum can too be just development reconstructive under generative limits of the wavelets used for the analyses.

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