On resolution/error trade-offs in incoherent scatter radar measurements



[1] In this work, we investigate the performance of amplitude modulated coding schemes in incoherent scatter radar (ISR) measurements in terms of statistical estimation error, range resolution, and signal-to-noise ratio. We approach this goal by formulating the inherent trade-off between estimation error and resolution as mathematical measures for model order selection. These trade-offs are examined on numerical experiments with several amplitude modulated waveforms with different duty cycles. We demonstrate that compared with an unmodulated long pulse, reduced statistical estimation error with similar range resolution, or finer range resolution with similar estimation accuracy can be obtained by incorporating coding schemes.

1. Introduction

[2] Accurate and efficient inversion of incoherent scatter radar measurements with the goal of estimating ionospheric parameters such as electron density, electron temperature, ion temperature, ion composition, etc., remains challenging despite the development of the exact forward theory [e.g., Dougherty and Farely, 1960, 1961; Farley, 1966; Hagfors, 1961, 1971] more than four decades ago. The reason lies in part in the inherent range smearing of information from one altitude over a number of altitudes, which is due to the length of transmitted pulse. Existing modulation techniques aim at elimination of range smearing from non-zero lags of autocorrelation function (ACF) measurements. Examples include double pulse and multiple pulse techniques [Farley, 1969, 1972], alternating codes [Lehtinen and Haggstrom, 1987] and random codes [Sulzer, 1986b] which have been successfully used in high resolution E-region measurements. For F-region measurements with high gain radars such as the Arecibo radar, however, these modulation techniques amplify the measurement noise in the form of uncorrelated clutter from unwanted ranges [Holt et al., 1992]. In such cases, one can use the multiradar ACF technique [Sulzer, 1986a] which essentially divides the radar power into several independent un-coded long pulse transmissions at different frequencies to trade signal-to-noise ratio (SNR) ( the ratio of the power originating from one bit to the background and system noise power) for time resolution.

[3] Un-coded long pulse measurements, on the other hand, suffer from high correlation between ACF lag estimate errors. This correlation plays an important role in ISR error analysis, that is, in determining the uncertainty in estimated parameters. First, in order to provide an accurate error analysis one has to account for the complete error covariance matrix [Huuskonen and Lehtinen, 1996], whose computation and inversion is computationally prohibitive especially in the high SNR scenarios, where the error covariance matrix is far from diagonal. In addition, when SNR is high, lag/range integration does not work as effectively as it would for fully independent observations in terms of improvement in statistical accuracy [Huuskonen et al., 1996; Lehtinen et al., 1997]. Because the estimates are highly correlated in high SNR, the integration of signals from N adjacent altitudes does not yield an equation image-fold increase in estimation accuracy as would be the case for fully independent observations.

[4] Since plasma ACF estimates from long-pulse measurements are not readily available from the measured signal ACF, any proper analysis technique is required to take the range smearing into account. Nikoukar et al. [2008] have developed a computationally efficient, hybrid methodology for analysis of long pulse data, where the smearing of information is removed from the data via a set of 1-D deconvolutions followed by nonlinear least squares optimization techniques applied to the deconvolved ACFs. The deconvolution allows for the retrieval of plasma ACF at individual altitudes separated by the distance determined by lag resolution (fine resolution grid). The results presented in the above work, however, show that in long pulse measurements and in the presence of noise in data, it is not possible to obtain estimated parameters in the same fine resolution grid. The reason is that in order to overcome the measurement noise, one is required to incorporate regularization techniques which introduce dependencies into adjacent estimated parameters, hence reducing the effective range resolution.

[5] The purpose of the work presented in this paper is to investigate the performance of amplitude modulated coding schemes in F-region ISR experiments in terms of statistical estimation error, range resolution, and signal-to-noise ratio. For this purpose, we consider several amplitude modulated codes with a fixed length with different duty cycles. Our goal is accomplished by formulating the inherent trade-off between estimation error and resolution as mathematical measures for model order selection [Konishi and Kitagawa, 2008; Akaike, 1973]. Subsequently, numerical simulations are performed to verify the improved performance of the amplitude modulated (AM) codes guided by our analytical framework rendering optimal range resolution in comparison with long pulse modulation.

[6] In section 2, we describe the structure of the error covariance matrix and show how it is affected by the shape of the modulation waveform. In section 3 we develop analytical approaches to determine the fundamental attainable resolution based on model order selection. We proceed by presenting numerical estimates of fundamental range resolution as a function of SNR for various AM waveforms with different on-off ratios (duty cycles) in section 4. The last section provides a summary and conclusion.

2. Error Covariance and Modulation Waveform

[7] A necessary requirement for ISR error analysis is that the correlations between the lag estimate errors are calculated accurately. In this section, we illustrate how the envelop of the modulated waveform affects the structure of the error covariance matrix. We also show the performance of lag/range integration for several AM waveforms in terms of improvement in statistical accuracy as a function of SNR and ion temperature, which is indicative of incoherent scatter spectrum width.

[8] The received signal v(t) is a sum of two signals, the radar echo z(t) and the noise n(t). The radar echo is a Gaussian stochastic process because it is the sum of a large number of independent elementary contributions within the scattering volume (i.e., central limit theorem [Papoulis, 1986]). Additionally, the echo is a wide sense stationary stochastic process (over a finite time when the ionosphere does not change significantly). The noise process is Gaussian, stationary and independent of the radar echo. Since both these processes have zero mean and are Gaussian, they are specified by their correlation functions. Let us denote the correlation functions of the radar echo, the noise, and the received signal at time t and time lag t′ − t by k(t, t′), kn(t, t′), and K(t, t′), respectively, as follows:

equation image
equation image
equation image
equation image

where E[.] and * denote the expected value and conjugation operations, respectively. In practice, however, we obtain estimates of the correlation function denoted by M(t, t) by calculating the cross products (z(t) + n(t))(z(t′) + n(t′))* over repetitions of the experiment. In other words, we form the following sum, as ACF lag estimates, over different scans, when the same radar pulse is transmitted repeatedly:

equation image

where sc denotes different scan counts and N presents the total number of pulse transmissions. Since N is finite and the ionosphere is continuously subject to change, these estimates of ACF lags deviate from the true values. For the received signals at time t and time lag t′ − t, these deviations can be defined as follows:

equation image

[9] Lehtinen and Haggstrom [1987] have shown that these deviations are not independent from one lag or one altitude to another, and that the correlation can be expressed in terms of the true expected functions, k(t, t′) and kn(t, t′), or K(t, t′). Using the fourth-moment theorem for Gaussian processes [Gihman and Skorohod, 1980], we can express the covariance between the real (Re) and imaginary (Im) parts of the errors in ACF estimates from time t and time lag t′ − t, and time u and time lag u′ − u as [Lehtinen and Huuskonen, 1996]:

equation image
equation image
equation image
equation image

These expressions imply that the covariance between the estimates M(t, t′) and M(u, u′) does not depend explicitly on the corresponding expected plasma ACFs, K(t, t′) and K(u, u′), but rather on the expectation of the ACF evaluated at the other four possible permutations. Therefore, evaluation of the full error covariance matrix becomes more computationally expensive, especially in high SNR scenarios where the matrix becomes far from diagonal. For example, the errors between ACF from adjacent heights or adjacent lags are always correlated because of zeroth lag terms that will appear in error expressions (7) to (10).

[10] Figure 1 illustrates examples of modulations with different duty cycles. These examples include a long pulse, and several amplitude modulated codes with duty cycles ranging from 1 to 0.75 to 0.64 to 0.57 to 0.42 from left to right. In order to visualize the effect of the transmitted waveform on correlation between errors, in Figure 2 we show the error correlation coefficients of lags for these waveforms. Note also that lag resolution is set to 10 μs and a hypothetical ionosphere (oxygen only with no drift velocity) and constant parameter profiles (Ti = Te = 1000°K) are assumed. From the plots in Figure 2, it is clear that as the duty cycle of a code decreases, the correlation length reduces to fewer lag neighbors. Note that the modulations used in this simulation are chosen such that they provide information about all ACF lags with 10 μs resolution up to 250 μs. The AM4 waveform is the 5-pulse modulation [see Farley, 1972] with 20 μs lag resolution.

Figure 1.

Examples of modulations with different duty cycles (1, 0.75, 0.64, 0.57, and 0.42 for pulses in Figures 1a–1e, respectively). These modulations will be subsequently used in simulations throughout the paper.

Figure 2.

Correlation coefficients between lag estimate errors for amplitude modulated pulses shown in Figure 1.

[11] When the data is integrated over a few lags (ranges), one expects the variance of ACF lag estimate errors to be reduced. This reduction in variance which is inversely related to statistical accuracy, however, becomes more pronounced for codes with lower duty cycles. Figure 3 shows this error variance reduction when data is integrated over three consecutive lags as a function of both SNR and ion temperature. In long-pulse measurements where the errors are highly correlated, the integration of signal over three lags results in variance reduction (improvement in estimation accuracy) by a factor of 1.1 instead of 3 in the case of fully independent observations. The improvement in estimation accuracy gained by lag integration becomes more pronounced as the on-off ratio of the modulated pulse is decreased.

Figure 3.

Improvement in statistical accuracy due to reduced variance when data are integrated for three consecutive lags for five pulses in Figure 1 as a function of ion temperature and SNR. The improvement in accuracy can be up to 45% with respect to the long pulse. This improvement implies that the same accuracy can be achieved with the reduction of integration time by up to 45%.

[12] Figure 4 shows the reduction in variance of lag estimate errors as a function of both SNR and ion temperature when measurements with different modulations are integrated over three consecutive ranges (separated by 10 μs). At SNR = 1, compared to the long pulse, AM3 and AM4 show 52% and 63% improvement in statistical accuracy, respectively. This essentially means that compared to the long pulse, the integration time required to achieve a certain level of estimation error for a fixed range resolution in high SNR (SNR > 0.4) is 52% and 63% shorter with AM3 and AM4, respectively. Equivalently, for a fixed integration time, same estimation error levels will be obtained over finer range resolution grids. This issue is addressed from a model order selection standpoint in the next section.

Figure 4.

Improvement in statistical accuracy due to reduced variance when the data are integrated over three consecutive ranges for five pulses in Figure 1 as a function of ion temperature and SNR. The improvement in accuracy can be up to 65% with respect to the long pulse.

[13] It should be mentioned that the duty cycle of amplitude-modulated codes affects sensitivity to the background noise. An example of normalized variance of the zeroth lag error (relative error) as a function of SNR is presented in Figure 5 for pulses with different duty cycles. The same level of variance is reached in higher SNRs as the on-off ratio of the code decreases, confirming the higher sensitivity of measurements to background and receiver noise for pulses with smaller on-off ratios. In this work we focus on the most relevant SNR scenarios where the AM scheme is not affected adversely by increased sensitivity to low SNR. In addition, lower duty-cycle waveforms implicitly result in the absence of certain lags in the ACF measurements, and the lack of certain ACF lags can introduce large errors in estimation. Therefore, in this work, we consider 0.41 as the lowest duty cycle for the AM pulses. Note that the waveforms used in this study (AM1, AM2, AM3, and AM4) are chosen based on a computationally feasible exhaustive search over all possible waveform configurations with a fixed duty cycle. The search criterion is to minimize condition number of the corresponding convolution matrices. This criterion guarantees the presence of all lags in measurements which is necessary for estimation accuracy.

Figure 5.

Normalized variance of the zeroth lag errors as a function of SNR for pulses with various duty cycles.

3. Fundamental Resolution in ISR Measurements

[14] In this section we develop a framework, based on model order selection in parametric estimation and linear regression, for quantifying resolution in ISR measurements. This problem essentially can be expressed as finding the optimal grid density over which the physical parameters are most accurately estimated from the measurements. For a nonanalytical solution, the unknown ionospheric parameter profiles must be discretized and can typically be represented adequately by a weighted sum of a finite number of basis functions. In the hybrid technique developed by Nikoukar et al. [2008] for computationally efficient estimation of ionospheric parameters from ISR measurements, models of plasma ACF lag profiles at various discretizations can be viewed as grids with different densities.

3.1. Model Order Selection Framework

[15] In the hybrid technique, the physical parameters are estimated by first computing plasma ACF lag profiles (denoted by mτ for τ = 0, Δτ, ⋯, where τ and Δτ represent time lag and time lag increment, respectively) from the received signal ACF (denoted by yτ). To find the optimal resolution, we consider the discretization of the plasma ACF lag profile at lag τ, mτ, as

equation image

which is equivalent to parametrization of mτ with n known basis functions {ϕi}i=1n and the corresponding unknown coefficients {θτi}i=1n. In this work, we consider {ϕi} as the set of unit rectangular boxes which constitutes an ortho-normal basis for the plasma ACF lag profiles. Different discrete models can then be considered as a linear combination of unit rectangular boxes with different widths. Here, the larger the widths of these basis functions, the lower the number of required functions to specify each ACF lag profile and hence the coarser the resolution. Examples of discretization of an ACF lag profile with large-width and small-width boxes are depicted in Figures 6a and 6b, respectively. We should mention that orthogonal functions have been used in statistics for estimating probability distributions [e.g., see Kendell and Stuart, 1979; Sharif and Kamalabadi, 2005].

Figure 6.

Examples of discretization of an ACF lag profile with (a) large-width basis function (low resolution) and (b) small-width basis function (high resolution).

[16] Note that the discretization grid density for the measurement ACF is fixed and determined by the time lag increment. As shown by Nikoukar et al. [2008], when the grid density of plasma ACF is also determined by the time lag increment, the relationship between the measured ACF and the plasma ACF (which is unknown) is expressed in the matrix framework as

equation image

where Pτ is the convolution matrix constructed based on the ambiguity function at a given time lag τ and equation image represents the measurement error. In (12), yτ and θτ are n × 1 vectors, where n is the number of altitudes of interest (with spacing equal to Δr = equation image) and Pτ is an n × n matrix. We can further combine all such equations for all time lags and form an overall matrix equation as follows:

equation image

where the combined forward model A is a block diagonal matrix with each convolution matrix as its diagonal blocks, given by

equation image

with T the pulse length. The measurement and unknown vectors can be represented by y = [y0; yΔτ; …] and θ = [θ0; θΔτ; …], respectively. In this case, the data and unknowns are (n × equation image) × 1 vectors and A is a (n × equation image) × (n × equation image) matrix.

[17] For models with other grid densities, the relationship between the measurements and unknowns is represented in a similar fashion. The only difference would be in the forward model matrices and the representation of the unknowns in terms of basis functions. As the resolution becomes coarser, the number of unknowns decreases while the number of data points remains the same. For example, if the resolution becomes k times coarser, i.e. the width of the basis function ϕ(k) becomes k times larger, the unknown vector θ(k) becomes (n × equation image) × 1 (as opposed to (n × equation image) × 1. The forward matrix A(k) would then be obtained by averaging each k columns of A to yield a (n × equation image) × (n × equation image) matrix. Note that the subscript (k) denotes the resolution which is coarser by a factor of k.

[18] With the assumption of the Gaussian errors, the probability distribution of the k-discrete model can be described as:

equation image

where Σϵ represents the error covariance matrix of measurements.

3.2. Model Order Selection Method

[19] Having identified a set of discrete models for ACF lag profiles, we now need to address the selection of an optimal model that estimates the true ionospheric profile as accurately as possible from the available data. To accomplish this, first a criterion for model evaluation must be constituted. Subsequent optimization of the desired criterion would then yield the optimal discrete model. Here, for the model evaluation criterion we consider “generalized information criterion” (GIC) which has been widely used in model order selection problems with noisy measurements [Konishi and Kitagawa, 2008]. GIC is a criterion that consists of two terms. The first term controls the data fidelity while the second term controls the model order. The optimal model order is obtained when there is a balance between the two terms. In what follows GIC is introduced.

[20] The information criterion, in general, is defined in terms of Kullback-Leibler (K-L) information (difference between the predictive distribution defined by the model f(k) and the true distribution g, I(g, f(k)) = ∫ g(y) log equation imagedy) as an indicative measure of the “information” lost when f(k) is used to approximate g [Cover and Thomas, 1991; Akaike, 1973; Konishi and Kitagawa, 2008]. The K-L distance can be simplified as follows

equation image
equation image

where Eg[.] denotes the expectation with respect to the true distribution g. The first expectation, Eg[log[g(y)]], is a constant that depends only on the unknown true distribution, and is clearly not known. With treating this unknown as a constant (C), only a measure of relative K-L distance is possible up to the constant C as

equation image

[21] In model selection based on the information criterion, in the first step we estimate unknown parameters for each candidate parametric model. We next construct a statistical model f(k)(y|equation image(k)) by replacing the unknown parameter θ(k) contained in the probability distribution of the parametric model by its estimate equation image(k). The goodness (or badness) of each model then is determined by evaluating −Eg[log[f(k)(y|equation image(k))]].

[22] Although the above expectation is not computable (since the true model g is not known), it can be estimated by the log of likelihood (equation image) of each model, described by [Burnham and Anderson, 1998]

equation image

[23] In practice, where there is inherent measurement noise, −Eg[log[f(k)(y|θ(k))]] is estimated through penalized log likelihood of each model. An example of such modified log likelihood function which guarantees the smoothness of estimated profiles in candidate models is

equation image

where L is the first-order or second-order gradient matrix imposing smoothness constraint and λ is a parameter that controls the weight between data fidelity and the constraint. Moreover, T denotes the transpose operation.

[24] With the current setting, the estimated solution, equation image(k), can be obtained by solving the following equation:

equation image

[25] The generalized information criterion for model selection is defined as [Konishi and Kitagawa, 2008]

equation image

where Tr(.) denotes the trace operation, and R(ψp) and Q(ψp) are (n × equation image) × (n × equation image) matrices, respectively, given by

equation image
equation image

[26] For the problem at hand, R(ψp) and Q(ψp) are given by

equation image
equation image

[27] The term on the right-hand side of equation (22) (bias term) is expressed as

equation image
equation image
equation image

where Mλ = Σϵ−1/2A(ATΣϵ−1A + λ2LTL)−1ATΣϵ−1/2. The term Mλ relates the predictive and actual data (equation image and y, respectively), as equation image = Mλy, and is called the influence or smoothing matrix. The trace of this matrix is always less than the number of model parameters and indicates the effective model order, effective number of parameters, or degree of freedom. This reduction in the number of parameters is due to the smoothness constraint (regularization) which brings some dependencies into neighboring parameters.

4. Numerical Estimates of Range Resolution

[28] In this section, we present the results of numerical resolution analysis for the five modulations shown in Figure 1. The profiles of physical parameters used for simulations are shown in Figure 7, and the time lag increment is set to 10 μs. To generate the simulated signals we first generate random signals based on the incoherent scatter spectrum at each altitude. We then multiply the signals by the envelop of the transmitted pulse, form time correlations, and accumulate the resulting signals over many pulse transmissions. (For more detailed explanation on data simulation steps, see Nikoukar et al. [2008].)

Figure 7.

Profiles of a hypothetical ionosphere used in simulations.

[29] We consider three different SNR regimes (high, SNR > 0.4; medium, 0.08 < SNR < 0.2; and low, 0.02 < SNR < 0.07). The SNR altitude profiles and the corresponding average errors are shown in Figure 8. Note that the relative error in SNR altitude profile (ratio of (SNR error/ SNR)) increases dramatically as SNR decreases (from 0.027 in the high SNR to 0.17 in the low SNR regime at 500 km). It should be noted that we do not consider the case of very low SNR regime (SNR < 0.02) in this study. The reason is that in the very low SNR regime the correlation between lag estimate errors diminishes even for long pulse modulation. Therefore, the main advantage of the amplitude modulation, which is rendering reduced correlation between lag estimate errors, is lost. The increased sensitivity of the amplitude modulated measurements to the background noise makes this coding scheme less favorable in such low SNR regimes.

Figure 8.

(a) High, (b) medium, and (c) low signal-to-noise ratio profiles used for ISR numerical experiments. Average errors in profiles are shown with thin lines.

[30] We first start with the estimation of range resolution for long-pulse measurements in the high SNR regime (see Figure 8a). In order to determine the range resolution, we consider the models with basis functions with widths 10, 20, 30, 50, 60, 100, 150, and 300 μs, corresponding to 1.5, 3, 4.5, 7.5, 9, 15, 22.5, and 45 km, respectively. (These are noted by models A(1), A(2), A(3), A(5), A(6), A(10), A(15) and A(30), respectively. A(1) is the model with the finest resolution, and A(k) is the model with resolution k times coarser.)

[31] For each model, the solution is obtained via regularization based on (20) in the work of Nikoukar et al. [2008] for various regularization parameters and the expression of GIC is computed from (22) for each model and each regularization parameter. Figure 9 shows values of GIC for the model A(1) for which the width of the basis function is equal to the lag resolution (1.5 km). As expected, GIC first decreases with increasing regularization parameter and then increases. This indicates that GIC can be also used for the selection of the optimum regularization parameter, λ for a fixed model. The smallest parameter which results in minimized GIC is the optimal λ and is depicted by ‘o’ in Figure 9a.

Figure 9.

GIC values for long-pulse measurements for the data with SNR shown in Figure 8a as a function of (a) regularization parameter and (b) resolution. The optimal regularization parameter and optimal resolution are shown by an open circle in Figures 9a and 9b, respectively.

[32] Once the optimum λ is determined, we can obtain the true resolution based on Tr(Mλ), the trace of the corresponding influence matrix. As explained previously the trace of the influence matrix for a specific λ is representative of the number of free parameters in the plasma ACF lag estimation problem. The optimal resolution is obtained as follows:

equation image

where n × equation image is the total number of unknowns in the lag estimation problem when model A(k) is used. In a hypothetical case where the measurements are noise-free, no regularization is necessary and the number of free parameters is equal to the number of unknowns for each model. In this case, the resolution is equal to the width of the model basis function, as expected. In realistic cases, however, the degree of freedom is considerably smaller than the number of unknowns, and hence the resolution differs significantly from the width of the basis function for long-pulse measurements. Figure 9b shows the GIC values as a function of the resolution for the long-pulse measurements. The optimal resolution shown by an o is approximately 23 km, which is nearly half of the distance covered by the length of the transmitted pulse. This result is in agreement with that presented by Lehtinen and Huuskonen [1996] where the optimal grid density was obtained via a full-profile analysis technique.

[33] Figure 10a shows the variation of GIC for models A(1), A(2), A(3), A(5), A(6), A(10), and A(15) as a function of range resolution. As in the previous case, for all these models, GIC first decreases with an increase in regularization parameter and then increases, with minimum GIC occurring at resolution close to 23 km. We note here that the minimum GIC decreases as we consider models with lower grid densities. The minimum of these values occurs for model A(15) when regularization parameter is set to 0, and the resolution is 22.5 km. This result indicates that although the number of unknown parameters differs in different models, the effective number of parameters, and hence the resolution, remains nearly constant for all models. The reason is that the true resolution is implicit in the expression of GIC through the bias term (see (27)). The GIC for A(30) is presented in Figure 10b. As seen in this figure, GIC for all regularization parameters (resolutions) is significantly higher than those for other models. The smallest resolution offered by this model is equal to the length of the transmitted pulse, which is much coarser than the optimal resolution. This implies that this model is not appropriate as a deconvolution model. This explains the poor performance of the height-by-height analysis method, where the physical parameters are estimated with spacing equal to the length of the transmitted pulse [see Nikoukar et al., 2008].

Figure 10.

GIC values for long-pulse measurements as a function of range resolution for models (a) A(1), A(2), A(3), A(5), A(6), A(10), and A(15) and (b) A(30). All models except A(30) offer an optimal resolution at nearly 23 km. The resolution achievable by A(30) is too coarse even with no regularization applied.

[34] We now focus on the estimation of range resolution for other amplitude modulations and SNR regimes shown in Figures 1 and 8, respectively. For our analysis, we consider GIC values for one model only (A(1) with the highest density). Because of regularization, the effective model order corresponding to the optimal λ in A(1) is equal to the number of independent variables in other models.

[35] Plots in Figures 11a11d show GIC values as a function of range resolution for ISR measurements with AM1–AM4, respectively, for the high SNR scenario (SNR > 0.4 in Figure 8a). By comparison of these plots with those in Figure 9b, we realize that as the on-off ratio of the modulated waveform decreases (from 1 to 0.41), finer range resolutions become achievable from ISR measurements (from 22.5 km to 8.3 km).

Figure 11.

GIC as a function of range resolution for the SNR profile shown in Figure 8a for AM1–AM4.

[36] We repeat the same analysis for medium and low SNR scenarios (Figure 8) with results shown in Figures 12 and 13 and summarized in Table 1. It can be seen that resolutions for the long-pulse modulation and AM1 (with on-off ratio = 0.75) remain constant for the three SNR regimes at 22.5 km and 17.8 km, respectively. For AM2, the resolution first decreases from 13.2 km to 15.3 km with SNR dropping from high to medium and then remains the same. For AM3, the resolution decreases from 9 to 10 to 14.1 km for the three SNR regimes. In the case of measurements with AM4, range resolution decreases from 8.3 km to 14 km with lowering SNR. The reason for variation in the optimal range resolution for modulations with smaller duty cycles is that when SNR is low, the relative errors in estimated ACF lags become higher. This requires higher regularization parameters to stabilize the deconvolution solutions, and hence results in coarser optimal range resolution.

Figure 12.

GIC as a function of range resolution for the SNR profile shown in Figure 8b for long pulse and AM1–AM4.

Figure 13.

GIC as function of range resolution for the SNR profile shown in Figure 8c for (a) long pulse and (b–e) AM1–AM4.

Table 1. Summary of Resolutions Achievable With Different Modulation Schemes and Different SNR Levelsa
 SNR > 0.40.08 < SNR < 0.2SNR < 0.07
  • a

    Resolutions are expressed in terms of the transmitted waveform length.

Long pulse0.50.50.5

[37] Note that the range resolution for AM3 and AM4 is very similar in medium and low SNR regimes. This shows that further lowering the on-off ratio of the modulated pulse not only has no significant effect on improving range resolution, but also makes the measurements more sensitive to the background noise level. This is an important result which should be considered in optimal amplitude modulated waveform design.

[38] Once the optimal resolution is determined, we construct a forward model with basis function width equal to the resolution. Deconvolution, then, is performed using least squares techniques. No regularization is necessary in this case, as the effect of the regularization parameter is already taken into account in the choice of the optimal model. The details of inversion are given by Nikoukar et al. [2008]. To visualize how this mechanism works, we present estimated physical parameters using long-pulse and AM3 measurements for the high and low SNR cases. According to the results of Table 1, A(15) is the optimal model for long-pulse measurements in both cases, whereas A(6) and A(10) are optimal models for AM3 measurements in high and low SNR scenarios, respectively. Figures 14 and 15 show samples of estimation results while Tables 2 and 3 present averages of estimation errors using 50 runs of Monte-Carlo type of simulations for high and low SNR scenarios, respectively. Based on Tables 2 and 3, we realize that although the range resolutions differ significantly especially for the high SNR level, estimated parameter errors remain nearly equal. Note that this result is the direct consequence of improved statistical accuracy for integrated data in lag or range which is more pronounced for amplitude modulation with lower on-off ratios. This is advantageous in situations where we need high resolution and low integration time in a high or medium SNR scenario when high-resolution techniques such as multipulse or alternating codes would not be preferable.

Figure 14.

Estimated physical parameters (solid lines) for (top) long-pulse and (bottom) AM3 measurements in a high SNR regime. The optimal resolution for AM3 is improved by a factor of 2.5, while the estimation errors are nearly the same. Original parameters are denoted by dashed lines.

Figure 15.

Estimated physical parameters (solid lines) for (top) long pulse and (bottom) AM3 measurements in a low SNR regime. The optimal resolution for AM3 is improved by a factor of 1.5, while the estimation errors are nearly the same. Original parameters are denoted by dashed lines.

Table 2. Estimated Errors in Physical Parameters for a High SNR Situation for 50 Runs of Monte Carlo Simulations
 Electron Temperature (K)Ion Temperature (K)Electron Density (m−3)
Long pulse29.138.22.0e10
Table 3. Estimated Errors in Physical Parameters for a Low SNR Situation for 50 Runs of Monte Carlo Simulations
 Electron Temperature (K)Ion Temperature (K)Electron Density (m−3)
Long pulse43.248.61.9e10

[39] It is worth mentioning here that considering a low-resolution model for the deconvolution can reduce computations significantly by reducing the number of unknown parameters while producing accurate results. This reduction in computation can be extremely useful in real-time data analysis. Note that other methods, such as full-profile analysis, can also be applied to amplitude-modulated measurements for ionospheric parameter estimation.

5. Conclusion

[40] In this work we have developed a framework to determine the optimal range resolution for different modulation techniques in ISR experiments by incorporating mathematical measures from model order selection literature. To accomplish this, we have demonstrated the inherent trade-off between ISR parameter estimation error and resolution. We have compared the performances of several amplitude modulated pulses and a long pulse in ISR data inversion. The analyses of numerical simulation of incoherent scatter measurements show that depending on the duty cycle of the transmitted AM waveform, the optimal range resolution can be improved by up to a factor of 3, and 2.5 in high and medium SNR scenarios, respectively, compared to a long pulse.

[41] The results of this work provide a basis for optimal waveform design for F-region experiments for a high SNR regime. The next step is to determine the optimal on-off ratio with predetermined fixed lag resolution, and to design the optimal configuration of codes which will result in minimum parameter estimation errors. This is the subject of a future work.