Continuous wavelet transform (CWT) - wavelet function Y (a,b) is continuous

1. The argument on p. 8 (the first paragraph in section 2.2) is not convincing. The author uses the CWT for a discrete number of scales such as in the case of the DWT. For instance, the analyses performed in Chapter 4 are plotted for a discrete scales (the smallest scale considered is 40 in Figure 4.31). Would the DWT do a similar analysis as the CWT? If yes, would the computational time required to carry out the DWT be shorter compared to the CWT?

Continuous wavelet transform (CWT) - wavelet function (a,b) is continuous

- the set of functions (a,b) is highly redundant

- we can compute the wavelet transform in any chosen scale a and time b

- optimal method for signal analysis and signal processing

Discrete wavelet transform (DWT) - wavelet function ij= (ai,bj) is discrete, ai,bj are properly given

- critical sampling that a discrete set of wavelet coefficients still retains the total information from a signal

- we have lost the covariance by dilation and translation of the CWT and the redundancy of the wavelet coefficients

- optimal method for compression and whenever we need a decomposition of a signal to (orthogonal) basis

2. The edge effect documented in Fig. 2.3, 4.28, 4.29, etc. is the price paid for using the continuous 2-D Mexican-hat wavelets. Which reasons led the author to apply the 2-D Mexican-hat wavelets for the analysis of mantle-flow patterns? Did he consider and test other types of wavelets for this purpose?

There are two main reasons of the edge effects:

1. The CWT is computed with help of the FFT. The analyzed signal is periodical extended during the computation. Therefore long-scale wavelets can also see extended parts of the signal (see Fig. 2.3d).

2. The analyzed signal should vanish on the edges. Sometimes signal demeaning (which helps to avoid false effects in large scales) can cause a false effects in small-scale wavelet spectrum on the edges.

The edge effect documented in Fig. 2.3 demonstrates an influence of the periodical extension of the signal upon the large-scale wavelet spectrum. It is not connected with use of Mexican-hat wavelet itself. Figures 4.28, 4.29, etc. show thermal-boundary layers which are very expressive in the cases of high Rayleigh numbers.

Reasons for Mexican-hat wavelet:

- real function, suited for detection of spatial (or time) structures in a field (or signal)

- even function, detects locations of peaks, not edges (as odd functions do)

- minimal number of secondary waves in the wavelet function, it means each peak in the signal will be only tripled

- one vanishing moment, wavelet spectrum is not influenced by a global linear trend of the signal

Fig. 2.4: Wavelet transforms of the box-like transforms.

Fig. 2.5: Wavelet functions and their Fourier spectra

Fig. 4.28, Fig. 4.30: Scalograms of temperature deviations for given Rayleigh numbers

3. Equation (2.13) relates the Fourier period T and the wavelet scale a for the DOG wavelets. Why (2.13) does not depend on o such as T in eq. (2.10). Should eq. (2.13) reduce to (2.10) for m=0. If yes , how can it be shown?

Definition of the Morlet wavelet: o= -1/4 eioe-1/22 (2.9)

Related Fourier period T: T=(4ao+(2+ o2)1/2

Definition of the DOG wavelet: o= (-1)m+1(m+1/2)dm/dm(e-1/22) (2.11)

Related Fourier period T: T=(2a)/(m+1/2)1/2 (2.13)

Parameters o and m are both related with the number of waves in the wavelet function. But the Morlet and DOG wavelets have different structures, so o and m cannot be related in such easy way.

Remark:

‘Extended’ definition of the Morlet: o= co(eio-e-1/2o

2) e-1/22

- it exactly satisfies ‘wave’ condition

- analytical relation between the Fourier period T and wavelet scale a does not exist

Limit o = 0: oMORLET= -i oDOG1

4. Why is definition (2.14) of the 2-D Mexican-hat wavelets not reduced to definition (2.11) for a 1-D case?

1-D Mexican-hat wavelet:

o= -2.3-1/2.-1/4 d2/d2(e-1/22) = 2.3-1/2.-1/4 (1- ) e-1/22 (2.11)

2-D isotropic Mexican-hat wavelet:

o= -(2)-1/2 2(e-1/2|2) = (2)-1/2 (1- |) e-1/2|2 (2.14)

They both satisfy:

- zero wave (or admissibility) condition

- zero first moment

- unit L2 norm

5. Why is Figure 2.3(b) not axisymmetric?

Panel (b) demonstrates the effect of the edges. During computation of the CWT (with help of the FFT) the signal is periodically extended. Large-scale wavelets can be also influenced by extended part of the original function. Other scalograms in the thesis are corrected to this effect.

6. The term the amplitude power of the magnitude of the wavelet spectrum used in the 2nd paragraph on p.10 is difficult to understand intuitively. Should it be interpreted as the amplitude of the wavelet spectrum? According to my view, eq. (2.5) and the following statement are not trivial to see. Can the author present a simple derivation of (2.5) during the defense?

Yes, it is a mistake, the amplitude power of the magnitude of the wavelet spectrum should be written as the amplitude of the wavelet spectrum (or also the magnitude of the WS).

The 2nd paragraph on page 10:

7. What is a periodic box, the term used in the 3rd paragraph on p.10? Should it be interpreted as a periodic extension of a function outside a box? How far?

Yes, as the signal f(x) on a periodic box we understand a signal originally given on a box but then periodically extend (due to the FFT).

9. How should we understand the spatial resolution T=1s (p.14)? Should it be interpreted as the resolution in time domain.

Yes, it is better to use words ‘temporal resolution’ (author meant ‘spatial’ as an opposite to a Fourier domain, in a general sense).

10. The author uses the term the wavelet spectrum energy (p.17) and wavelet power spectrum (p.18) for the same quantity E2. I guess the better term would be power wavelet spectrum since the term ‘wavelet power spectrum’, used by the author, is understood as the WT of the power spectrum of a signal.

Yes, the term power wavelet spectrum is better.

8. How should we understand relatively thin in physical space (p.13, the last sentence)? Relatively to what?

We can choose such dilatations of the wavelet functions of different kinds that all the wavelets will have the same resolution in a physical space (e.g., we have chosen 1s resolution in the figure).

Because the wavelets have different shapes, we can compare their thickness in a physical domain. Then wavelets relatively thin to the others have better temporal (or spatial) resolution (but worse resolution in a spectral domain).

11. It is difficult to understand the behavior kmax as a function of time in Figure 2.7(d). How is the jump in kmax at time=275 interpreted in terms of the original double-box functions? There is no dramatic change of this function at time=275. Vice versa, the changes of the original function between times 160 and 220 are not plotted in Figure 2.7 (c) and (d). Why? I believe Figure 2.7 badly illustrates the power of Emax/ kmax method. Can the author explain that figure in a better way than is presented in the thesis?

Figure 2.7 illustrates the technique, how to get Emax and kmax quantities from the original wavelet spectrum. The power of the Emax/kmax method is illustrated in Figure 2.8.

That is why:

- a long-scale part of the wavelet spectrum was used in Figure 2.7. Then the wavelet spectrum is relatively simple and the extraction of the Emax and kmax from the spectrum can be easily demonstrated. The jump in the kmax distribution corresponds to the edge of the box - of course, we have used too huge hammer for analysis of the box, so the location of its edge is not perfect

- the changes of the original function for times lesser than 180 are not plotted in Figure 2.7. Because the double-box function is symmetrical (for time=190) and the used DOG2 wavelet is even, the power wavelet spectrum is also 190-symmetrical. We have preferred a simple spectrum only from a part of the signal - Figure 2.7(b) is better visual then.

Fig. 2.7: Algorithm for Emax/kmax determination.

Fig. 2.8(c): Emax/kmax distribution for the small-scale range of wavenumbers k. The power wavelet spectrum is normalized.

11. The reference state is not defined in section 3.2. Later on, one can implicitly deduce that the reference state means the initial conductive state. It would help for easier reading if this state is introduced immediately at the beginning of section 3.2.2.

12. If and are taken from (3.4), (3.5) and (3.14), the system of equation (3.1)-(3.3) is not consistent because of gravity acceleration g. Hence, g must be approximated by the initial value go even in (3.1)-(3.3) to formulate the problem consistently.

14. In the first line on p. 26, the words equation of continuity (3.26) should be replaced by the divergence-free constraint (3.26)

18. The word viskosity is misspell in the caption to Table 3.1.

Thank you for constructive comments.

13. I do not understand why eqs. (3.26)-(3.27) together with (3.6) and (3.7) represent equilibrium system (the paragraph before section 3.2.4). How the equilibrium system is define? References? The author may have thought on a consistent system. The next sentence in this paragraph is also difficult to understand. Why is the temperature deviation the only independent variable in (3.28)? What about the velocity field and coupling terms between temperature and velocity in both (3.27) and (3.28)? The author may have had in mind an explicit scheme of solution to this system of equations. But, in general, all three equations in (3.26)-(3.28) are coupled to each other.

The nonlinear systems are usually described in a form dy/dt = F(y), where F is a nonlinear operator. The only time derivation in the system (3.26)-(3.28) belongs to the time deviation - therefore we look at as on the only independent variable and the rest of equations (3.26)-(3.27) is considered to represent an equilibrium system with the mapping čv.

The paragraph with equations (3.26)-(3.28) from the thesis:

15. The statement after (3.34) is confusing. How can we exchange the scalar quantity by the vector v? How to modify, for instance, the dot products of v with gradient of temperature in (3.28)?

The sentence The energy conservation (3.28) is of the same form except that the stream function (3.30) is used instead of velocity v is badly written.

Rather it should be as follows:

The energy conservation (3.28) need not any rearrangement except of filling the formula (3.30) for a velocity v.

16. The statement after eq. (3.39) follows immediately from the condition (3.35) without considering (3.39). Demonstrate it.

Condition on boundaries r=r1,r2: /= 0 (3.35)

Condition on boundaries =0,: /r= 0 (3.39)

Then =0 on all boundaries.

If we only consider the condition (3.35), then there can exist different constants on the boundaries given by r=r1,r2. Let choose (r1)=0, (r2)=r2. Then we can write r,r,+r, with r1)=0, r2)=0. The velocity v=(1/r2sin/-1/rsin/r0)=vo+(0,-1/rsin,0).

In the case of non-zero constants on the top bottom boundaries there exist a global (one-direction) flow in a medium.

17. What does the abbreviation the ADI method in the first paragraph on p. 28 mean?

The ADI method means the Alternating Direction Implicit method (Douglas and Rachord, 1956).

Douglas, J. Jr. and H.H. Rachford, On the numerical solution of heat conduction problems in two and three space variables, Trans. Amer. Math. Soc., 82, 421-430, 1956.

19. Was there any motivation to choose the initial temperature deviation in the forms (3.49) or (3.50)? It would help for easier understanding if there is a reference to Figure 3.3 (p.36) that plots the initial temperature deviations.

Yes, as we had used the code developed by Moser (1994) for computing the axisymmetrical thermal convection, we also kept original initial temperatures in the cases of our low Rayleigh number runs. Such we were able to check our first computations.

On the other hand, high Ra runs were started from the last stages of finished low Ra runs, to reduce the computational time needed for bypassing their transition state.

Moser J., Mantle Dynamics and Rotation of the Earth, PhD thesis, Charles University, Prague, 1994.

20. Is it proveable that the symmetrical initial temperature distribution generates a symmetrical flow pattern? Or is it only an intuitive statement? (p.30, after (3.50), p.46, the 2nd paragraph)

21. What is the difference between the strange attractor (p.31) and the chaotic attractor (Fig.3.2)?

It is no difference.

22. How is Xim in eq. (3.55) defined? Is it simply Xm(ti) defined by (3.54)?

Yes, Xim=Xm(ti).

23. Why is the attractor of the velocity field studied by means of the total kinetic energy, which is a global quantity, while the attractor of temperature field is studied by means of the surface heat flow, which is not a quantity characterizing a temperature distribution globally. One objection is that the attractor of velocity field will differ from that of temperature, which is confirmed by Figure 3.12. Was there a motivation to define a global quantity for the velocity field, and a surface quantity for the temperature filed?

We suppose only one attractor in the phase space of the convection system. We do not search characteristics of the attractor right in the phase space but we consider time-series of quantities which depend on phase variables. Then we can use so-called reconstruction of the phase space and we are able to find the (correlation) dimension of the attractor.

Of course, some quantities can reflect better a behavior of all the phase variables, some worse. There is no wonder that we need shorter time-series for finding the attractor dimension from a global quantity as the kinetic energy is than from the semi-global Nusselt number quantity.

24. Why did the computations shown in Figure 3.12 (d) not continue to establish a longer time series for surface heat flow?

We computed 3,200.000 time-steps in the case Ra=105. But although space structure of the convection system is rather simple, the time behavior embodies a high level of complexity. Also other runs had to be computed...

25. Why are the time series in Figure 3.13 longer than in Figure 3.14? Is the intention of the author in the near future to extent of these time series to compute the correlation dimension?

In both runs the same number of time-steps was computed, about 2,000.000. But the higher Rayleigh number the more turbulent flow, and it needs higher grid sizes of computed area and smaller time-step in the each iteration. That is why the second run (Ra=106) has shorter time-series than the first one (Ra=5.105). If the assumption is correct that the attractor dimensions in these runs would be even higher than in the case of Ra=105, the extent of the time-series appear to be almost impossible.

26. It is highly probable that a large-scale part of the power wavelet spectra and the Fourier spectra are influenced by a too-short length of time series (p. 46, the last paragraph). Why has not this been tested?

In the wavelet analysis, we use cone-of-influence lines (see Figures 2.2b-c) showing you parts below which the spectra can be influenced due to edge effects. The lines are not drawn in Figures 3.15 and 3.16 but they were computed and we can say that spectra below period 0.1 are trustworthy (except of boundaries).

27. Which method is used to compute the Newton convection in the Cartesian box? (p.82, the 1st paragraph). References? The description of the viscosity of the Newtonian liquid is ambiguous. How large is the viscosity contrast between the bottom and top of the box? 10 or 300?

I am sorry, the reference about the method is written later, in the 2nd paragraph (it is Ten et al., 1996). Also other works (Ten et al., 1997; Ten et al., 1998; Ten et al., 1999) describe the used method.

Ten et al., 1997: The time-dependent velocity fields used in the mixing come from the extended Boussinesq, base-heated convection models with viscous and adiabatic heating included. The surface temperature To is 0.1 and the surface dissipation number is 0.05. The depth-dependence of the viscosity is a factor of ten across the layer with a thermal expansivity decrease by a factor 3. The temperature-dependent viscosity contrast is near 300 across the layer, which have an aspect-ratio of two.

Ten et al., 1999: The rheology establishes the power-law dependence ij’ = A exp(BT-Cz) ij, ij’ is a strain rate tensor, ij is a stress tensor, T is temperature, z is depth, A is a material constant, and B and C define temperature and depth dependencies. The constants B and C have been chosen in order to obtain a temperature-dependent viscosity of about 300 and a depth-dependent viscosity increase of about 10 across the layer.

Ten A., Yuen D.A., Larsen T.B. and A.V. Malevsky, The evolution of material surfaces in convection with variable viscosity as monitored by a characteristics-based method, Geoph. Res. Lett., 23, 2001-2004, 1996.

Ten A., Yuen D.A., Podladchikov Yu.Yu., Larsen T.B., Pachepsky E. and A.V. Malevsky, Fractal features in mixing of non-Newtonian and Newtonian mantle convection, Earth Planet. Sci. Lett., 146, 401-414, 1997.

Ten A., Podladchikov Yu.Yu., Yuen D.A., Larsen T.B. and A.V. Malevsky, Comparision of mixing properties in convection with the particle-line method, Geoph. Res. Lett., 25(6), 3205-3208, 1998.

Ten A., Yuen D.A. and Yu.Yu. Podladchikov, Visualization and analysis of mixing dynamical properties in convecting systems with different rheologies, Electronic Geosciences, 4(1), 1999.

28. The 2nd paragraph on p.82 brings very detailed information that cannot tell anything to a reader. This detailed information is unballanced with nearly no information provided about modeling of the Newton flow. I guess that the first two paragraphs should removed from the text and substituted by adequate references, or the modeling of the Newton flow should be described in a more understandable way.

The second paragraph does not brings very detailed information, it only informs a reader about computational techniques employed to monitor the evolution of the field. A reader can find more information in the referenced work Ten et al., 1996 (or also Ten et al., 1997).

2nd paragraph:

A spline-based technique in combination with a characteristic-based method (Malevsky and Yuen, 1991) is employed to monitor the evaluation of the field (Ten et al., 1996). A resolution of 3000 by 1500 bi-cubic splines is used. The field is advected with the fourth-order Runge-Kutta method.

29. We can believe or not all statements in the first two paragraphs on p.83. There are no references on a figure, or on literature.

Two paragraphs on p.83 present author’s concept about a behavior of the wavelet spectrum when there is present not only continuous signal but also discontinuities - there exist so-called HGSAs (high-gradient surrounded areas) in the field. The idea presented in these paragraphs seemed to be simple from a veiw of the author, he apologies that he did not include any references on figures.

30. Why does the author introduce the terms scale mode (Table 4.1), modes (the caption to Figure 4.14) and scale (the titles of panels in Figure 4.14, as well as wavelet modes (Fig. 4.26.)? If I understand correctly, all of these terms denote the same quantity a, introduced by (2.1) and named a scale.

A quantity a is really introduced by (2.1) and is named a (wavelet) scale. Scale mode (also wavelet mode or only mode) is a logarithmic measure of the scale a. Whenever a mode is mentioned in the thesis, a reader also gets the reference to Table 4.1, where the modes are recomputed to related resolutions of the wavelets (what is much familiar to a reader).

Why to use modes instead of scales:

- logarithmic representation of the scale is natural for wavelets, because it corresponds to the multiplicative nature of the dilation parameter (Farge, 1992)

- there exist methods how to calculate the CWT from the DWT, and vice versa. As the scale used in the DWT is powered (the mostly used power is 2, so-called dyadic grid), there is tendency to also keep the CWT scale powered.

- it is highly recommended (e.g., Farge, 1992) to visualize wavelet spectra with a logarithmic scale, a linear scale causes that small-scale behavior of the spectrum (which is in general the most interesting to study) is completely flattened

- in our case the scale was computed with help of integer modes. Then is, e.g., better to speak about mode 4 than about scale 0.216.

Farge M., Wavelet transforms and their applications to turbulence, Annu. Rev. Fluid. Mech., 24, 395-457, 1992.

31. I would not introduced the term local wavelet spectrum (p.86) since it confuses the reader. Or, is there any different between wavelet spectrum, originally introduced in Chapter 2, and the local spectrum?

No, by the term local wavelet spectrum it is thought the wavelet spectrum, the word ‘local’ was (maybe improperly) used as an opposite to the term global wavelet spectrum.

34. What is meant by small-scale glasses in the 6th paragraph on p. 91? Does the author want to focus on anything or does he want to filter-out some frequencies?

A meaning of small-scale glasses is that one will take so fine glasses which allow him to see only small-scale structure. So we can understand the small-scale glasses as a band-pass filter, which will filtered-out all frequencies except of small scales.

33. The 3rd paragraph on p. 91. The term stable part is inaccurate in this context. The author means a constant. I agree that the wavelet modes 5 and 7 corresponds to a mixed field, but not wavelet mode 3. This statement is confirmed in the next paragraph.

In the 3rd paragraph, there are showed some criteria of mixing. Because it is relative what to consider as a mixed medium, a possible definitions were described. Definition with help of constant GWS cannot fight the absolute mixing (with GWS equals to zero). It is only stage when the medium can not be mixed better (during some acceptable time).

32. I do not agree that there is a large difference in feature between panels (b) in Figures 4.14-4.18. That is why, I cannot agree with the statement that the last time steps t5 reveals good mixing for large scales.

t1

t2

t3

t4

t5

Maxima in times t4 and t5 are flatter than in the previous time-steps.

The number of maxima is smaller.

But: observation is relative, computation of the similarity coefficients of the wavelet spectra is needed.

35. How may the presence of numerical instabilities in Figure 4.31d be detected? From which feature(s)? (p.97, the 3rd paragraph)