We present an analysis of the spectral characteristics of 1-hop HF radar ground scatter and - and 1-hop ionospheric scatter as measured by the Super Dual Auroral Radar Network. Our objective is to determine criteria that separate signals scattered from the ground and the ionosphere. We find that for both ground scatter and ionospheric scatter that the probability density function of backscatter Doppler velocity decreases exponentially with velocity, but with significantly different e-folding velocities for the two types of backscatter. We use this observation to separate the total probability density of Doppler velocity and spectral width into two component distributions. This process yields the posterior probability that a signal of given Doppler velocity and spectral width is ground scatter. The resulting criterion for classification of a particular signal as ground scatter, v < 33.1 m/s + 0.139w − (0.00133 s/m)w2, significantly reduces the probability that a signal will be erroneously classified as ionospheric scatter, while only moderately increasing the probability that an ionospheric scatter signal will be erroneously classified as ground scatter. Finally, we validate the ground scatter probability function by demonstrating that the backscatter virtual height increases as expected with increasing probability of ground scatter.
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 The Super Dual Auroral Radar Network (SuperDARN) is a global network of high-frequency (HF), over-the-horizon radars for the purpose of space physics research [Greenwald et al., 1985, 1995]. SuperDARN is designed to map ionospheric plasma convection in the polar regions. In the standard operating mode, sequential 16-beam scans are synchronized to start every 2 min with an integration time of 7 s per beam. From the transmission of a multiple-pulse scheme in the high-frequency band, the SuperDARN radars measure the autocorrelation function (ACF) of the backscattered signal in range gates of 45 km at ranges typically extending from 180 km to 3555 km. The backscattered signals are produced by coherent scatter by field-aligned electron density irregularities in the case of -hop or 1-hop propagation paths (ionospheric scatter) or land or sea surface roughness in the case of 1-hop propagation paths (ground scatter). In each range gate, this ACF is routinely analyzed by a basic method (referred to as FitACF) which extracts the signal power p, the line of sight mean Doppler velocity v of the irregularities, and the spectral width w of the Doppler power spectrum directly from the ACF [Villain et al., 1987; Hanuise et al., 1993; Baker et al., 1995, Appendix A]. Note that the spectral parameter data used in this study were calculated using the 2001 contemporary version of FitACF. The SuperDARN antenna array comprises a main array and an interferometer array. The cross-correlation function (XCF) between the main and interferometer arrays is analyzed to determine the elevation angle of arrival ɛ of the backscattered signal [Baker and Greenwald, 1988].
 In a common application, the line-of-sight velocity measurements are fit to an eighth-order spherical harmonic function of ionospheric electrical potential [Ruohoniemi and Baker, 1998]. Despite attempts to filter out ground backscatter by removing data with v < 30 m/s and w < 35 m/s, Chisham and Pinnock  found that significant contamination by ground scatter remains, which can lead to spurious results. The criteria for removal of ground scatter used by Chisham and Pinnock  were derived from a report by Baker et al.  in which they concluded that the spectral characteristics of ground scatter are as follows:
Since there have been no other studies of the criteria for identifying ground scatter following that of Baker et al.  and since the study of Chisham and Pinnock  indicates an ongoing problem with the identification of ground scatter, we undertake in this study a reevaluation of the criteria to identify ground scatter in the SuperDARN data.
2.1. Probability Density Functions
 The fundamental difficulty with separation of signals of ionospheric scatter and ground scatter origin is that in terms of line of sight velocity and spectral width, there is no distinguishing feature of ionospheric scatter [Baker et al., 1988]. That is, the distribution of w and v values produced by ionospheric scatter covers the entire range v > 0 and w > 0 up to the maximum values measurable and overlaps with the distribution of w and v values produced by ground scatter. Therefore, one can only speak of the probability P that a given echo is of ground scatter origin G given particular values of width w and velocity v. In Bayesian statistics, this probability is written as P(G∣w, v) and is referred to as the posterior probability of ground scatter conditional upon the values of w and v. This probability is distinguished from to the probability of ground scatter regardless of the values of w and v, which is written as P(G) and is referred to as the prior probability of ground scatter. Similarly, P(w, v∣G) is the posterior probability of w, v conditional upon the echo being of ground scatter origin G and P(w, v) is the prior probability of w, v regardless of the origin of the echo. These four probabilities are related by Bayes' theorem [cf. Schmidt, 1969],
 To determine the relevant probabilities necessary to calculate P(G∣w, v), we acquire data from the Kapuskasing, Ontario, Canada HF Radar (49.39°N, 82.32°W geographic coordinates; 60.06°N, 9.22°W AACGM magnetic coordinates) and the Saskatoon, Saskatchewan, Canada HF Radar (52.16°N, 106.53°W geographic coordinates; 61.34°N, 45.26°W AACGM magnetic coordinates) over the following time intervals: 25 March 2001 0000–2400 UT, 27 March 2001 0000–2400 UT, 11 June 2001 0000–2400 UT, and 22 December 2001 0000–2400 UT. These intervals, 48 h under equinoctial conditions and 24 h each under summer and winter solstice conditions, were chosen to eliminate seasonal and local time biases. The specific dates were chosen following a visual inspection of the data to select intervals with a high occurrence rate of both ground and ionospheric backscatter. These intervals yield a total of 386,081 individual ACFs and XCFs.
 We first determine P(w, v) by calculating the joint histogram of the backscatter spectral parameters w and v in 10 m/s × 10 m/s bins and normalizing by the total number of ACFs to obtain the probability density function. We assume that the total prior joint probability distribution function P(w, v) is the sum of two component distributions functions
This assumption does ignore a third possibility, which is that of mixed scatter, in which case scattering from both the ground and ionosphere occurs at the same range along two different ray paths. However, unless the signals from the different paths are of comparable power, we expect the autocorrelation function to display the characteristics of the dominant signal. Therefore, the signal may be classified as being predominately of ionospheric scatter or ground scatter origin, and we neglect the case of equal strength mixed scatter as being of low probability. Using the total probability theorem, equation (2) can be written as
Since the posterior probability density functions are normalized to 1, we obtain P(G) and P(w, v∣G) from
 Based on observations by Breech et al.  that the probability density function of the electric field in near-Earth space obeys an exponential distribution, we assume a similar distribution of the ionospheric electric field and consequently v. We allow the probability density function to vary freely with w. Therefore,
where we determine the values of AI(w) and bI(w) by least squares fitting. A priori, we have no similar information on the functional dependence of P(w, v∣G) on v. Therefore, after least squares fitting of candidate functions to the actual distribution, we assess the merit of each candidate function on the basis of a chi-square goodness-of-fit test. Once P(w, v∣G), P(G), and P(w, v) are determined, we calculate the posterior probability P(G∣w, v) of ground scatter from equation (1). We then establish a criterion for identification of a particular signal as ground scatter based on P(G∣w, v) for that signal.
2.2. Error Probabilities
 Once we determine a criterion for identification of a particular signal as ground scatter, we examine the probability of false positive errors (actual ionospheric scatter erroneously identified as ground scatter) and false negative errors (actual ground scatter erroneously identified as ionospheric scatter) in the identification of ground scatter in the SuperDARN dataset. The probability of a false positive error is given by
and the probability of a false negative error is given by
Both of these probabilities are a measure of the overlap between P(w, v∣G) and P(w, v∣I).
 We consider the P(G∣w, v) = 0.5 contour to be the optimal threshold for routine identification of a particular signal as ionospheric scatter or ground scatter. In the case of P(G) ≈ P(I), using the criterion P(G∣w, v) ≥ 0.5 to identify a signal as ground scatter minimizes the total error P(false positive) + P(false negative). Choosing a higher or lower threshold on P(G∣w, v) to identify a signal as ground scatter reduces or increases, respectively, the false positive rate while having the opposite effect on the false negative rate. In certain applications, however, such a trade off may be desirable. Therefore, we also analyze the effect of different criteria on the false positive and false negative rates.
2.3. Sensitivity of Results to P(G)
 Experienced users of SuperDARN data are aware that the prior probability of ground scatter P(G) depends on environmental factors that vary with time, such as operating frequency and ionospheric density. Under the assumption that these factors affect only the prior probability of ground scatter and not the posterior distributions of v and w in the case of ground or ionospheric scatter, P(w, v∣G) or P(w, v∣I), equations (1) and (3) permit us to recalculate the criterion for identifying a signal as ground scatter for different hypothetical values of P(G) other than the value of P(G) actually observed in this study. In this way, we study the sensitivity of the results obtained to P(G).
2.4. Validation Using Backscatter Virtual Height
 Finally, we use the additional information present in the cross-correlation functions to calculate the backscatter virtual height to provide a physical validation of the posterior probability of ground scatter P(G∣w, v) produced by this method. Backscatter virtual height is the apparent height at which backscatter appears to occur assuming a straight-line propagation path from the scattering to the receiver. For given range r and elevation angle ɛ of arrival of the ray at the receiver, the virtual height is h = r sin ɛ. Yeoman et al. [2008a, 2008b] observed that for -hop F region ionospheric scatter the most likely virtual height h* as a function of range r is given by
This model does not apply to ground scatter. As seen in Figure 1, the virtual height of ground scatter at range r will be greater than the virtual height of ionospheric scatter at the same range. Therefore, if P(G∣w, v) is a valid indicator of the probability of ground scatter, then we expect that the average relative virtual height anomaly (h − h*)/h*will increase with increasing probability of ground scatter P(G∣w, v). Note, however, that the efficacy of this validation will be limited by the aliasing of elevation angles above a certain critical angle that is dependent upon the radar operating frequency and the separation of the main and interferometer antenna arrays [Baker and Greenwald, 1988]. At an operating frequency of 10 MHz, which was the dominant operating frequency during the data intervals chosen for study, the critical angle is 45° for both the Saskatoon and Kapuskasing radars. This effect appears as a random error with a systematic negative bias in the determination of the virtual height. In the next section we present the result of these analyses.
3.1. Probability Density Functions
 The probability density function P(w, v) is shown in Figure 2. As can be seen in Figure 2, for any fixed value of w, including w ≈ 0, where ground scatter is expected to predominate, the probability density function decreases monotonically with increasing v. Therefore, we limit our consideration of ground scatter component distribution functions PG(w, v) to those that share this property. Four functions that we consider are the exponential, AG(w)e−bG(w)v, the Gaussian with zero mean, AG (w)e−bG(w)v2, the power law, AG (w)v, and the Lorentzian AG (w)(1 + bG (w)v2). We fit PI(w, v) + PG(w, v) to P(w, v) by least squares fitting, where PG(w, v) is represented by each of the above function in turn and PI(w, v) is represented by an exponential distribution (as in equation (6)) in each case, and we calculate χ2. The results are presented in Table 1. Of the four candidate functions, only the sum of two exponential distributions is not significantly different from the actual distribution at the 5% significance level (α = 0.05).
Table 1. Chi-Square Goodness of Fit Test of Candidate Probability Distribution Functions to P(w, v) and Significance Level α for Each Value of Chi-Square
PG (w, v)
Degrees of Freedom
2 × 10−8
 The inverse of the best fit parameters bG and bI are shown in Figure 3. The inverse of the exponential parameter can be interpreted as the average velocity of each component of the distribution. As can be seen in Figure 3, one component has a relatively small average velocity and the other has a large average velocity which is consistent with typical ionospheric velocities. We identify the low average velocity component with ground scatter G and the high average velocity component with ionospheric scatter I. With this identification, we present the posterior distributions P(w, v∣G) and P(w, v∣I) in Figures 4 and 5, respectively. For this particular data set, equation (4) yields P(G) = 0.39. It should be noted that we do not claim that this value of the a priori probability of ground scatter is a generally applicable result. However, because of the large differences between P(w, v∣G) and P(w, v∣I) seen in Figures 4 and 5, we do not expect that small variations in P(G) will have a significant effect on the location of the P(G∣w, v) = 0.5 contour. We show P(G∣w, v) in Figure 6.
 As described above, we consider the P(G∣w, v) = 0.5 contour to be the optimal threshold for routine identification of backscatter as ground scatter or ionospheric scatter. This contour is well represented by the least squares best fit polynomial v = 33.1 + 0.139w − 0.00133w2. Below this parabola, any particular signal is more likely to be of ground scatter origin than of ionospheric scatter origin. Therefore, we classify backscatter that satisfies the following criterion:
as ground scatter. We now consider the effect of these definitions in application.
3.2. Error Probabilities
 Using equations (7) and (8) and the threshold for identification of a signal as ground scatter (equation (10)), we calculated the probability of false positive and false negative errors. Similar calculations are carried out for the criteria presented by Baker et al.  and Chisham and Pinnock . These results are presented in Table 2. It can be seen in Table 2 that when compared to the criteria used by Chisham and Pinnock , the criterion presented in this paper significantly reduces the probability of false negative errors (ground scatter erroneously identified as ionospheric scatter) that they identified as adversely affecting the SuperDARN data products. However, when compared to the criteria used by Chisham and Pinnock , the criterion presented in this paper moderately increases the probability of false positive errors (ionospheric scatter erroneously identified as ground scatter). When compared to the criteria used by Baker et al. , the criterion presented in this paper moderately reduces the probability of false negative errors while not affecting the probability of false positive errors. It should be noted that these error levels are a consequence of the P(G∣w, v) contour that is chosen as the threshold of identification of a signal as ground scatter and may be adjusted according to the needs of the data user. We repeat the analysis above several times, each time using a different threshold value of P(G∣w, v) as the criterion for identifying a signal as ground scatter. The results are presented in Table 3. For example, the false positive rate is 7% if one requires P(G∣w, v) ≥ 0.9 to identify a signal as ground scatter. However, the corresponding false negative rate is 26%. Conversely, the false negative rate is 1% if one only requires P(G∣w, v) ≥ 0.1 to identify a signal as ground scatter. The corresponding false positive rate is then 37%. Some applications may make such a trade-off desirable, however, and we therefore include the parameters of the best polynomial fit to the respective contour lines.
Table 2. Probability of False Positive and False Negative Errors Versus Criteria for Identifying Ground Scatter
Table 3. Probability of False Positive and False Negative Errors and Parameters of Best Fit of Polynomial v[m/s] = a0 + a1w[m/s] + a2w[m/s]2 to Contour Line Versus Threshold Value of P(G∣w, v) Used to Identify Identifying Ground Scatter
P (False Positive) (%)
P (False Negative) (%)
3.3. Sensitivity of Results to P(G)
 The results presented above depend on the prior probability of ground scatter P(G), which in this study is 0.39. We test the sensitivity of the criterion to identify ground scatter to P(G) by holding P(w, v∣G) and P(w, v∣I) constant in equations (1) and (3) and varying P(G). The results of this analysis are presented in Table 4. It can be seen in Table 4 that all parameters vary with P(G). However, except for the smallest values of P(G), the variation in the parameters a0 and a2 is relatively small. The parameter a1 is the most sensitive to variations in P(G). Note that for P(G) = 0.1, no threshold was found. For such a small prior probability, P(G∣w, v) ≤ 0.5 for all values of v and w. That is, for such a small prior probability of ground scatter, it is always more likely that a given signal is of ionospheric scatter rather than ground scatter origin. In order for a user to take advantage of this analysis, however, it would be necessary to have an independent means of estimating the prior probability of ground scatter.
Table 4. Parameters of Criterion to Identify Ground Scatter v[m/s] ≤ a0 + a1w[m/s] + a2w[m/s]2 Versus Hypothetical Prior Probability of Ground Scatter P(G)
3.4. Validation Using Backscatter Virtual Height
 Finally, as a validation of P(G∣w, v) we present a graph of the average relative virtual height anomaly in bins of width 0.05 versus P(G∣w, v) in Figure 7. As expected, the virtual height is only a few percent higher than the model for signals with low probability of ground scatter and 33% higher than the F region virtual height when the probability of ground scatter is high. This analysis validates our identification of ground scatter. Note, however, that the standard deviation of the virtual height is large in all bins. This effect is likely due in large part to the contribution of the aliasing of the elevation angle mentioned above to errors in the virtual height. Therefore, inverting the process, and using virtual height as a proxy for the probability of ground scatter would not produce a reliable indicator of ground scatter.
 From the fact that the observed joint probability density of backscatter spectral width w and Doppler velocity v does not differ significantly from the expected sum of two component distribution functions and from the fact that the backscatter virtual height displays the expected dependence on ground scatter probability, we conclude that the method presented above validly determines the probability that backscatter of given spectral width w and Doppler velocity v is ground scatter. We further conclude that using a criterion to identify ground scatter that is based on a 50% or greater probability that a particular signal is ground scatter yields less contamination of ionospheric data by ground scatter than the criteria of Baker et al.  and Chisham and Pinnock , while not significantly increasing the probability of discarding ionospheric scatter as suspected ground scatter. However, we cannot conclude that this result will be robust when applied to SuperDARN stations other than the two used in this study.
 Our analysis yielded the result that the optimal criterion for identification of a signal as ground scatter is v[m/s] ≤ 33.1 + 0.139w[m/s] − 0.00113w[m/s]2. The location of the P(G∣w, v) contour depends, however, on the prior probability of ground scatter, P(G). If a data user is able to make an independent estimate of P(G), then the criterion may be adjusted in accordance with the results of the sensitivity analysis presented in section 3. Finally, the authors are aware from personal experience with SuperDARN radar data that the prior probability of ground scatter varies greatly between stations due to differences in terrain or surface type: ground, sea, or ice. If P(G) is the only factor that varies between stations, then the results of this analysis are applicable to other stations as well. If, however, P(w, v∣G) depends on surface type as well, then it would be necessary to repeat this analysis for each station.