SEARCH

SEARCH BY CITATION

Keywords:

  • Incoherent scatter radar;
  • inverse problems;
  • parameter estimation

Abstract

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Incoherent Scatter Radar Equation and Ambiguity Functions
  5. 3. Inversion of Incoherent Scatter Radar Measurements—Proposed Method
  6. 4. Numerical Results
  7. 5. Summary and Conclusion
  8. Acknowledgments
  9. References

[1] We present a computationally efficient, near-optimal approach to the estimation of ionospheric parameters from incoherent scatter radar measurements. The method consists of removing the range smearing of ionospheric autocorrelation function via a set of 1-D deconvolutions and performing nonlinear least squares fitting on the deconvolved autocorrelation functions. To stabilize the solution in the presence of noise, we incorporate regularization techniques. The computational cost is reduced significantly by estimating the ionospheric parameters at individual altitudes, in comparison to full-profile-type analysis, which attempts to estimate ionospheric parameters at all altitudes simultaneously. The performance of the new technique is evaluated in a numerical example and is shown to give estimates of almost equal quality as the full-profile technique but at a 95% reduction in computation.

1. Introduction

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Incoherent Scatter Radar Equation and Ambiguity Functions
  5. 3. Inversion of Incoherent Scatter Radar Measurements—Proposed Method
  6. 4. Numerical Results
  7. 5. Summary and Conclusion
  8. Acknowledgments
  9. References

[2] Incoherent scatter radar (ISR) is the most comprehensive ground-based technique for studying the Earth's ionosphere. The incoherent scatter echo is the result of the scattering of electromagnetic energy, radiated from the radar, by electron density fluctuations in the ionospheric plasma, which in the most common case are influenced by much slower, massive positive ions. The frequency spectrum of the received signal provides information about electron and ion temperatures, ion composition and velocity. The analytical relationship between the spectrum and these parameters has been well established in the literature [e.g., Dougherty and Farely, 1960, 1961; Farley, 1966; Hagfors, 1961, 1971]. With the estimation of these parameters from incoherent scatter (IS) measurements, one can deduce many further ionospheric parameters such as electric field strength, conductivity and current, neutral air temperature, and wind speed.

[3] Although the exact forward theory of incoherent scatter was established more than four decades ago, inversion, i.e., the extraction of parameters from incoherent scatter spectra, has remained an open problem primarily because of two major factors. The first complication stems from the fact that variation of different plasma parameters may give rise to similar changes in the IS spectrum [Vallinkoski, 1988]. For example, the distinction between changes of the spectra due to ion composition or temperature ratio is very difficult. The same is true for ion mass and ion temperature for fixed temperature ratio. The second factor is the range smearing of information from one altitude over a number of altitudes, which is due to the length of transmitted waveform. An example is shown in Figure 1 where different lags of the autocorrelation function (ACF) are plotted prior to and after imposing the range ambiguity for an uncoded modulation of length 280 μs with a 430 MHz radar (thin and thick curves, respectively). The estimation of ionospheric parameters from the range-smeared ACF would result in a greater ion temperature than electron temperature, which is not physical.

image

Figure 1. Normalized theoretical and measured ACF for a long uncoded pulse transmission and a constant ionosphere (thin and thick curves, respectively). Range smearing makes the shape of ACF so distorted that results of the fit by standard nonlinear least squares methods become poor. Notice that the above ACFs have been generated using a radar frequency of 430 MHz for a single-ion ionosphere (oxygen). The lag spacing is 10 μs, and an uncoded long pulse of length 280 μs is considered for transmission.

Download figure to PowerPoint

[4] Traditionally, incoherent scatter analysis has been based on performing nonlinear least squares fitting at individual altitudes (height-by-height or gated analysis) [Lehtinen et al., 1996], where in the most common case the spacing between the altitudes is determined by the pulse length. The important assumption in this technique is that plasma parameters are considered constant in all regions which have influence on an individual set of spectral or ACF measurements. The effects of the transmitted waveform are then accounted for in the form of various corrections to ACF lag estimates. This method, although simple and fast, is based on an unrealistic assumption (constant parameter profiles for each range-gate) and suffers from coarse resolution of the parameters, as well as a bias which is introduced into the profiles [Holt et al., 1992; Lehtinen et al., 1996].

[5] Another method is the full-profile analysis [Holt et al., 1992; Lehtinen et al., 1996], which attempts to determine the complete altitude profiles of parameters simultaneously. The basic estimation procedure is to first calculate theoretical spectra on the basis of an initial parameter grid (followed by compensating for the smearing of information across the range, if applicable). Subsequently, the difference between the measured and predicted ACFs is used to update the plasma parameter values iteratively. The technique is optimal in the sense that all information including the range smearing effect as well as the covariance between the lag-estimate errors, can be incorporated into the analysis. The main limitation of the method, on the other hand, is its significant computational cost. Attempts have been made to reduce the cost by exploiting interpolation techniques from a coarse parameter grid to a fine range grid [Holt et al., 1992; Lehtinen et al., 1996]. The accuracy of the method, however, is highly dependent on the resolution of the parameter grid. A very fine spatial resolution, which is required to approximate fine details of the ambiguity function or to capture sharp gradients of parameter profiles, increases the number of parameters of the optimization search space and thus makes the technique more computationally expensive. One basic reason is due to the requirements of any nonlinear optimization technique, one of which is the computation of the derivative of the minimization function with respect to the search variables (parameters in the coarse grid). Even though analytical expressions of the derivatives of the lags of the theoretical autocorrelation function with respect to the ionospheric parameters are available, they cannot be exploited in the optimization procedures of the full-profile techniques. Therefore, forward difference, which slows down the speed of computation, is the only method that can be used for derivative calculation.

[6] In this paper, we develop the theory of a new hybrid inversion technique which aims at obtaining estimates that are close to optimal at a fraction of computational cost. The technique is based on a correction to the effect of the transmitted waveform on the ACF lag profiles through a deconvolution process, and subsequent estimation of parameters from the plasma ACF at individual altitudes.

[7] Note that some coding schemes such as alternating codes [Lehtinen and Haggstrom, 1987] can be utilized to eliminate the range smearing from the non-zero lags of ACF measurements analytically (provided that the medium coherence is maintained over code transmission cycles). These modulation techniques, however, increase the noise in the form of uncorrelated clutter from unwanted ranges and as such are not suitable for high-gain radars such as Arecibo radar. In such cases, one can use the coding technique developed by Sulzer [1986] which essentially divides the radar power into several independent uncoded long-pulse transmissions at different frequencies. In this paper we focus on such situations and investigate the performance of the proposed approach on simulated data using the long-pulse modulation. The performance comparison of different analysis techniques on ISR experiments is the subject of a future publication.

[8] The paper is organized as follows: section 2 describes background information on the incoherent scatter radar equation as well as the concepts of range smearing and ambiguity. In section 3, we present a detailed description of our proposed method and its justification using the radar equation. Section 4 investigates the performance of the new method on simulated data and compares the corresponding estimation results with those of common methods currently used. The last section provides a summary and conclusion.

2. Incoherent Scatter Radar Equation and Ambiguity Functions

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Incoherent Scatter Radar Equation and Ambiguity Functions
  5. 3. Inversion of Incoherent Scatter Radar Measurements—Proposed Method
  6. 4. Numerical Results
  7. 5. Summary and Conclusion
  8. Acknowledgments
  9. References

[9] In any plasma correlation measurement the received voltage from an altitude in the ionosphere contains information from not only that particular altitude but also a number of altitudes below. According to the incoherent scatter radar equation, the transmitted waveform plays an important role in how this information is combined over the range. In what follows, the radar equation is derived and ambiguity functions are described.

[10] Let us denote the envelope and the length of the transmitted waveform in the radar experiment by V0(t) and T, respectively. Suppose the pulse is a baseband waveform modulated by a sinusoidal wave of frequency ω0. The corresponding wavelength and wave vector will be λ0 = equation image and k0 = equation image, where c is the speed of light and equation image is the direction of wave transmission. The returned signal is then sampled with a period δt (Figure 2).

image

Figure 2. Transmission and reception scenario in a typical incoherent scatter radar. A pulse (modulated or unmodulated) of length T is transmitted through the ionosphere, and the returned signal is sampled in δt intervals.

Download figure to PowerPoint

[11] The received scattered signal from the electron density fluctuations in the ionosphere can be represented by [Kudeki, 2003]:

  • equation image

where n(r, t) represents the fluctuating component of the plasma electron density at range r = requation image along the direction equation image (the proportionality factor depends on the classic electron radius, and antenna-related factors such as radiation resistance of the antenna, antenna gain along the direction of transmission, transmitted power, etc.). Note that the received signal spectrum is centered at the carrier frequency, ω0, instead of zero, since the antenna output is modulated by the carrier signal. At the detector, the received signal is converted to the baseband. The exponential term in (1) represents this conversion.

[12] Since there are many scattering electrons in the ionosphere, the scattered signals (or received voltages) can be treated as Gaussian random variables to a very good approximation, according to Central Limit Theorem [Papoulis, 1986]. Therefore, instead of working directly with the received voltage values, we model their joint statistics in the form of the autocorrelation function (ACF), 〈v(t)equation image〉, where τ represents the time lag, and 〈.〉 and equation image denote the expected value and conjugation operations, respectively.

[13] From (1), the ACF of the received voltage can be expressed as:

  • equation image

where r′ = requation image′ denotes the range r′ along the direction equation image′. Within the integrand above 〈n(r, t)equation image〉 is the space-time ACF of the density fluctuation n(r, t). With the assumption that the fluctuation ACF vanishes rapidly with an increasing magnitude of xr′ − r, we can proceed as:

  • equation image

[14] The inner integral represents a spatial Fourier transform, as follows:

  • equation image

where R(k, r, τ) denotes the ACF of electron density fluctuations at altitude r, time lag τ and the wave vector k(=−2k0) and is directly related to ionospheric parameters [e.g., Farley, 1966]. Inserting (4) into (3) yields:

  • equation image

[15] Equation (5), which demonstrates the relationship between the received voltage and the target in a statistical sense, is called the incoherent scatter (or soft target) radar equation. Inspection of this equation brings us to the two following observations:

[16] 1. Incoherent scatter radar equation holds for each lag of ionospheric ACF independently from other time lags. Therefore, to evaluate the plasma ACF at time lag τ, we need to consider only the same lag of the received signal ACF.

[17] 2. Using an incoherent scatter radar, it is not possible to achieve point estimations of the plasma ACF. Instead, weighted averages of this quantity over a finite range interval are obtained. The weights are merely dependent on the modulated waveform and vary from lag to lag.

[18] The functions describing the averaging operation on the underlying plasma ACF are called soft target radar ambiguity functions, (pτ(t) = V0(t)equation image). These functions essentially indicate that the signal coming from range r contains information from several altitudes, where the altitude interval is equal to the distance covered by the product of the waveform and its shifted version. Note that the 0th lag contains the most range smearing of information. As we move to farther lags, this altitude interval decays as the common part between the pulse and its shifted version diminishes.

[19] Note that in a more general case, the ambiguity functions are dependent on both range and time lag, as developed in [Lehtinen and Huuskonen, 1996]. This dependence on time lag is caused by a non-ideal receiver, whose impulse response contains the time average of previous samples. In this work, however, we assume the receiver has a sufficiently narrow impulse response, and as such the ambiguity in the lag direction is negligible.

[20] Also note that our definition of the soft target ambiguity function differs from that of hard target radar applications where it is defined as [see for example Blahut, 2004]:

  • equation image

In this equation χ(τ, ν) is the hard target ambiguity function in terms of a delay (range) resolution variable, τ, and a Doppler resolution variable, ν. The hard target radar equation is used as a measure of detectability of two hard targets with separation in range and velocity. The difference between the two definitions is due to different filters used in radar receivers. In a hard target application normally a matched filter is used, whereas in soft target radars the received signal is usually over-sampled. Through the rest of this paper we use the term “ambiguity function” to refer to soft target radar applications only.

3. Inversion of Incoherent Scatter Radar Measurements—Proposed Method

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Incoherent Scatter Radar Equation and Ambiguity Functions
  5. 3. Inversion of Incoherent Scatter Radar Measurements—Proposed Method
  6. 4. Numerical Results
  7. 5. Summary and Conclusion
  8. Acknowledgments
  9. References

[21] In this section, we develop an efficient hybrid technique for the inversion of incoherent scatter radar measurements. We exploit both simplicity of height-by-height analysis and accuracy of full-profile methods through considering the full model of ambiguity to present a simple, fast and accurate technique without the limitations of each of these common, currently used methods. For this purpose, we revisit the forward (direct) model of the incoherent scatter process as the form of the 1-D convolution of the ionospheric ACF across range for each lag. We then present the matrix framework of convolution. The inversion technique is then formulated as the deconvolution of the lag profiles followed by the minimization of a least squares cost function. Regularization methods for performing the deconvolution are also discussed.

3.1. Forward Model

[22] This section describes the discretization of the radar equation (section 2) and the corresponding matrix framework of the problem. Let r′ = equation imager and rewrite the radar equation as:

  • equation image

where equation image refers to the altitude from which the signal is received (reference altitude). We discretize and approximate both sides of the above equation by a Reimann sum, as below:

  • equation image

where Δr = equation image, and i and j are indexing terms. Note that i can range from jequation image = jequation image = jNR to j, where NR is the number of altitudes in each range-gate. This discretization is necessary since eventually the data and the resolution of the final parameter grid, Δr, are restricted by the sampling time of the receiver.

[23] Following the notation introduced in section 2, we replace V0equation image in (8) by pτ(i) to obtain:

  • equation image

[24] Note that the above equation describes the relationship between the input and output of a linear time-invariant system, where the ISR ACF at a certain time lag, τ, is the input, and the measured voltage ACF at the same time lag is the output. Moreover, the impulse response of the system is expressed as:

  • equation image

where Δτ represents the time lag increment. Notice that pτ(i) preserves its form over different range gates. Therefore the index i can be considered to vary over a range gate only, rather than being dependent on a particular altitude j.

[25] The above scheme can be visualized as incorporating the effects introduced by the transmitted pulse into a number of low-pass filters, each of which affects only one lag profile. These effects take place in the form of weighted averaging of the lag profiles, where the weights are determined by the ambiguity function at each time lag τ. Thus the filter shape at each lag in the time domain is determined by the product of the pulse shape and its shifted version.

[26] Once we envision range smearing as a filtering system, we can describe the relationship between its input and output, i.e., the plasma ACF at individual altitudes and the measured voltages ACF, as a convolution process (The relationship between input and output of a linear time-invariant system can be expressed by the system impulse response through a convolution sum, as below: y(i) = equation imageh(ij)x(j) = equation imageh(j)x(ij) = h(i) * x(i) for i = −∞, ⋯, +∞ where x(i), y(i) and h(i) are the input, output and impulse response of the system at instant i, respectively. * represents the convolution operator.), that is:

  • equation image

With expansion of (11) with respect to all possible values of t and fixed value of τ, we can represent this convolution relationship in a matrix framework as:

  • equation image

where

  • equation image

[27] Notice that yτ is the vector of measurement ACF at time lag τ from all altitudes (lag profiles). Similarly, mτ includes the altitude profile (for n altitudes) of the true plasma ACF at the same time lag. Pτ is the convolution matrix which relates the measurement ACF and plasma ACF at time lag τ.

[28] A more complete model of ISR measurement can be rewritten as:

  • equation image

Where ετ represents the measurement error at time lag τ. In general this error is signal-dependent especially when the backscattered signals are strong because of high electron densities or a high transmitted power. Because of this dependence, it is not possible to obtain an estimate of lag profiles using only one single transmission. Instead, the measured ACFs should be added for several pulse transmissions to improve statistical accuracy (integration). The data is typically integrated for several seconds. Although signal-dependent, the noise gains Gaussian characteristics as signal is integrated over many pulses (according to Central Limit Theorem). Therefore it can be well described by its mean, μ = 〈ετ〉, and covariance matrix, equation image = 〈ετετT〉. A nonzero mean would indicate a bias. Without loss of generality the mean can be considered as zero since a bias is simply a constant term that can be taken into account. The covariance matrix will be diagonal if the errors in different lags are independent. Otherwise it will have non-zero off diagonal elements.

3.2. Inverse Model

[29] In the previous sections we established the relationship between the plasma ACF at individual altitudes and the received voltage ACF as a convolution process, where the shape of the convolving function is dependent on the pulse envelope as well as the time lag values. We exploit this property in our proposed inversion method, a detailed description of which is presented in the following subsections.

3.2.1. Deconvolution

[30] The major motivation for the deconvolution of the lag profiles is to remove the range smearing from the measured signal ACF and obtain the plasma ACF at single altitudes. The elimination of range ambiguity allows us to use analytical derivatives of the theoretical ACF lags with respect to ionospheric parameters, as opposed to forward differences, in least squares optimization algorithms and hence to reduce the computational cost significantly. Furthermore, when using deconvolution methods, one does not require the imposition of unrealistic assumptions on the parameter profiles, such as stationarity for the whole range-gate as it is the case in height-by-height analysis.

3.2.1.1. Inverse Filtering and Least Squares Solutions

[31] We model the measured signal ACF as the convolution of plasma ACF with the corresponding ambiguity functions at various time lags. In the frequency domain, this convolution amounts to a multiplication of the plasma spectra with the Fourier transform of the ambiguity functions. Therefore a naive approach to the ISR deconvolution (known as inverse filtering, e.g., Blahut [2004]) would be to divide the spectra of the measured signal by the Fourier transform of the ambiguity function, Pτ(f), as:

  • equation image

where Yτ(f), Mτ(f) and equation imageτ,inv(f) represent the Fourier transforms of altitude profiles of the measured signal, true plasma and estimated plasma ACFs at time lag τ in the frequency domain, respectively. The inverse filter Pτ−1(f) is the inverse of the Fourier transform of the ambiguity function, i.e., Pτ−1(f) = equation image.

[32] The advantage of the inverse filter is that it requires only the convolution filter as a priori knowledge, and it allows for perfect deconvolution in the absence of noise. However, several problems exist with this inversion technique. First, the inverse filter may not exist because Pτ(f) could be zero at selected frequencies. Second, the convolution filter may take very small values, close to zero, at some frequencies causing the noise term to dominate the estimated signal in equation (14) [Lagendijk and Biemond, 2000; Karl, 2000; Blahut, 2004].

[33] In situations where the inverse filter does not exist or the signal power spectrum is not available, the reasonable solution is to apply least squares techniques to the convolution problem, yτ = pτ * mτ + ετ, or equivalently yτ = Pτmτ + ετ in the matrix framework. In this method, the estimate is defined as the least squares fit to the observed data:

  • equation image

where ∣∣yτPτmτ∣∣2 = (yτPτmτ)T(yτPτmτ) and arg denotes the argument producing the minimum (as opposed to the minimum itself).

[34] In cases where the data points are not equally reliable or independent, we need to consider the effect of the data covariances through the inclusion of the covariance matrix by using weighted least squares objective functions, defined as follows:

  • equation image

where ∣∣yτPτmτ∣∣equation image2 = (yτPτmτ)Tequation image−1(yτPτmτ) and equation image is the error covariance matrix of the altitude profiles of the measured signal ACF at time lag τ.

[35] Nevertheless, similar to the inverse filtering technique, the least squares method does not provide a stable solution since it amplifies noise in data. To overcome the noise sensitivity of the inverse filter and least squares techniques, we can use the Wiener filter which provides the optimal linear estimator in the sense of mean square error (MSE) between the ideal and the deconvolved profiles. In the spectral domain, the Wiener filter can be described as [see, e.g., Lagendijk and Biemond, 2000]:

  • equation image

where HWiener(f) denotes the frequency response of the Wiener filter, equation image represents the complex conjugate of Pτ(f), and equation image(f) and equation image(f) represent the power spectrum of the ideal lag profile and noise, respectively.

[36] For frequencies where equation image(f) ≪ equation image(f) the Wiener filter approximates the inverse filter:

  • equation image

On the other hand, for the frequencies where equation image(f) ≫ equation image(f), the Wiener filter acts in favor of suppression of noise according to equation (17).

[37] The main difficulty associated with Wiener filter is that this method requires the knowledge about the spectral or second-order statistics of both noise and signal. Although various methods have been developed to estimate the power spectrum of the unknown signal from the measured signal [see, e.g., Jain, 1981; Lagendijk and Biemond, 2000], the signal-dependency of the method makes it less favorable. In what follows, we present a systematic view to the problem of noise suppression in the deconvolution through regularization techniques.

3.2.1.2. Regularization

[38] The drawbacks of the inverse filtering and least squares methods mentioned previously (such as instability in the face of perturbations to the data), raise the need for regularization. Through regularization, we impose a priori knowledge about the underlying process to stabilize the solution in the presence of noise and to permit the identification of physically reasonable estimates of parameters of interest. A regularization method can be considered as a modified least squares technique, where the modifications appear in the form of additional constraints to the residual norm defined in (15) as side constraint norms. More precisely, we can represent the regularized estimate as the solution to the following minimization problem:

  • equation image

where λi and Ci are the ith regularization parameter and regularization functional, respectively. The first term controls data fidelity (i.e., how closely the solution fits the data), whereas the second term (the regularization term) controls how well the solution matches our prior knowledge. The role of regularization parameter can be viewed as controlling the tradeoff between the impact of data and the impact of a priori knowledge on the solution.

[39] The most common regularization method is the Tikhonov regularization with a quadratic functional [Demoment, 1989; Karl, 2000]. The general expression for the Tikhonov method is

  • equation image

where the most common choice for L is the discrete representation of the gradient operator enforcing a roughness penalty and hence a smoothness constraint. As an example, the discretized first-order gradient operator can be represented as:

  • equation image

In this case ∣∣Lmτ∣∣2 is a measure of variability of the estimate. Therefore the overall functionality of the method can be visualized as penalizing large gradients of the solution, resulting in smoother lag profiles where the degree of smoothness depends on the value of regularization parameter.

[40] Although inclusion of other types of regularization functionals is certainly possible, this quadratic functional provides considerable improvements in the quality of deconvolved lag profiles. The solution to the minimization in (19), i.e. the Tikhonov regularized estimate, can be obtained as the solution to the following set of equations:

  • equation image

[41] Although the above deterministic view of regularization provides an expression for the estimate, it lacks the ability to provide measures of the solution uncertainty. To obtain the estimation errors, we need to cast the inversion problem into a statistical framework by choosing an appropriate prior statistical model for mτ and seeking a maximum a posteriori probability (MAP) estimate [Demoment, 1989; Kamalabadi et al., 1999].

[42] In order to apply the MAP estimation to our inversion problem (equation (13)), we first need to view the noise, ετ, and the unknown lag profile, mτ, as random vectors. Then the MAP estimate of mτ can be expressed as the vector which maximizes the posteriori density p(mτyτ). Using Bayes rule and the monotonicity properties of the logarithm, we obtain:

  • equation image

where ln(.) represents the natural logarithm. Notice that, similar to the deterministic optimization objective function (19), the above optimization problem consists of two terms: the first term is data-dependent and is called the log-likelihood function while the second term is dependent only on mτ and is called the prior model.

[43] Let us now consider the case of Gaussian statistics for noise (see section 3.1) and prior model:

  • equation image
  • equation image

where mτ ∼ ��(0, equation image) denotes that mτ is a Gaussian distributed random vector with mean 0 and covariance equation image. Under these Gaussian assumptions and upon substitution in (23) we obtain:

  • equation image

[44] Comparison of equations (20) and (26) yields that they are equivalent under certain condition. This equality condition is only related to the prior model, since the data fidelity terms are equal. The appropriate prior model corresponds to

  • equation image

Where L is a discrete approximation of the gradient operator. According to this prior model, mτ itself is Gaussian distributed with covariance equation image = λ2(LTL)−1. In addition, the increments of mτ are uncorrelated with variance λ2I, thus mτ itself corresponds to a Brownian motion type of model.

[45] We can now provide an expression for the associated measure of uncertainties, using the statistical framework and through the error covariance matrix Σequation image = 〈eτeτT〉, where eτ = mτequation imageτ. It has been shown that for the MAP estimate in (23) the error covariance is given by [Demoment, 1989]

  • equation image

Notice that the diagonal entries of Σequation image are the variances of individual estimation errors. This error covariance matrix is used as an uncertainty level of the deconvolved lag profiles in the next step of our proposed method, which is the nonlinear least squares optimization procedure. We still need to consider the choice of the regularization parameter as it has a direct effect on the solution.

3.2.1.3. Choice of Regularization Parameter

[46] Regularization inherently involves the trade-off between the fidelity to the data, as measured by residual norm ∣∣yτPτequation imageτ∣∣, and fidelity to a priori information, as measured by the side constraint norm. Since regularization parameter, λ, controls this trade-off, an important part of any regularization technique is to find a reasonable value for λ. While various methods of choosing a proper regularization parameter exist (like visual inspection, discrepancy principle, generalized cross-variation techniques, etc. Karl [2000]), here we focus on the L-curve method, which does not require the knowledge of the noise characteristics.

[47] Since regularization parameter controls the relative weight of the residual and side constraint norms, it would seem natural to choose a regularization parameter on the basis of the behavior of these two terms as λ varies. Such a plot of side constraint norm versus the residual norm as λ varies has a characteristic L shape (hence, the name L-curve) and has been considered as a means of choosing regularization parameter [Hansen, 1998]. An L-curve consists of vertical and horizontal parts, where the vertical part corresponds to unregularized estimates (where the solution is dominated by the amplifying noise and small changes in λ has a large effect on the solution), and the horizontal part corresponds to overregularized solutions. In this region, changes to λ affect the solution weakly, but produce large changes in the residual error.

[48] The regularization parameter is chosen to be the corner between the horizontal and vertical portions of the curve. This corner defines the transition between the over- and underregularization and thus represents the balance between these two extremes.

3.2.2. Nonlinear Least Squares (NLLS) Optimization

[49] Once the range smearing effect of the transmitted waveform is removed via deconvolution, we perform LS optimization to estimate ionospheric parameters. We minimize the following objective function for each altitude separately:

  • equation image

where m is the deconvolved (plasma) ACF at a certain altitude and is obtained from mτ for all values of τ, a represents the parameter vector (electron temperature (Te), ion temperature (Ti), electron density (Ne), ion composition (p),…) and g is the nonlinear incoherent scatter function which relates the ionospheric parameters to the plasma spectrum, or equivalently its autocorrelation function. Moreover, Σequation image is the error covariance matrix of the deconvolved ACF and is different from that of the measured voltage lag profiles (convolved ACF) in (13). In any nonlinear least squares estimation, the errors play an important role since they provide information about the reliability of each data point. In the case of independent and equal errors, the error covariance matrix can be written as Σequation image = σ2I, where σ2 denotes the error, and equation (29) can be simplified as:

  • equation image

The situation, however, becomes more complicated when the errors are correlated, as it is the case for plasma ACF measurements. In order to make (29) as simple as (30) in the general case of dependent errors, we need to factor out the inverse of deconvolved ACF error covariance matrix (Σequation image) by Cholesky decomposition into matrices D and DT such that Σequation image−1 = DTD. With this factorization, (29) can be rewritten as:

  • equation image

which has a similar structure to that of (30) and hence is easy to implement.

4. Numerical Results

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Incoherent Scatter Radar Equation and Ambiguity Functions
  5. 3. Inversion of Incoherent Scatter Radar Measurements—Proposed Method
  6. 4. Numerical Results
  7. 5. Summary and Conclusion
  8. Acknowledgments
  9. References

[50] In this section we investigate the performance of the hybrid technique on a numerical experiment and compare the corresponding estimation results with those of common methods currently used in terms of mean-squared error and computation time.

[51] In data simulation, we take into account the fluctuations of both the incoherent scatter signal and the receiver noise by simulating the received signals. The data simulation algorithm is summarized as follows:

[52] 1. Generate the altitude profiles of the ionospheric parameters. For simplicity, we restrict the ions in the ionosphere to oxygen only, which is a realistic assumption in the lower F region and around the height of peak electron density. Figure 3 shows an example of three parameter profiles (profiles of electron and ion temperatures and electron density) used in the simulations.

image

Figure 3. Profiles of a hypothetical ionosphere used in the simulations. For simplicity, this ionosphere is composed of a single ion species. This is a realistic assumption in the lower F region and near the height of peak electron density.

Download figure to PowerPoint

[53] 2. Generate noisy voltage samples on the basis of the parameters at each altitude and the transmitted pulse shape, which itself requires the following steps:

[54]    a. Form the incoherent scatter spectrum at individual altitudes on the basis of parameter profiles.

[55]    b. Generate two sets of random spectra based on of the square root of the spectrum at each altitude. With this configuration, the generated signal will have stronger variations whenever the spectrum itself is stronger (consideration of signal fluctuation).

[56]    c. Form the complex spectra.

[57]    d. Transform into the time domain.

[58]    e. Multiply the complex ACF by the pulse shape.

[59] 3. Add voltage samples from different heights.

[60] 4. Add random noise to the voltage samples generated in (3) (This random noise represents sky and receiver noise and is independent from the signal).

[61] 5. Form the lag-profiles.

[62] 6. Accumulate over many pulse transmissions.

[63] Note that a radar frequency of 430 MHz and an uncoded long pulse of length 280 μs are used in simulations. The lag spacing is set to 10 μs, and the data is accumulated over 12000 pulse transmissions, which yields an error of about 4% in the lags of the measured ACF.

[64] In the implementation of the hybrid technique, we apply Tikhonov regularization with the first-order gradient operator (refer to (21)) on the lag profiles in the first step. The deconvolved lag profiles are thus obtained by solving the set of equations denoted in (22) for each time lag (with integer multiple of lag spacing (Δτ)) individually. From various approaches that exist to solve the equations in (22), we use conjugate gradient method, which is a fast converging iterative method [Golub and Van Loan, 1989]. In the second step of the method, the Levenberg-Marquardt algorithm [Levenberg, 1944; Marquardt, 1963] is used as the NLLS optimization procedure.

[65] The proper value of the regularization parameter, λ, is achieved by means of L-curve shown in Figure 4 along with the proper regularization parameter (λ ≃ 25.6), identified by the dashed lines. Figure 5 illustrates the estimated parameter profiles for this regularization parameter. While using smaller values of this parameter results in a much larger variation in the estimated parameters, using larger values introduces biases into the estimation results (see Figure 6). The ionospheric parameter estimation errors are also plotted in Figures 7a7c as a function of regularization parameters, where the errors are measured as equation image with aij and equation imageij denoting the ith ideal and estimated ionospheric parameter at altitude j, respectively, and n the number of altitudes. According to this figure, there is a range of regularization parameters over which the parameter estimation errors are minimum (or within 10% of the minimum) and the proper regularization parameters selected by L-curve fits well in this range. Figure 7d also illustrates the computation time required to complete the analysis on a 2.4 GHz workstation.

image

Figure 4. L-curve for Tikhonov regularization applied in the first step of the proposed method. As the regularization parameter increases, the curve moves from the upper left to the lower right. The best choice of the regularization parameter corresponds to the point of maximum curvature (corner) of the L-curve. This point (λ = 25.6) is identified by the dashed lines.

Download figure to PowerPoint

image

Figure 5. The proposed hybrid method estimation results (thick curves). Original profiles are represented by the thin lines. The regularization parameter has been chosen on the basis of the corner point in Figure 4.

Download figure to PowerPoint

image

Figure 6. Estimated ion and electron temperature profiles for small and large regularization parameters (thick curves). The thin curves show the original profiles. While using a very small value of this parameter results in a much larger variation in the estimated parameters, using a large value introduces biases into the estimation.

Download figure to PowerPoint

image

Figure 7. Performance of the hybrid technique (in terms of estimation error and computation time) as regularization parameter varies. The estimation errors for the regularization parameter selected by L-curve are denoted by dashed lines.

Download figure to PowerPoint

[66] The height-by-height estimation results, which are obtained by the application of triangular weighting prior to LS optimization, are illustrated in Figure 8. Note that the fitting is performed only at parameter grids specified by squares, and the estimated parameter profiles (dotted curves) are produced by linear interpolations on these grids. The coarse resolution achieved by this method is evident from the plots.

image

Figure 8. Height-by-height analysis estimation results. Note that the LS fitting is performed only at parameter grids specified by squares. The estimated parameter profiles (dotted curves) are produced by linear interpolations on these grids. The original profiles are given by solid lines.

Download figure to PowerPoint

[67] The optimal parameter estimation results of the full-profile technique is illustrated in Figure 9. The resolution of the parameter grid in this optimal case is 26 km. The optimality is determined on the basis of the performance of the technique over a wide range of parameter grid resolutions. The plots in Figures 10a10d illustrate the errors in estimated parameters as well as the timing required to complete the analysis (on a 2.4 GHz workstation) for parameter grids with spatial resolutions from 10 to 60 km (corresponding to 30- to 6-point parameter grids). These errors are minimum (or very close to minimum) over a limited range of spatial grid resolutions. This range of optimal grid densities varies slightly for different ionospheric parameters. However, in general we can consider the interval between 21 and 29 km to be within 10% from the optimal. For comparison, we also plot the estimated temperatures for the two spatial resolutions of 57 km and 12 km in Figure 11. The high variation of the estimated parameters from the true parameters in both cases, because of underfitting or overfitting of the solution to the noisy data, is evident from the plots.

image

Figure 9. Optimal full-profile estimated parameters obtained for a parameter grid with 26-km spatial resolution (dotted curves). Original profiles are represented by solid lines.

Download figure to PowerPoint

image

Figure 10. Performance of the full-profile technique (in terms of estimation error and computation time) as a function of parameter grid resolution.

Download figure to PowerPoint

image

Figure 11. Full-profile estimated temperatures for two different grid parameters (spatial resolution of 57 km in Figure 11a and 12 km in Figure 11b). The original profiles are represented by solid lines. The high variation of the estimated parameters from the true parameters in both cases due to underfitting or overfitting of the solution to the noisy data is evident from the plots.

Download figure to PowerPoint

[68] From Figures 7a7c, 8 and 10a10c, it is clear that the hybrid technique is able to provide faithful estimates of the ionospheric parameters that are nearly as good as the ones given by the full-profile technique (provided that the optimal regularization parameter and grid density are chosen) while it outperforms the height-by-height method in terms of estimation accuracy. Two main differences between the performances of the two methods, however, can be inferred from the plots in Figures 7d and 10d. First, the hybrid method requires nearly constant computational time for various regularization parameters, unlike the full-profile technique where the analysis time is linearly dependent on the parameter grid density. Second, the hybrid technique reduces the computation time to nearly 95%. The restriction of the optimization search space to the number of ionospheric parameters at each range and hence the ability to utilize analytical derivatives of the ACF lags with respect to the physical parameters serves as the main key to the efficiency of the method.

[69] Note that in our simulation setup, the errors in lag estimates become highly correlated. However, we include only the variances of the errors (diagonal elements of the error covariance matrix) in the above analysis as the computation of the full covariance matrix is not tractable, especially as the number of altitudes increases. The estimation errors noted in Figures 7a7c and 10a10c have been obtained numerically using Monte-Carlo-type simulations. Therefore no underestimation of errors occurs as is the case when errors are obtained analytically using the diagonal error covariance matrix [Huuskonen and Lehtinen, 1996]. Nevertheless, the high correlations in the parameter profiles estimated by the hybrid method are in part consequences of ignoring these error covariances.

5. Summary and Conclusion

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Incoherent Scatter Radar Equation and Ambiguity Functions
  5. 3. Inversion of Incoherent Scatter Radar Measurements—Proposed Method
  6. 4. Numerical Results
  7. 5. Summary and Conclusion
  8. Acknowledgments
  9. References

[70] In this work, we have presented an efficient approach to the estimation of ionospheric parameters from incoherent scatter radar measurements. The proposed hybrid method is based on removing range smearing of information across the range and performing nonlinear least squares optimization on the retrieved ACFs at individual altitudes. To stabilize the deconvolution solution in the presence of noise we have utilized regularization techniques, where the proper regularization parameter has been found by means of L-curve. Through simulations, we have demonstrated that once the optimal grid density and regularization parameters are chosen, the hybrid technique achieves nearly equal estimation quality as the full-profile technique in a fraction of computational cost. Nevertheless, the parameter profiles estimated by the proposed method show high correlations from one altitude to the next. These correlations are in part consequences of ignoring error covariances of the lag-profiles. Developing coding schemes other than a simple long-pulse transmission which may improve the quality of estimation by lowering these correlations is the subject of a future study.

References

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Incoherent Scatter Radar Equation and Ambiguity Functions
  5. 3. Inversion of Incoherent Scatter Radar Measurements—Proposed Method
  6. 4. Numerical Results
  7. 5. Summary and Conclusion
  8. Acknowledgments
  9. References
  • Blahut, R. (2004), Theory of Remote Image Formation, Cambridge Univ. Press, New York.
  • Demoment, G. (1989), Image reconstruction and restoration: Overview of common estimation structures and problems, IEEE Trans. Acoust. Speech Signal Process., 37, 20242036.
  • Dougherty, J., and D. Farely (1960), A theory of incoherent scattering of radio waves by a plasma, Proc. R. Soc. London Ser. A, 259, 79.
  • Dougherty, J., and D. Farely (1961), A theory of incoherent scattering of radio waves by a plasma. II, Proc. R. Soc. London Ser. A, 263, 238.
  • Farley, D. (1966), A theory of incoherent scattering of radio waves by a plasma 4. The effect of unequal ion and electron temperatures, J. Geophys. Res., 71(17), 40914098.
  • Golub, G., and C. F. Van Loan (1989), Matrix Computations, Johns Hopkins Univ. Press, Baltimore, MD.
  • Hagfors, T. (1961), Density fluctuations in a plasma in a magnetic field, with applications to the ionosphere, J. Geophys. Res., 66(6), 16991712.
  • Hagfors, T. (1971), A theory of collision dominated electron density fluctuations in a plasma with applications to incoherent scattering, Phys. Fluids, 14, 1143.
  • Hansen, P. C. (1998), Rank-Deficient and Discrete Ill-Posed Problems, SIAM, Philadelphia, PA.
  • Holt, J., D. Rhoda, D. Tenebaum, and A. van Eyken (1992), Optimal analysis of incoherent scatter radar data, Radio Sci., 27, 435.
  • Huuskonen, A., and M. Lehtinen (1996), The accuracy of incoherent scatter measurements: Error estimates valid for high signal levels, J. Atmos. Terr. Phys., 58, 453.
  • Jain, A. K. (1981), Advances in mathematical models for image processing, Proc. IEEE, 69, 502.
  • Kamalabadi, F., W. C. Karl, J. L. Semeter, D. M. Cotton, T. A. Cook, and S. Chakrabarti (1999), A statistical framework for space-based EUV ionospheric tomography, Radio Sci., 34, 437448.
  • Karl, W. (2000), Regularization in image restoration and reconstruction, in Handbook of Image and Video Processing, edited by A. C. Bovik, pp. 141160, Academic Press, Orlando, FL.
  • Kudeki, E. (2003), ECE 458 Lecture Notes, Application to Radio Wave Propagation, Univ. of Illinois, Urbana, IL.
  • Lagendijk, R. L., and J. Biemond (2000), Basic methods for image restoration and identification, in Handbook of Image and Video Processing, edited by A. C. Bovik, pp. 125139, Academic Press, Orlando, FL.
  • Lehtinen, M., and A. Haggstrom (1987), A new modulation principal for incoherent scatter measurements, Radio Sci., 22, 625.
  • Lehtinen, M., and A. Huuskonen (1996), General incoherent scatter analysis and GUISDAP, J. Atmos. Terr. Phys., 58, 435.
  • Lehtinen, M., A. Huuskonen, and J. Pirttila (1996), First experiences of full-profile analysis with GUISDAP, Ann. Geophys., 14, 1487.
  • Levenberg, K. (1944), Method for the solution of certain problems in least squares, Quart. Appl. Math, 2, 164.
  • Marquardt, D. (1963), Algorithm for least squares estimation of nonlinear parameters, SIAM J. Appl. Math, 11, 431.
  • Papoulis, A. (1986), Probability, Random Variables and Stochastic Processes, McGraw-Hill Book, New York.
  • Sulzer, M. (1986), A phase modulation technique for a sevenfold statistical improvement in incoherent scatter data-taking, Radio Sci., 21, 737.
  • Vallinkoski, M. (1988), Statistics of incoherent scatter multiparameter fits, J. Atmos. Terr. Phys., 50, 839.