Geophysical modelling with Colombeau functions: Microlocal properties and Zygmund regularity
Abstract
In global seismology Earth’s properties of fractal nature occur. Zygmund classes appear as the most appropriate and systematic way to measure this local fractality. For the purpose of seismic wave propagation, we model the Earth’s properties as Colombeau generalized functions. In one spatial dimension, we have a precise characterization of Zygmund regularity in Colombeau algebras. This is made possible via a relation between mollifiers and wavelets.
1 Introduction
Wave propagation in highly irregular media. In global seismology, (hyperbolic) partial differential equations the coefficients of which have to be considered generalized functions; in addition, the source mechanisms in such application are highly singular in nature. The coefficients model the (elastic) properties of the Earth, and their singularity structure arises from geological and physical processes. These processes are believed to reflect themselves in a multifractal behavior of the Earth’s properties. Zygmund classes appear as the most appropriate and systematic way to measure this local fractality (cf. [2, Chap.4]).
The modelling process and Colombeau algebras. In the seismic transmission problem, the diagonalization of the first order system of partial differential equations and the transformation to the second order wave equation requires differentiation of the coefficients. Therefore, highly discontinuous coefficients will appear naturally although the original model medium varies continuously. However, embedding the fractal coefficient first into the Colombeau algebra ensures the equivalence after transformation and yields unique solvability if the regularization scaling is chosen appropriately (cf. [7, 10, 4]). We use the framework and notation (in particular, for the algebra and for the mollifier sets) of Colombeau algebras as presented in [11].
An interesting aspect of the use of Colombeau theory in wave propagation is that it leads to a natural control over and understanding of ‘scale’. In this paper, we focus on this modelling process.
2 Basic definitions and constructions
2.1 Review of Zygmund spaces
We briefly review homogeneous and inhomogeneous Zygmund spaces, and , via a characterization in pseudodifferential operator style which follows essentially the presentation in [3], Sect. 8.6. Alternatively, for practical and implementation issues one may prefer the characterization via growth properties of the discrete wavelet transform using orthonormal wavelets (cf. [8]).
Classically, the Zygmund spaces were defined as extension of Hölder spaces by boundedness properties of difference quotients. Within the systematic and unified approach of Triebel (cf. [13, 15]) we can simply identify the Zygmund spaces in a scale of inhomogeneous and homogeneous (quasi) Banach spaces, and (, ), by and . Both and are Banach spaces.
To emphasize the close relation with mollifiers we describe a characterization of Zygmund spaces in pseudodifferential operator style in more detail.
Let and choose , symmetric and positive, if , if , and strictly decreasing in the interval . Putting for then defines a function . Finally we set
and note that if then . We denote by the set of all pairs that are constructed as above (we usually suppress the dependence of on and in the notation).
We are now in aposition to state the characterization theorem for the inhomogeneous Zygmund spaces as subspaces of . It follows from [14], Sec. 2.3, Thm. 3 or, alternatively, from [3], Sec. 8.6. Note that all appearing pseudodifferential operators in the following have independent symbols and are thus given simply by convolutions.
Theorem 1.
Assume that and and choose arbitrary. Let then belongs to the inhomogeneous Zygmund space of order if and only if
(1) 
(Note that we made use of the modification for in [14], equ. (82).)
Remark 2.

defines an equivalent norm on . In fact that all norms defined as above by some are equivalent can be seen as in [3], Lemma 8.6.5.

If then is the classical Hölder space of regularity . Denoting by the greatest integer less than it consists of all times continuously differentiable functions such that is bounded when and globally Hölder continuous with exponent if .

Due to the term the norm is not homogeneous with respect to a scale change in the argument of .

In a similar way one can characterize the homogeneous Zygmund spaces as subspaces of modulo the polynomials . A proof can be found in [12], Sec. 3.1, Thm. 1. We may identify with the dual space of , the Schwartz functions with vanishing moments, by mapping the class with representative to . Assume that and and choose as constructed above and let and . Then belongs to the homogeneous Zygmund space of order if and only if
(3) (Note that we use the modification for in [12], equ. (16).)
2.2 The continuous wavelet transform
Following [2] we call a function with a wavelet. We shall say that it is a wavelet of order () if the moments up to order vanish, i.e., for .
The (continuous) wavelet transform is defined for () by ()
(4) 
where we have used the notation and . By Young’s inequality is in for all and defines a continuous operator on this space for each .
If we can define for directly by the same formula (4). If then can be extended to as the adjoint of the wavelet synthesis (cf. [2], Ch. 1, Sects. 24, 25, and 30) or directly by convolution in formula (4).
Remark 3.
If is a polynomial and it is easy to see that . In fact, , , and are in and . Since is in the Fourier transform is smooth and vanishes of infinite order at . But has to be a linear combination of derivatives of implying . Therefore the wavelet transform ‘is blind to polynomial parts’ of the analyzed function (or distribution) . In terms of geophysical modelling this means that a polynomially varying background medium is filtered out automatically.
2.3 Wavelets from mollifiers
The Zygmund class characterization in Theorem 1 (and remark 2,(v)) used asymptotic estimates of scaled smoothings of the distribution which resembles typical mollifier constructions in Colombeau theory. In this subsection we relate this in turn directly to the wavelet transform obtaining the wellknown wavelet characterization of Zygmund spaces.
Let with and define the function by
(5) 
Then is in and is a wavelet since a simple integration by parts shows that
and if we have if and only if . Therefore defined by (5) is a wavelet of order if and only if the mollifier has vanishing moments of order .
Furthermore, by straightforward computation, we have
(6) 
yielding an alternative of (5) in the form .
If arbitrary and , are the unique Schwartz functions such that and , then straightforward computation shows that and satisfy the relation (5). Therefore since is then a real valued and even wavelet we have for
Hence the distributions in the Zygmund class can be characterized in terms of a wavelet transform and a smoothing pseudodifferential operator by and . We have shown
Theorem 4.
Let . A distribution belongs to the Zygmund class if and only if
(7) 
Remark 5.

Observe that the condition on implies that and hence can never have compact support. If this characterization is to be used in a theory of Zygmund regularity detection within Colombeau algebras one has to allow for mollifiers of this kind in the corresponding embedding procedures. This is the issue of the following subsection. Nevertheless we note here that according to remarks in [5, (2.2) and (3.1)] and, more precisely, in [9, Ch.3] the restrictions on the wavelet itself in a characterization of type (7) may be considerably relaxed — depending on the generality one wishes to allow for the analyzed distribution . However, in case and a function a flexible and direct characterization (due to Holschneider and Tchamitchian) can be found in [1], Sect. 2.9, or [2], Sect. 4.2.
3 Colombeau modelling and wavelet transform
3.1 Embedding of temperate distributions
We consider a variant of the Colombeau embedding that was discussed in [4], subsect. 3.2. As indicated in remark 5,(i) we need to allow for mollifiers with noncompact support in order to gain the flexibility of using wavelettype arguments for the extraction of regularity properties from asymptotic estimates. On the side of the embedded distributions this forces us to restrict to , a space still large enough for the geophysically motivated coefficients in model PDEs.
Recall ([4], Def. 11) that an admissible scaling is defined to be a continuous function such that , , and if (fixed) as .
Definition 6.
Let be an admissible scaling, with , then we define by
(8) 
where
(9) 
is welldefined since is clearly moderate and negligibility is preserved under this scaled convolution. By abuse of notation we will write for the standard representative of .
The following statements describe properties of such a modelling procedure resembling the original properties used by M. Oberguggenberger in [10], Prop.1.5, to ensure unique solvability of symmetric hyperbolic systems of PDEs (cf. [10, 7]). The definition of Colombeau functions of logarithmic and bounded type is given in [11], Def. 19.2, the variation used below is an obvious extension.
Proposition 7.

is linear, injective, and commutes with partial derivatives.

: .

If then is of type, i.e., there is such that for all there exist and :
(10) 
If then is of bounded type and its first order derivatives are of type.
Proof.
ad (i),(ii): Is clear from in as and the convolution formula.
ad (iii): Although this involves only marginal changes in the proof of [10], Prop. 1.5(i), we recall it here to make the presentation more selfcontained.
Let with () then with
where the expression within brackets on the r.h.s. is bounded by some constant , dependent on and only but independent of , as soon as with chosen appropriately (and dependent on , , , and ). Therefore the assertion is proved by putting and .
ad (iv): is proved by similar reasoning ∎
In particular, we can model a fairly large class of distributions as Colombeau functions of logarithmic growth (or logtype) thereby ensuring unique solvability of hyperbolic PDEs incorporating such as coefficients.
Corollary 8.

If then and

If for then is of type. In particular, there is an admissible scaling such that and all first order derivatives () are of logtype.
3.2 Wave front sets under the embedding
One of the most important properties of the embedding procedure introduced in [4] was its faithfulness with respect to the microlocal properties if ‘appropriately measured’ in terms of the set of regular Colombeau functions ([4], Def. 11). But there the proof of this microlocal invariance property heavily used the compact support property of the standard mollifier which is no longer true in the current situation. In this subsection we show how to extend the invariance result to the new embedding procedure defined above.
Theorem 9.
Let , an admissible scaling, and with then
(11) 
Proof.
The necessary changes in the proof of [4], Thm. 15, are minimal once we established the following
Lemma 10.
If and with then .
Proof.
Using the shorthand notation and we have
Hence we need to estimate terms of the form when . Let be a closed set satisfying and put . Since is a temperate distribution there is and such that
implies that each term in the sum on the righthand side can be estimated for arbitrary by
if varies in . Since we obtain
with a constant depending on , , , , and but still arbitrary. Choosing , for example, we conclude that has a uniform growth over all orders of derivatives. Hence it is a regular Colombeau function. ∎
Referring to the proof (and the notation) of [4], Thm. 15, we may now finish the proof of the theorem simply by carrying out the following slight changes in the two steps of that proof.
Ad step 1: Choose such that in a neighborhood of and write
The first term on the right can be estimated by the same methods as in [4] and the second term is regular by the lemma above.
∎
3.3 The modelling procedure and wavelet transforms
Simple waveletmollifier correspondences as in subsection 2.3 allow us to rewrite the Colombeau modelling procedure and hence prepare for the detection of original Zygmund regularity in terms of growth properties in the scaling parameters.
A first version describes directly but involves an additional nonhomogeneous term.
Lemma 11.
If has the properties and () then defines a wavelet of order and we have for any
(12) 
Proof.
A more direct mollifier wavelet correspondence is possible via derivatives of instead.
Lemma 12.
If with then for any with
(13) 
is a wavelet of order and for any we have
(14) 
Proof.
Let then which proves the first assertion. The second assertion follows from
with the shorthand notation . ∎
Both lemmas 11 and 12 may be used to translate (global) Zygmund regularity of the modeled (embedded) distribution via Thm. 7 into asymptotic growth properties with respect to the regularization parameter. To what extent this can be utilized to develop a faithful and completely intrinsic Zygmund regularity theory of Colombeau functions may be subject of future research.
4 Zygmund regularity of Colombeau functions: the onedimensional case
If we combine the basic ideas of the Zygmund class characterization in 2.3 with the simple observations in 3.3 we are naturally lead to define a corresponding regularity notion intrinsically in Colombeau algebras as follows.
Definition 13.
Let be an admissible scaling function and be a real number. A Colombeau function is said to be globally of Zygmund regularity if for all there is such that for all we can find positive constants and such that
(15) 
The set of all (globally) Zygmund regular Colombeau functions of order will be denoted by .
A detailed analysis of in arbitrary space dimensions and not necessarily positive regularity will appear elsewhere. Here, as an illustration, we briefly study the case and in some detail. Concerning applications to PDEs this would mean that we are allowing for media of typical fractal nature varying continuously in one space dimension. For example one may think of a coefficient function in to appear in the following ways.
Example 14.

Let be constant outside some interval and equal to a typical trajectory of Brownian motion in ; it is wellknown that with probability those trajectories are in whenever . This is proved, e.g., in [2], Sect. 4.4, elegantly by wavelet transform methods.

We refer to [16], Sect. V.3, for notions and notation in this example. Then similarly to the above one can set in , in and in let be Lebesgue’s singular function associated with a Cantortype set of order with (constant) dissection ratio . Then belongs to with . (The classical triadic Cantor set corresponds to the case and .)
We have already seen that the Colombeau embedding does not change the microlocal structure (i.e., the wave front set) of the original distribution. We will show now that also the refined Zygmund regularity information is accurately preserved. If we denote by the set of all times continuously differentiable functions with the derivatives up to order bounded. Note that is a strict superset of .
Theorem 15.
Let and . Define such that then we have
(16) 
In other words, in case we can precisely identify those Colombeau functions that arise from the Zygmund class of order within all embedded bounded continuous functions.
Proof.
We use the characterizations in [1], Thms. 2.9.1 and 2.9.2 and the remarks on p. 48 following those; choosing a smooth compactly supported wavelet of order we may therefore state the following^{1}^{1}1Note that we do not use the wavelet scaling convention adapted to spaces here: belongs to if and only if there is such that
(17) 
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