Computers & Geosciences 25 (1999) 963±978 The scale invariant generator technique for quantifying anisotropic scale invariance G.M. Lewis a, 1, S. Lovejoy a,*, D. Schertzer b, S. Pecknold a aDepartment of Physics, McGill University, 3600 University St., MontreÂal, Que., Canada H3A 2T8 bLMD (CNRS) Boite 99, Universite P.&M. Curie, 4 Pl. Jussieu, Paris 75252 Cedex 05, France Received 14 October 1997; received in revised form 25 June 1998; accepted 20 August 1998 Abstract Scale invariance is rapidly becoming a new paradigm for geophysics. However, little attention has been paid to the anisotropy that is invariably present in geophysical ®elds in the form of dierential strati®cation and rotation, texture and morphology. In order to account for scaling anisotropy, the formalism of generalized scale invariance (GSI) was developed. Until now there has existed only a single fairly ad hoc GSI analysis technique valid for studying dierential rotation. In this paper, we use a two-dimensional representation of the linear approximation to generalized scale invariance, to obtain a much improved technique for quantifying anisotropic scale invariance called the scale invariant generator technique (SIG). The accuracy of the technique is tested using anisotropic multifractal simulations and error estimates are provided for the geophysically relevant range of parameters. It is found that the technique yields reasonable estimates for simulations with a diversity of anisotropic and statistical characteristics. The scale invariant generator technique can pro®tably be applied to the scale invariant study of vertical/horizontal and space/time cross-sections of geophysical ®elds as well as to the study of the texture/morphology of ®elds. # 1999 Elsevier Science Ltd. All rights reserved. Keywords: Anisotropy; Scale invariance; Analysis technique; Texture; Multifractal 1. Introduction pic assumptions and models, geophysical ®elds are gen- erally highly anisotropic. For example, in the The use of scale invariance in the study of geophy- atmosphere, ocean and earth interior, they are dieren- sics Ð even if only implicitly in the form of fractal tially strati®ed due to gravity. Furthermore, clouds, geometry Ð is becoming widespread. Unfortunately, ridges in sea ice, fault planes in earthquakes and existing scale invariant models and analysis techniques mountain ranges in topography Ð to name a few Ð (whether mono or multifractal) usually assume self- all have scale-dependent preferred directions arising from the Coriolis force, external stresses, or other similarity (hence isotropy). In contrast to these isotro- boundary conditions which lead to dierential ro- tation. The full scope of the scale-invariant symmetry principle has therefore been drastically underestimated; * Corresponding author. Fax: +1-514-398-8434. E-mail address: [email protected] (S. Lovejoy) perhaps an extreme example being the atmosphere, 1 Present address. Department of Mathematics, University where the use of outmoded isotropic scaling notions of British Columbia, Vancouver, BC, Canada. has lead to the prediction of a ``meso-scale gap'' 0098-3004/99/$ - see front matter # 1999 Elsevier Science Ltd. All rights reserved. PII: S0098-3004(99)00061-8 964 G.M. Lewis et al. / Computers & Geosciences 25 (1999) 963±978 between the large and small scales which is not compa- type of anisotropy (e.g. self-anity); quite general tible with modern scaling analyses or theories (see forms of scale invariance must be considered. In sev- Lovejoy et al., 1993). eral papers, Schertzer and Lovejoy (1983, 1984, 1985, Scale invariance is a symmetry respected by systems 1987, 1989, 1991) present a formalism called whose large and small scales are related by a scale Generalized Scale Invariance (GSI), which de®nes the changing operation involving only the scale ratio: they notion of scale in anisotropic scaling systems. The have no characteristic size. As with the familiar sym- physical motivation for GSI is that the dynamics metries associated with energy and momentum conser- should determine the appropriate notion of scale; it vation, the scaling symmetry must a priori be assumed should not be imposed from without. to hold. Only when a speci®c symmetry breaking Other researchers have seen the need for studying mechanism can be shown to exist should one invoke the anisotropy of geophysical ®elds. Fox and Hayes the existence of fundamental length scales. While it is (1985), VanZandt et al. (1990), and Pilkington and true that in self-similar scaling small structures look Todoeschuk (1993) independently proposed the intro- the same as large ones, this is no longer true in aniso- duction of dierent scaling exponents in dierent direc- tropic scaling, hence phenomenological classi®cations tions. Although at ®rst sight this is appealing, it turns of structures can be quite misleading. Although many out to be quite incompatible with a scaling generation geophysical ®elds exhibit no symmetry-breaking mech- of anisotropy. For the mechanism to be independent anism over a wide range of scales, for purely phenom- of the absolute scale, and to depend only on the scale enological reasons (i.e. phenomena look dierent at ratio (relative scale), the scale changes must satisfy dierent scales) a scale bound/nonscaling approach is group properties (see Section 2). Since this is not the often adopted; e.g. a phenomenological school exists in case for this approach, any underlying dynamics will meteorology which hypothesizes the existence of dier- be fundamentally dependent, rather than independent, ent dynamical mechanisms every factor of two or so in of size. scale Ð in spite of the fact that the underlying Recently, starting in the turbulence literature, an (Navier±Stokes) equations are scaling between a small idea closely related to GSI has been in vogue: extended viscous scale (of the order of a mm) and a large scale self-similarity (ESS). GSI uses an anisotropic notion of of planetary dimensions. Furthermore, since they are scale which is physically determined by the dynamics also scaling, boundary conditions (e.g. topography, (especially the dierential rotation and strati®cation of Lovejoy and Schertzer, 1990; Lavalle e et al., 1993) will structures). Similarly, ESS uses a scale de®ned by the not break the scaling. statistics (in turbulence, the third order velocity struc- Not only are the usual theoretical scaling notions ture function), in this case, the idea is to (somewhat) isotropic, so are the corresponding data analysis tech- take into account the dissipation which destroys the niques. For example, a common tool used to study scaling with respect to the usual scale notions. In both geophysical scale invariance is the isotropic energy cases, rather than using an academic (Eulerian) scale, spectrum, E(k ), (where k=|k| and k is the wave num- one attempts to let them be determined by the ber), which is obtained by angle integrating (i.e. inte- dynamics. See Schertzer et al. (1997) for further discus- grating out) the angular energy density. If a ®eld is sion of the relation of ESS to GSI. isotropic and scaling, then E(k )0k b, where b is the To date, the only data analysis technique which can scale invariant spectral exponent. Therefore, the aniso- handle both dierential rotation and strati®cation is tropy of a scaling ®eld (if it is not extreme) may be the Monte-Carlo dierential rotation method (P¯ug et ``washed-out'' by the smoothing eect of the inte- al., 1991 and 1993; Lovejoy et al., 1992; see also gration. Thus, a power-law isotropic energy spectrum Lovejoy et al. (1987) for a discussion of the anisotropic can indicate approximate scaling without implying iso- ``elliptical-dimensional sampling technique''). However, tropy (see Lovejoy et al., 1993, for a discussion of this this technique has many problems (discussed later) and in cloud radiances). The same is true of box-counting the new scale invariant generator (SIG) technique out- and other fractal or multifractal analysis techniques lined here is a considerable improvement. which use similarly shaped boxes or circles at dierent SIG quanti®es anisotropic scale invariance by esti- scales, hence (implicitly) isotropic scale changes. On mating the GSI parameters. It was developed to study the other hand, an apparent break in the scaling of an the scaling of the spectral energy density in geophysical isotropic spectrum or other statistic may be spurious; phenomena (yet with little eort it can be more gener- it may simply imply anisotropic scaling. Similarly, spec- ally applied). The scaling is a statistical property, tra of one-dimensional cross sections of anisotropic therefore large data sets (large ranges of scales) are processes may also show spurious breaks in the scal- needed to ensure accurate results. For this purpose, ing. satellite images of geophysical phenomena are often Although geophysical ®elds are a priori scale invar- studied. Because of these large data sets, a major chal- iant, there is usually no reason to assume a speci®c lenge of developing an analysis technique is to make it G.M. Lewis et al. / Computers & Geosciences 25 (1999) 963±978 965 way that a characteristic size is not introduced to the system. Usual (isotropic) scale invariant systems use the length of vectors to determine their scale and there- fore vectors of equal length are of the same scale. Vectors of dierent scales are related by a magni®- cation (scaling) factor which depends on the ratio of the lengths of the vectors. See top left of Fig. 1. Generalized scale invariance (GSI) moves away from this restrictive case, allowing more general concepts of scale and scale changing operations. GSI is a formal- ism which states the most general conditions under which a system can be scale invariant. A GSI system requires three elements: 1. The unit ball, B1, which de®nes the unit vectors. In general, B1 will be de®ned by an implicit equation: B1 fxj f1 x < 1g; @B1 fxj f1 x1g 1 where @B1 is the ``frontier of the unit ball'', and f1 (a function of position, x; bold will denote vector quantities) is the ``scale function''.