ESA Contract 13945/99 Technical management by R. Jehn, ESOC. September 1, 2000

Transcription

1 MEASUREMENTS OF SMALL-SIZE DEBRIS WITH BACKSCATTER OF RADIO WAVES WP 1: Definition ofa Concept to Detect Small Size Debris Huuskonen A., Lehtinen M., and Markkanen J. Sodankylä Geophysical Observatory, University ofoulu, Sodankylä, Finland ESA Contract 13945/99 Technical management by R. Jehn, ESOC September 1, 2000 EUROPEAN SPACE AGENCY CONTRACT REPORT The work described in this report was done under ESA contract. Responsibility for the contents resides in the author or organization that prepared it. i

2 Abstract The purpose ofthe contract is to study the detection ofsmallsize debris by the radars ofthe European Incoherent Scatter Facility (EISCAT). We have approached the problem firstly by using the radar equation to obtain an estimate ofthe smallest detectable debris with the radars. Secondly we have produced a model ofmoving debris and calculated how accurately the the distance, radial velocity and radial acceleration ofdebris can be determined. We have found that the EISCAT UHF radar, operating at 933 MHz, is best suited for debris detection ofall the EISCAT radars. A conservative estimate tells that, at 1000 km distance, the radar is able to detect debris ofsize of1.4 cm. At this size the distance is obtained with an accuracy ofabout 200 m and the velocity to an accuracy better than 1 ms 1. For larger diameters the accuracy is considerably better. The results are based on the assumption that the signal is coherent for 0.1 s. If the coherency is valid for a longer time, considerably smaller debris are seen, and the larger debris are measured to a better accuracy. An important finding ofthe study is that the phase coding and the phase changes have a great effect on the accuracy ofthe distance. We have also considered a system in which the EISCAT UHF antenna is equipped with a 5 GHz transmitter system. As the scattering cross section for small particles increases strongly with diminishing wavelengths, this kind ofa system would be even better, detecting debris of0.7 cm in size. ii

3 Contents 1 Summary 1 2 Analysis by the radar equation Radar equation for point targets Scattering cross section ofa perfectly conducting sphere Radars and simple estimates Analysis by statistical inversion Model Results as a function of diameter Results as a function of velocity and acceleration Effect ofthe phase flips iii

4 List of Figures 1 The scattering cross section divided by the geometrical cross section for a perfectly conducting sphere. The thick line follows first the Rayleigh approximation and then the optical approximation. The dashed line shows where the approximations give equal results Scattering cross section ofa perfectly conducting sphere Debris echo measured by the EISCAT Svalbard Radar and a model curve fitted to the data The transmitter envelope (env), receiver impulse response (p) and the effective pulseform (p env) for a simple code with sampling time indications Relative error ofthe amplitude and and absolute error ofthe range as the function of the sphere diameter. The upmost curve at the left end corresponds to the first radar in the legend etc Absolute error ofthe velocity and the acceleration as the function ofthe sphere diameter. The upmost curve corresponds to the first radar in the legend etc The inverse ofthe variance ofthe range measurement as a function ofthe number ofphaseflips. Results are shown with open circles, and a fitted line in red. The phase codes are shown on the right ofthe results List of Tables 1 Radar properties and estimates ofthe smallest detectable size at 1000 km The diameter in cm which corresponds to the indicated risk levels ofaccepting a false echo The absolute error ofamplitude A and the relative error of range R, velocity v and acceleration a for the diameters given on the left for the EISCAT UHF radar iv

5 1 Summary The purpose ofthis work is to study how space debris can be detected with the radars ofthe European Incoherent Scatter Facility (EISCAT). The radar system consist ofa tristatic UHF radar and a monostatic VHF radar near Tromsø, Norway and ofthe EISCAT Svalbard Radar near Longyearbyen at Svalbard. The tristatic UHF system has additional receiver stations at Sodankylä, Finland, and Kiruna, Sweden. We have approached the problem by two methods. In the first we have used the radar equation and a simple model ofthe debris which assumes a perfect spherical shape. This analysis gives an estimate for the smallest detectable size. The second, and more complete analysis, is based on a model ofa spherical debris described in terms ofits diameter, distance, radial velocity and radial acceleration. The error analysis by using the statistical inversion theory is able to give an estimate ofthe smallest detectable size, as well as the accuracy to which the distance, radial velocity and radial acceleration ofthe object is determined. Contrary to the radar equation method, the error analysis method is able to take the transmitter waveform details into account. We found that the EISCAT UHF radar is best suited for debris detection ofall the EISCAT radars. A conservative estimate tells that, at 1000 km distance, the radar is able to detect debris ofsize of1.4 cm. At this size the distance is obtained with an accuracy ofabout 200 m and the velocity to an accuracy better than 1 ms 1. For larger diameters the accuracy is considerably better. The results are based on the assumption that the signal is coherent for 0.1 s. If the coherency is valid for a longer time, considerably smaller debris are seen, and the larger debris are measured to a better accuracy. An important finding ofthe study is that the phase coding and the phase changes have a great effect on the accuracy ofthe distance. We have also considered a system in which the EISCAT UHF antenna is equipped with a 5 GHz transmitter system. As the scattering cross section for small particles increases strongly with diminishing wavelengths, this kind ofa system would be even better, detecting debris of0.7 cm in size. 1

6 2 Analysis by the radar equation 2.1 Radar equation for point targets The power observed at the radar receiver is determined by the radar equation, which for point targets at the antenna optical axis has the following simple form P r = P tg 4πR 2 σ A e 4πR 2 (1) where P r is the power at the receiver input, P t is the transmitter power, G is the transmitter antenna gain, R is the range to the scattering point, and A e is the effective area ofthe antenna. All three factors in the radar equation have a physical interpretation: 1. The power density at the scattering point. The transmitter power is concentrated by the antenna gain, which tells how efficiently the antenna directs the transmitter energy. The power density diminishes with the usual range square law. 2. The scattering cross section tells how efficiently the object scatters the radio wave. It is a function ofthe particle size, orientation, and the direction between the incident and the scattered wave. 3. The receiver antenna collects the scattered wave with an effective area A e. The range square law is again employed. The effective area is related to the antenna gain G by A e = Gλ2 4π, (2) where λ is the wavelength. The two equations combined give the radar equation in a form which is also valid for directions off the optical axis, if the gain G is thought as a function of the directional angles: P r = P tg 2 λ 2 σ (4π) 3 R 4. (3) 2.2 Scattering cross section of a perfectly conducting sphere The radar scattering cross section depends on the form and the size of the particle in a complex manner. For the modelling purposes we assume that the debris particle is a sphere. We do this because the cross section for a sphere can be calculated from a simple formula, found in standard textbooks. The formula states that for small diameters d, i.e. small when compared to the wavelength, the scattering cross section is proportional to d 6, and for large diameters it is equal to the geometrical cross section ofthe sphere, 2

7 σ / π r d / λ Figure 1: The scattering cross section divided by the geometrical cross section for a perfectly conducting sphere. The thick line follows first the Rayleigh approximation and then the optical approximation. The dashed line shows where the approximations give equal results Cross section d / λ Figure 2: Scattering cross section ofa perfectly conducting sphere. 3

8 1 4 πd2. The first region is called to Rayleigh region, and the latter the optics region. In between there is the resonance or Mie region, in which the scattering cross section shows an oscillatory behaviour. A simple rules says that the Rayleigh approximation is valid ifthe diameter is less than one fifth of the wavelength and the optical approximation when the diameter is larger than 10 times the wavelength. The cross section is shown in Figures 1 and 2. The thick line in Figure 1 gives a simplifying approximation, in which the first part follows the Rayleigh approximation and the latter the optical approximation and the resonance region is neglected altogether. This is reasonable because in this study the radars operate mostly in the Rayleigh region. The high frequency radars operate in the resonance region, but there is no point in using the curve in full, as the true debris particles are not perfect spheres, and the result is approximative in any case. Hence the scattering cross section in the model calculations obeys the formula 1 σ 1 = 9(πd 4πd2 λ )4, d < λ π (4) 3 σ 1 = 1, d > λ 4 πd2 π (5) 3 The upper formula tells that the Rayleigh approximation is used whenever the diameter is less than roughly one fifth of the wavelength. A look at Figure 1 shows that the approximation is indeed valid for small diameters and the deviations from the curve from the full formula are very small. 2.3Radars and simple estimates Table 1 lists the three EISCAT radars under study, and three other radars included for comparative purposes. The radars are listed in the order of increasing frequency. The three other radars, denoted by Fgan, Luosto and UHF/5G in Table 1, are the Tracking and Imaging Radar ofthe Research Establishment for Applied Science (FGAN) in Germany, a weather radar ofthe Finnish Meteorological Institute under construction at Luosto, close to Sodankylä, and the third one is a radar system, which would combine the 32 m antenna ofthe EISACT UHF system at Sodankylä with a 3 kw transmitter to operate at about 5 GHz. The relevant properties ofthe radar transmitters, i.e. the frequency, the wavelength, the transmitter power, the gain ofthe antenna and the maximum duty cycle are given in the first block oftable 1. In the center block we calculate the received power at the receiver input fora3cmdebris at at the range of1000 km, based on Eq. 1. The intermediate steps include the incident power density at the scattering point, i.e. the first factor ofthe radar equation, and the calculation ofthe scattering cross 1 See e.g. Morchin, Radar Engineer s Sourcebook, Artech House, 1993, p.95 4

9 Table 1: Radar properties and estimates ofthe smallest detectable size at 1000 km Radar properties: VHF ESR UHF Fgan UHF5G Luosto Frequency, f[mhz] Wavelength, λ [m] Transmitter power, P t [MW] Antenna gain, G [db] Maximum duty cycle [%] Received power for a debris ofsize of3 cm at a distance of1000 km: Incident power density [mw/m 2 ] Scattering cross section, σ [mm 2 ] Received power, P r [10 18 W] in [dbm] Detectable size: Noise temperature, T [K] Noise power/mhz, [10 15 W] in [dbm] Samples in 0.1 seconds Detectable power, P r [10 18 W] in [dbm] Detectable size [cm] section. We can note that the first four radars are able to produce a higher incident power density than the others, whereas the scattering cross section is largest for the three last radars having the shortest wavelengths. Thus it is not surprising that the Fgan radar is able to get the highest received power ofall the six radars. The original and modified UHF systems come next, with some 6 db lower received power. Our aim is to find out an estimate for the smallest detectable debris. We define it here as being the limit where the received power is equal to the background noise power after integration. We assume that the debris signal is coherent for at least 0.1s and assume that the debris signal is integrated coherently, whereas the background noise adds up incoherently, as is always the case. This means that, for the debris signal, the amplitude after integrating N samples is N-fold, and for the background signal, it is the power which is N-fold. Thus the detectable debris size is one which produces the received power P r such that P r N 2 = k B TB W N,or P r = k B TB W /N, (6) where k B is the Boltzmann constant, T is the receiver noise temperature, 5

10 and B W is the receiver bandwidth. We assume a bandwidth of1 MHz, which is consistent with the sampling interval of1 µs. The number ofsamples N is twice that given in the table, because a complex sample corresponds to two real samples. By using the scattering cross section from Eq. 5 and the radar equation in Eq. 1 one can calculate the detectable debris size. The result with some intermediate steps is shown in the lowest part oftable 1. The result is that the EISCAT UHF is the most efficient ones ofthe existing radars, and that the Fgan radar comes close by. The other radars are weaker for detection of small space debris particles. It appears that our special UHF/5G system, which combines a high frequency and low power transmitter with a large antenna would be by far the most efficient system. 3Analysis by statistical inversion 3.1 Model We specify the scattering object by five parameters A, φ o,r o,v o,a o, which denote the amplitude and phase ofthe signal, the range from the radar to the scatterer, the radial velocity ofthe scatterer, and the radial acceleration ofthe object at a suitably chosen reference time t o. The model signal is s(t) =Ab(t) e iφ(t). (7) where functions φ(t) and b(t) depends on the model parameters in a way to be described below. The formula is best understood in conjunction with Figure 3, which shows example data measured with the EISCAT Svalbard radar. The figure shows how two consecutive 660 µs pulses with a separation of4000 µs hit a debris, whose signature is seen clearly in the measured inphase and quadrature components ofthe signal, given in blue and green. On top ofthe debris signal, random fluctuations caused by the random thermal noise are seen. The red smooth line shows a best fit model curve to one of the components. Here A is simply the amplitude ofthe fitted curve, φ(t) its phase as a function of time, and b(t) is needed to tell ifthe debris is scattering the radio wave or not. The phase factor is very simple. Because of the Doppler effect, the frequency of the radio wave changes when it is scatters from a moving debris. This can also be interpreted as a phase change which depends on the distance R(t) R o travelled by scatterer given by R(t) R o = v o (t t o )+ 1 2 a o(t t o ) 2. (8) In terms ofthe distance the phase as a function oftime is obtained from the formula φ(t) =φ o + 4π λ (R(t) R o). (9) 6

11 Figure 3: Debris echo measured by the EISCAT Svalbard Radar and a model curve fitted to the data The second step is to find the value ofthe envelope ofthe transmitted wave at each sample time. The key function to use is penv = p env, the convolution ofthe receiver impulse response with the envelope ofthe transmitted waveform. This is often called the effective pulseform, and it gives the combined effect ofthe transmitter and the filtering in the receiver. For stationary debris we would sample the effective pulseform at constant intervals corresponding to the sample interval. As the scatterer moves we have to correct for the motion of the debris as follows: τ =(t t o ) (R(t) R o). (10) c/2 Time values t specify the length of the time window to consider. In our calculations it is oflength of0.1 s and the time sampling interval is 1 µs. The amplitude of the signal can now be picked from the effective pulse form b(t) =(p env)(τ) (11) Figure 4 visualizes the meaning ofeq. 11. Starting from the bottom, we see the envelope ofa phase coded pulse, in which one ofthe bauds has a different phase from the others. The impulse response of the receiver is 7

12 p * env p env Increasing Xmission/sampling time Figure 4: The transmitter envelope (env), receiver impulse response (p) and the effective pulseform (p env) for a simple code with sampling time indications. one unit long. The effective pulse form is the convolution of the envelope with the impulse response. The triangles in red (pointing up) show how the pulse would sample a debris object with zero radial velocity. The sampling is done at one time unit intervals and the transmitter pulse advances by one time unit between each sample. The triangles in green (pointing down) show what happens ifthe debris object has a non-zero radial velocity away from the radar. The effect is exaggerated here to be visual. As the scatterer is moving away, the successive samples, which are again taken at one unit intervals at the receiver, correspond to slightly smaller time increments in the effective pulseform. The movement of the object partly compensates the movement ofthe transmitted waveform. The error covariance matrix Σ p ofthe model parameters is calculated in the standard manner from the equation Σ p =(D T Σ 1 m D) 1 (12) where D is a matrix formed by the partial derivatives of the model function in Eq. 7 with respect to the model parameters, D T is the transpose of D, and Σ m is the error covariance matrix ofthe measurements. In our case the measurements are independent, and hence Σ m is diagonal. Moreover, all the measurements have equal error and thus the formula simplifies to Σ p = σ 2 (D T D) 1 (13) 8

13 where σ is the standard deviation ofthe measurements. The fluctuations in the measurements are caused by the white noise component, which includes the sky noise and the noise originating from the receiving system. These are characterized by the noise temperature, which depends on the radar and may also depend on the antenna pointing direction. Representative values of the noise temperature and ofthe noise power in the assumed 1 MHz receiver bandwidth are given in Table 1. The noise power divides evenly into the in-phase and quadrature components ofthe signal and thus the variance of a measured point is σ 2 = k B TB w /2. (14) The errors ofthe model parameters are obtained as square roots ofthe diagonal elements ofσ p. Amplitude, relative error Legend: VHF ESR UHF UHF/5G Range, error [km] Diameter [cm] 10 1 Legend: VHF ESR 10 0 UHF UHF/5G Diameter [cm] Figure 5: Relative error ofthe amplitude and and absolute error ofthe range as the function of the sphere diameter. The upmost curve at the left end corresponds to the first radar in the legend etc. 9

14 Velocity, error [m/s] Legend: VHF ESR UHF UHF/5G Diameter [cm] Acceleration, error [m/s 2 ] Legend: VHF ESR UHF UHF/5G Diameter [cm] Figure 6: Absolute error ofthe velocity and the acceleration as the function ofthe sphere diameter. The upmost curve corresponds to the first radar in the legend etc. 3.2 Results as a function of diameter Figures 5 and 6 show the relative error ofthe model parameters as the function of the sphere diameter for four radars, the EISCAT VHF, UHF and ESR radars, and for our design system, where the EISCAT UHF antenna is equipped with a 5 GHz transmitter (labelled as UHF/5G). Two radars shown in Table 1 are omitted from the model calculations. These are the Luosto weather radar, because its performance is clearly inferior to the others, and the Fgan radar, because it is very close to the EISCAT UHF in its properties. Most ofthe experiments with the EISCAT radars use phase coded pulses, in which the transmitter phase is flipped at selected intervals. The error estimation model shows that the accuracy ofthe amplitude, velocity and the acceleration ofthe debris is not affected by the phase coding. That is, 10

15 Table 2: The diameter in cm which corresponds to the indicated risk levels ofaccepting a false echo. std and risk level 1-σ 2-σ 3-σ 4-σ radars <5% <0.5% <0.01 VHF ESR UHF UHF/5G the phase coding does not increase the accuracy. The range determination, on the other hand, benefits very much ifthe phase is flipped, or gaps added to the transmission, because the edges are what give the accuracy for the range determination. Hence all results will be given for phase coded pulses only. What is the detectable size then? A possible criterium is to use the diameter where the probable error ofthe amplitude is 100 %. Accepting this as the detectable size assumes quite a high risk offalse alarms. The risk is made smaller if2-σ, 3-σ or even 4-σ error levels are used as guidelines. These correspond to the diameter which gives 2, 3 or 4 times higher amplitude for the signal. The diameters and the corresponding risk levels ofseeing a false echo are given in Table 2. Ifwe take as an example the 0.5 % risk level, we note that the random thermal noise in the data would produce an echo of the given size once in every 1/0.005 = 200 cases, which means that a false alarm is expected once in every 20 s. This is a lower rate than the occurrence rate oftrue space debris echoes. A limited study ofmeasurements made in November 1999 with the EISCAT Svalbard radars showed that clear debris echoes can be seen as often as once in seconds. Also we can reject many ofthe false echoes by studying adjacent 0.1 second intervals, because a true debris echo should be seen in many adjacent intervals. In this way even a higher raw false echo rate is acceptable, and the detection level may be taken from the 2-σ or 5% column. For the UHF radar it would be 1.43 cm. The corresponding sizes for the other radars are seen in Table 2. The errors for the other parameters at the chosen detection level can be read from the figures. Table 3 gives the errors for some selected diameters for the UHF radar. The diameter 1.4 cm, which is close to the 2-σ level, corresponds to an error of260 m for the range, 0.48 ms 1 for the velocity, and 37 ms 2 for the acceleration. Ofthese, the error ofthe velocity is without doubt sufficiently small and we can conclude that the velocity can be determined with a sufficient accuracy. The error for the acceleration causes on uncertainty of about ±2 11

16 ms 1 in the velocity within a 0.1 s detection window. Table 3: The absolute error ofamplitude A and the relative error ofrange R, velocity v and acceleration a for the diameters given on the left for the EISCAT UHF radar. diameter A R v a [cm] [m] [ms 1 ] [ms 2 ] Results as a function of velocity and acceleration The value ofthe radial velocity has a small effect on the error. Within the range ms 1, the error ofany ofthe model parameters varies by less than a factor of 0.05 (5 %) in maximum. Thus we may neglect the velocity in the model studies, and present the results for a single velocity value, which is 1000 ms 1. The values ofthe acceleration within the range ms 2 has a small effect on the results, but the effect is even smaller than that ofthe velocity. Thus all results shown are valid for any reasonable value of the radial acceleration the debris object may have. 3.4 Effect of the phase flips The results for range in Figure 5 were given for a phase coded pulse, because the phase coding increases the accuracy ofthe determination ofthe range. The effect is shown in detail in Figure 7 which shows the inverse ofthe variance ofthe measurements as a function ofphase flips. The number of phase flips is shown in the x-axis, and is also shown by from the transmitter pulse envelopes reproduced in the Figure. We can see in Figure 7 that the inverse variance increases linearly as the function ofthe number ofphase flips in the code. The best fit line tells that the increase is proportional to 1.9 times the number ofphase flips. This is result is very useful. We know that the inverse ofthe variance ofthe range determination increases linearly as the number ofphase flips increases, but that the other parameters are not affected by this. For instance, the mean number ofphase flips in the codes used for calculating the results in Figure 5 was slightly less than 12. As the maximum number ofphase flips would be 24 with the assumptions used, 12

17 Inverse variance Number of phaseflips Figure 7: The inverse ofthe variance ofthe range measurement as a function ofthe number ofphaseflips. Results are shown with open circles, and a fitted line in red. The phase codes are shown on the right ofthe results. the inverse variance could be doubled, leading to a 2 decrease in the error ofthe range determination. This, however, would make the code useless for any other purpose than the detection ofthe space debris. 13