The fractal character of turbulence degraded wave fronts C. Schwartz, G. Baum and E.N. Ribak Physics Department, Technion - Israel Institute of Technology, Haifa 32000 Israel ABSTRACT Atmospheric turbulence degraded wave fronts can be described as fractal surfaces from the Fractional Brownian motion family. Fractal character can be ascribed both to the spatial and temporal behavior. Implications of this description with regard to computer simulation and prediction are presented. 1. INTRODUCTION The phase of an atmospheric turbulence degraded wave front is a homogeneous and isotropic stochastic Gaussian process. This process is described well by a structure function which behaves as a power law over many scales of length in the so called "inertial range". This range lies between the inner scale which is of the order of a few millimeters and the outer scale which is of the order of the height above the ground. This structure function can be written as1: ro is the well known Fried length. This expression is obtained analytically using the Kolmogorov assumptions concerning the power spectrum of velocity and temperature fluctuations in a turbulent medium. The validity of this function (at least for the case of astronomical imaging) has been demonstrated in many observations2. From dimensional reasoning it turns out that the power spectrum of the wave fronts also has a power law dependence (which we shall call the Kolmogorov power spectrum) on the spatial frequency K: 2. FRACTIONAL BROWNIAN MOTION SURFACES The above relations resemble a mathematical entity called Fractional Brownian motion (FBm)3-4. A Brownian process B(L) (for example in one dimension) is a Gaussian stochastic process characterized by an incremental variance proportional to time: An extension is the FBm, Bh, for which: 476 / SPlE Vol. 1971 8th Mrcting on Oplical Engineering in Israel (1 992)
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The fractal character of turbulence degraded wave fronts
C. Schwartz, G . Baum and E .N. Ribak
Physics Department, Technion - Israel Institute of Technology, Haifa 32000 Israel
ABSTRACT
Atmospheric turbulence degraded wave fronts can be described as fractal surfaces from the Fractional Brownian motion family. Fractal character can be ascribed both to the spatial and temporal behavior. Implications of this description with regard to computer simulation and prediction are presented.
1. INTRODUCTION
The phase of an atmospheric turbulence degraded wave front is a homogeneous and isotropic stochastic Gaussian process. This process is described well by a structure function which behaves as a power law over many scales of length in the so called "inertial range". This range lies between the inner scale which is of the order of a few millimeters and the outer scale which is of the order of the height above the ground. This structure function can be written as1:
ro is the well known Fried length. This expression is obtained analytically using the Kolmogorov assumptions concerning the power spectrum of velocity and temperature fluctuations in a turbulent medium. The validity of this function (at least for the case of astronomical imaging) has been demonstrated in many observations2. From dimensional reasoning it turns out that the power spectrum of the wave fronts also has a power law dependence (which we shall call the Kolmogorov power spectrum) on the spatial frequency K:
2. FRACTIONAL BROWNIAN MOTION SURFACES
The above relations resemble a mathematical entity called Fractional Brownian motion (FBm)3-4. A Brownian process B(L) (for example in one dimension) is a Gaussian stochastic process characterized by an incremental variance proportional to time:
An extension is the FBm, Bh, for which:
476 / SPlE Vol. 1971 8th Mrct ing on Opl ical Engineering in Israel (1 992)
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