1

FRACTAL APPROACH OF THE IONOSPHERIC CHANNEL SCATTERING FUNCTION C. GOUTELARD, P. BRAULT LETTI-UNIVERSITÉ PARIS-SUD, ORSAY, FRANCE ___________

Abstract The applicability of fractal measurements to the ionospheric scattering function considered as a fractal object has already been demonstrated. Nevertheless in the high resolution method, the chaotic behavior of the object analyzed was deliberately restricted to the two dimensions of propagation delay and Doppler shift. Good results were obtained but we also realized that an important part of the information was missing in the 2-space procedure. A third dimension embodied in the scattering function, seen as a fractal, is the amplitude of the backscattered echo. Considering also the fact that a coordinate in the time-frequency domain can only have one corresponding amplitude, our fractal must be defined as an object of topological dimension 2 embedded in 3-space. With smooth variations, this fractal would be poetically called “a sheet” by some mathematicians.

I. – Generalities -

scattering function. The observation of the chaotic nature of the ionospheric channel rationally guided us to the application of a fractal approach in order to measure incoherent effects of the medium on the scattering function. This concept which has been presented in 1993 [3,] has been developed towards a tentative generalization. In this article we also relate of the specific problems we met in this generalization.

II. - Structure of the scattering function The study presented here relies on experimental measurements of the scattering function obtained by a backscattering method. This method is based on radar techniques where the ground acts as a target for the wave travelling through the ionosphere. It enables the measurement of ionosphere characteristics on a wide spatial range. The figure 1 displays a scattering function, obtained by backscattering, with its coherent part, constituting the skeleton of the function, and its incoherent part.

The propagation of electromagnetic waves in the ionosphere is made complex by the fact that this medium is inhomogeneous, anisotropic, turbulent , non-stationary and multipath. Usually, the ionospheric models lead to ray tracing computations which, in best cases, provide an acceptable representation of the behavior of the coherent part by considering only large and medium scale spatial variations as well as temporal variations and anisotropy. On the other way the turbulent part, mostly due to ionospheric plasma instabilities, is not taken into account in these representations. Theoretical models have been proposed [1,2] and give a statistical representation of the turbulent part of the medium as well as a solution by means of a “full wave” method, while geometrical optics methods become inadequate. Though obtained results are not representative in all the encountered cases, they provide encouraging representations to describe this turbulent part which introduces, in the scattering function, a non-coherent, random component whose effect is of capital consequence in telecommunication and teledetection systems. This article proposes a complementary aspect to this approach. Our first concern was the need to introduce the concept of measurement of the incoherent part of the

Figure 1 . Structure of a scattering function (with Bragg diffraction). “____” and “- - - -“ : theoretical structure (coherent part / ray tracing method) The coherent part illustrates the channel non-stationarity, the anisotropy and the Bragg diffraction on the sea and is in excellent agreement with the geometrical theory materialized, on this recording, by the dotted line. The non coherent part appears out of this lines. The non-coherent part, characterizing the chaotic behavior of the propagation medium, can be considered as the set of signals stretching out around a skeleton.

2 The observation of the scattering function provides the specialist with the global situation of the ionospheric medium. In this way one is able to qualify the ionospheric state of calm, perturbed or very perturbed. Our purpose, in introducing the fractal method, is to initiate a quantization of the ionospheric state leading to the notion of measurement.

III. - First approach of the fractal measurement III.1. - Advantage of the high resolution analysis The first approach of the fractal measurement was to simplify the representation of the scattering function by means of a high resolution analysis. This was made in order to extract a simpler structure in the form of the main components in the space time-Doppler, of the function constituting the image skeleton. This first work [3] has been conducted in a simplification purpose. It eliminates an important part of the scattering function, the amplitude, but reduces the object to be analyzed to a 2-space representation. Furthermore it also reduces, a priori, a part of the noise. The figure 2 displays the result of the analysis made on two scattering functions recorded the 17 January 1992 at 11h01 and 11h02 TU+1. One can notice that, though very close, the component traces evolve significantly between these two close instants. These results, found in all analyses, show the fidelity of the method and the important rate of variation of small scale instabilities.

means of (BCM).

the well-known Box-Counting Method

III.2. - The BCM algorithm The Box-Counting Method is a procedure of computation of the fractal dimension based on Bouligand-Minkowski’s definition. The analysis, made on the main components of the scattering function obtained by a high resolution method, has brought the following fractal dimensions : - for the E region, a dimension close to 0.9 - for the F region, a dimension close to 0.65. This first approach essentially applies to the scattering function which, submitted to perturbations, divides into parts. For a non-perturbed skeleton the dimension equals one and this dimension decreases as much as the perturbation increases.

IV. - Second approach of the fractal measurement The hopeful results demonstrated before inclined us to consider the non-coherent part of the scattering function in a more meaningful way. Our second approach of the fractal measurement consisted in keeping, on the high resolution 2D representation, all the components likely to contain a significant signal, and in computing the fractal dimensions on the different regions characterizing the perturbed and the non-perturbed zones of the scattering function. The figure 3 displays the high resolution analysis performed on the F region in the presence of a plasma instability visible around focusing (locus). This difference clearly appears in the measurement which gives : - a fractal dimension of 1.62 for the nonperturbed region - a fractal dimension of 1.03 for the perturbed region.

Figure 2. High resolution analysis of two scattering functions skeletons. Fractal dimensions : E region : 0.75 to 0.95, F region : 0.55 to 0.95 Obtained objects are obviously more simple than the initial scattering functions because the amplitude of their components is not taken into account. This moves back the object into a 2D space, so an important part of the noise is filtered out of the recording. We then have assimilated the backscattering echoes to fractal objects and have measured their dimension by

This analysis demonstrates the potential capability of the fractal approach in the detection and the measurement of perturbations and of the ionosphere global state. It replaces by quantitative data qualifiers such as “calm” or “perturbed” commonly used by the ionospherists. Nevertheless, it is judicious to notice that fractal dimensions vary accordingly to the type of object extracted from the scattering function. One observes that the nature of the selected components obviously modifies the fractal dimension. This result sets the problem of the choice of the object, which has been reduced to a 2D model for simplification reasons.

3 pulled downwards to the true topological dimension of the “digital” object, which is 0, the dimension of a point or pixel. In order for the BCM to see the object as a plane or a portion of plane, we tend to keep to the object a size of at least 30 x 30 pixels. • Second is the range of box sizes used to perform the computation of the fractal dimension. The experience shows that in order to avoid the asymptotic behavior for large and small boxes, one has to take, for the largest box, half the minimum size of the object in any of the 2 directions of the coordinate system propagation-Doppler. We also take not less than twice the resolution, which is one pixel. In this way good results have been observed on simple objects like an inclined plan, as well as on famous deterministic fractal patterns like the Sierpinski Gasket and the Von Koch curve. Figure 3 . Computation of the fractal dimension on the high resolution component by means of the BCM (BoxCounting Method). Case of a -perturbed ionosphere. Top figure : High resolution analysis Bottom figure : Fractal dimension Complete function : Df = 1.61 Non-perturbed part : Df = 1.67 Perturbed part : Df = 1.03 To this point of achievement we will consider that the scattering function is in fact a 3-space object and the ultimate step will consist in a computation of the fractal dimension on the whole function.

V. - Third approach (3D) of the fractal measurement The 3D approach, in which the amplitude of the components is now taken into account, has been initiated recently [4]. The analysis of the complete scattering function considered as a fractal object is obviously more complex in the 3-space than in the 2-space procedure. The accuracy of a simple 3D algorithm was first probed on deterministic fractals as test figures. A second important part of the program was to establish measurement criteria particular to the digital object studied and to its size, and to filter the background noise. Another point was to elucidate the behavior of the fractal dimension on different portions of the scattering function. We finally were able to bring new results on the metrology of the scattering function and of the ionosphere. V.1. - Measurement criteria In order to achieve a correct measurement, some critical measurement criteria have first to be clearly recognized and explained. • First is the distortion due to discretization, which is particularly perceptible when the size of the object analyzed is too small. For example, whenever the area of the echo went down to less than approximately 20 x 20 pixels, the dimension was

• Third is the centering of the echo. This one does not always have a perfect contour and has sometimes been partially truncated, especially in the case of incoherent F region. Care has also to be taken when delimiting the echo with a rectangular frame: the size of this frame in both x and y directions has to be a multiple of all the gauges, or boxe sizes, taken, in order to prevent any “side effect” that would affect the validity of the NP count and the regularity of the Log(NP)/Log(1/ε) curve. V.2. - Improvements in the metrology (synthetic image) To palliate to the distortion measurement on echoes of small size, a “synthetic” image was created with four copies of the echo put side by side. The result was an increase in the fractal dimension of about 5 to 10 %. This is a good proof of the distortion by discretization. V.3. - Dimension measurements on distinct parts of the scattering function According to the principle which would dictate that the fractal dimension is always greater than the topological dimension, the fractal dimension of our echo should always be comprised between 2 and 3. A first measurement made on the backplane outside the visible echoes of the scattering function, exhibits a dimension of 2.45. This, in fact, gives an evaluation of the dimension of the residual gaussian noise. Its high fractal dimension is characteristic of a very chaotic process. In comparison, a “raw” measurement on any of the echoes would give a fractal dimension between 1.9 and 2.2. This means that the information embedded in the echo is more “stable” than in the noisy background; but its true dimension is raised by influence of the high dimension of the noise superposed to the echo. V.4. - Background noise removal Our approach to obtain a real absolute value of the fractal dimension of an echo has been first to truncate

4 the amplitude of each echo under a threshold equivalent to the maximum amplitude of the residual noise all over the scattering function (fig. 4). We cut the “grass” [V. Wickerhauser]. Nevertheless, the annealing of the residual noise by truncature at basis also means the annealing of portions of the echo which have a smaller amplitude than the threshold (we also do not take into account another difficulty which is to remove the noise from the echo itself). The remnant problem is that the topological dimension of our object is lowered and can now be found between 0 and 2, because, seen in the Propagation-Doppler plane, the “sheet” of our scattering function may now be strewn with holes. Any echo can become made of several “patches” with nothing in between. The topological dimension of our noise filtered function is now obviously lower than 2. The function studied is not any more “dense” in the plane, and its fractal dimension decreases at the same time. To solve this problem, we attempted to preserve to the echo its topological dimension. To achieve it, any part of the echo that was “zeroed” in noise filtering would be replaced by a surface of uniform low amplitude of 1 pixel. But by doing so, we can observe that the fractal dimension is masked by the dimension of the plane added, and “pulled” towards 2. This is the same effect as the discretizing effect of the object (see above) where the real dimension of a too small object is perturbed by the dimension of its components, the pixels. The final idea is, of course, to use a more appropriate filtering that will only remove the residual, continuous spectrum gaussian noise. (Work has been initiated in this way by means of the wavelet transform).

of the dimension on the F region, while this one was subject to a masking phenomenon by a sporadic Es perturbation during the twenty minutes of recording (fig. 5). F E Es

Figure 5: Es travelling perturbation scheme (June 97) At start of recording the Es can be observed on the left part of the function (fig. 6 top). The dimensions computed on both E and F echoes were quite identical, at approximately 1.45. During twenty minutes, the high concentration Es perturbation evolved in such a way that it masked the emission beam that was reaching the F region (fig. 6 bottom). A filtering process using a smoothing 2D wavelet transform, associated with an amplitude contour plot, could display, at the end of the recording period, a very fragmented F region. The fractal dimension measured on the intact E region still was about 1.45, but the dimension of the F region had declined down to 1.1. This is a first application of the fractal metrology to perturbed transmissions and it shows that visible phenomena can be quantized. F IF N L 3 8 K .D O S lv l= 2

F d = + 2 ,5 6 H z 250

200

150

Fd= 0 H z 100

50

F d = -2 ,5 6 H z 50

T p= 0 k m

100

1000 km

150

200

250

3000 km

2000 km

F IF N L 5 9 K .D O S lv l= 2

F d= + 2 ,5 6 H z 250

200

150 F d= 0 H z 100

Figure 4: Noise truncature at basis on an F region.

50 F d= -2 ,5 6 H z

V.5. - Application to the quantification of evolution of a perturbation A recording made in June 97 at STUDIO, the ionosphere sounding facility of the LETTI near Paris, has been the object of an other interesting result. A fractal metrology performed on both the E and F regions of the echo was able to highlight the evolution

50 T p= 0km

100 1000 km

150 2000 km

200

250 3000 km

Figure 6 : Evolution of the scattering function in the scenario of an Es sporadic perturbation. Top : The Es perturbation echo is visible on the left (approx. 600 km). Bottom : Effect of the Es masking the beam towards the F region.

5 V.6. - Absolute values obtained for the E and F regions The application of a our former investigations on the metrology of the E and F regions finally brought two new results. Based on our criteria : minimum size of a numerical object, choice of framing for the different echoes, choice of box sizes, noise filtering, addition or not of extraneous objects (constant amplitude surfaces), identical topological dimensions, the dimension ∆f(E) found on E echoes has in general been found between 1.4 to 1.6. On F echoes the dimension has been found up to ∆f(F)= 2.05. For fractal objects of the same topological dimension, it proves that the F region can exhibit a much more chaotic nature. On the opposite, and if the echo (F in our example) is very fragmented due to a masking phenomenon by a sporadic Es region, the resulting decrease of its topological dimension (the echo is not anymore a surface) gives rise to a decrease of its fractal dimension. In this case ∆f(F) has been measured at 1.1.

VI. – Conclusion The fractal approach appears to be a viable solution to the quantization of the ionospheric state from a scattering function. The difficulty to interpret a measurement made on an object simplified by a 3 to 2-space projection tends, a priori, to make this method less robust. By considering the complete scattering function with its three dimensions of amplitude, propagation and Doppler, we make the method more robust. Nevertheless this method requires some circumspection and induces a higher computation complexity. According to the rapidly increasing power of computers, this last remark does not seriously affects this promising method.

Bibliography [1]

C. GOUTELARD : Analyse des structures fines des échos de sol obtenus en sondage ionosphérique par la méthode de rétrodiffusion. Application à la détection et à l’étude des perturbations itinérantes dans l’ionosphère. Thèse d’état. Université de Paris, mars 1968.

[2]

N. ZERNOV, V. GHERM : Analytical and numerical techniques for the description of the effects of HF propagation in the ionosphere with random and deterministic inhomogeneities. Workshop COST251, Paris, 16-18 Oct. 1998.

[3]

C. GOUTELARD, L. BARTHES, J. CARATORI : Comportement temporel et spatial des trajets multiples dans les propagations ionosphériques en incidence oblique. Mesure fractale des perturbations. 53ème Symposium AGARD SPP on “Multiples mechanisms propagation paths (MMPPs)”. Rotterdam (NL), 4-8 Oct. 1993.

[4]

P. BRAULT : Fractal analysis of the random propagation channels in the ionosphere. CNAM engineer thesis, Jan. 1998.