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Keywords:

  • inverse problem;
  • ionosphere;
  • radar

Abstract

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Merging of Backscatter Soundings With GPSII
  5. 3. Performance Evaluation
  6. 4. Conclusions
  7. Appendix A:: Derivation of RPR From Ray-Tracing Equations
  8. Acknowledgment
  9. References

[1] Over-the-horizon radar (OTHR) uses ionospheric reflection to propagate HF transmissions to long range (∼500–5000 km). The ionosphere acts as a dynamic “mirror” that varies diurnally, seasonally, and with the solar cycle. Geolocation of targets observed by OTHR (Coordinate Registration (CR)) requires accurate real-time ionospheric modeling and HF propagation calculations to convert radar-measured target signal delays and beam steers to geographical position. We merged our backscatter ionogram (BI) leading edge inversion algorithm CREDO with our more advanced ionospheric data assimilation capability, GPS Ionospheric Inversion (GPSII). The combined algorithm produces a dynamic model of electron density for a fixed geographical region. The model is consistent with BI leading edge data, vertical sounding data, as well as with absolute and relative total electron content (TEC) data from a number of GPS/LEO receivers. Incorporation of additional ionospheric data beyond conventional OTHR vertical and oblique backscatter soundings is expected to enhance the fidelity of real-time ionosphere models, resulting in improved OTHR Coordinate Registration metric accuracy. Initial tests of the OTHR CR supported by the new ionospheric inversion algorithm indicate noticeable improvement of CR accuracy in comparison with legacy techniques.

1. Introduction

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Merging of Backscatter Soundings With GPSII
  5. 3. Performance Evaluation
  6. 4. Conclusions
  7. Appendix A:: Derivation of RPR From Ray-Tracing Equations
  8. Acknowledgment
  9. References

[2] Over-the-horizon radar (OTHR) achieves wide area surveillance with long-range coverage (∼500 to 5000 km) by making use of the ionospheric reflection of high-frequency (HF) signals [Headrick and Skolnik, 1974; Headrick, 1990]. The ionosphere, however, is a dynamic “mirror” that varies diurnally, seasonally, and with the solar cycle. This makes the problem of geolocating OTHR targets difficult. Geolocation of targets observed by OTHR is referred to as Coordinate Registration (CR). Good CR requires an accurate model of the real-time ionosphere as well as high-fidelity HF propagation calculations to convert radar-measured target signal delays and beam steers to geographical position.

[3] Previously our team developed an OTHR CR capability called CREDO (Coordinate Registration Enhancement by Dynamic Optimization). The inversion of backscatter ionograms (BI) simultaneously with more conventional measurements [Fridman, 1998; Fridman and Nickisch, 2001; Nickisch et al., 1998] is an important block of CREDO. This inversion technique is a development of the methodology suggested by Fridman and Fridman [1994]. A review of alternative approaches to inverting of BI data can be found in this paper.

[4] CREDO uses OTHR vertical and oblique backscatter soundings to model the ionosphere by applying Tikhonov's methodology [Tikhonov and Arsenin, 1977] for solving ill-posed problems. We extended the methodology to multidimensional nonlinear inverse problems and adjusted it for fast numerical solution. The application of Tikhonov Regularization produces the smoothest ionosphere that is in agreement with the input data to within the data measurement error.

[5] More recently we created a Tikhonov-based ionospheric data assimilation capability called GPSII (GPS Ionospheric Inversion; pronounced “gypsy”) [Fridman et al., 2006, 2009]. GPSII is capable of ingesting data from GPS and LEO satellite beacons, in situ electron density (e.g., DMSP or CHAMP satellites), the Jason altimeter, the DORIS system, and vertical incidence sounders to derive a three-dimensional ionosphere model that is both spatially and temporally smooth, but is yet in agreement with all the input data to within the data measurement error.

[6] In this paper we address incorporation of oblique backscatter sounding data into the GPSII inversion algorithm. In other words we are extending GPSII with CREDO capabilities. The combined algorithm produces a dynamic model of electron density for a fixed geographical region. The model is consistent with a stream of BI leading edge data, vertical sounding data, as well as with streams of absolute and relative TEC data from a number of GPS/LEO receivers. Incorporation of additional ionospheric data beyond conventional OTHR vertical and oblique backscatter soundings is expected to enhance the fidelity of real-time ionosphere models, resulting in improved OTHR CR accuracy. We perform initial tests of the OTHR CR supported by the new ionospheric inversion algorithm.

2. Merging of Backscatter Soundings With GPSII

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Merging of Backscatter Soundings With GPSII
  5. 3. Performance Evaluation
  6. 4. Conclusions
  7. Appendix A:: Derivation of RPR From Ray-Tracing Equations
  8. Acknowledgment
  9. References

[7] Backscatter soundings are routinely performed by OTHR for propagation assessment. Figure 1shows a CREDO display of an OTH radar sounding (gray scale). The quasi-vertical ionogram (QVI) is in the upper left pane. The backscatter ionograms (BIs) for eight azimuthal directions occupy the remaining panes, one for each sounding sector. Extent of the horizontal axis for each BI pane is from 5 MHz to 28 MHz, while extent of the vertical axis is from 0 to 4940 km (in terms of one-way group path). Occasional bright vertical streaks are caused by interference from narrowband HF broadcasters. BI panes include overlays of the selected leading edge data (green boxes), CREDO fit to the selected data (red dots), and theoretical group delays for rays emitted at a set of frequencies with 0.5 degree step in elevation (yellow dots).

image

Figure 1. CREDO inversion result for a QVI and eight BIs as displayed by CREDO interface. See explanation of details in the text.

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[8] The leading edge information of BIs (minimum delay as a function of frequency) can be reliably extracted from the data and is employed by CREDO to produce a three-dimensional model for the downrange ionosphere. Our intention here is to extend GPSII to ingest the BI leading edge data in addition to all previously accommodated types of ionospheric measurements.

[9] Our approach to solving the ionospheric inverse problem is inherited from the GPSII algorithm [Fridman et al., 2006, 2009]. We represent the three-dimensional, time varying distribution of electron density in the ionosphere as

  • display math

where n0(rt) is a background model of the ionosphere, and u(rt) is an arbitrary function which will be determined as a result of the inversion procedure. The numerical solution will be performed over a four-dimensional spatial-temporal grid. The vector of values ofu(rt) in all nodes of this grid will be denoted U(we are using bold face only for vectors in three-dimensional physical space). The unknown vectorU is related to the vector of available measured quantities Y:

  • display math

[10] Here M is the measurement operator. It is a known nonlinear operator that relates the ionospheric model (which is specified by U) to theoretical estimates of each of the measured quantities, and vector η represents the noise of measurements. We will assume that the noise covariance matrix of measurement errors S = 〈ηηT〉 is known.

[11] The task of the ionospheric inverse problem is to resolve equation (2) with respect to U. In order to solve it within the GPSII framework it is necessary to provide L, the linearized version of the measurement operator M : L = δM[U]/δu. Once procedures for calculating the vector M[U] and the linear operator L are in place, the unknown vector of ionospheric modification U may be found iteratively using Tikhonov's regularization technique with the residual principle as described by Fridman et al. [2006, section 2.2]. This inversion technique takes into account errors of measurements characterized by the covariance matrix S and ensures that diverse data types (such as TEC and BI leading edge delays) are ingested simultaneously and with proper weights.

[12] Thus, in order to incorporate the BI leading edge data into GPSII we need to provide means for calculating the theoretical leading edge delay for any ionosphere represented by (1). This will augment M[U] with the components representing leading edge samples MLE[U]. We also need to augment L with LLE = δMLE[U]/δu which is the linearization of MLE. The former task is solved with the aid of numerical ray tracing [Coleman, 1997]. The later task is discussed in the next section.

2.1. Linearized Group Path Response Operator

[13] Principles for determining the response of the leading edge group path to infinitesimal variations of the ionospheric model were developed earlier by Fridman and Fridman [1994], Fridman [1998], and Fridman and Nickisch [2001]. It was demonstrated that the matrix approximation to the response operator LLEcan be obtained from solution of extended ray-tracing equations. The extended equations are conventional ray tracing equations [Haselgrove, 1957] augmented with derivatives of ray parameters over initial conditions (see Appendix A). Here we intend to formulate relationships that will allow for extending the algorithm to bistatic radar configurations.

[14] For a single leading edge data sample we introduce the linear group path response (GPR) operator as

  • display math

where P2denotes the group path for two-way propagation from the transmitter to ground surface and back to the receiver,ndenotes the three-dimensional distribution of electron density,θRx and βRx are the elevation and steer angles at the receiver. In view of (1) there is a straightforward relation between GPR and LLE. The functional derivative (3) is a Frechet derivative [Kolmogorov and Fomin, 1999] in the space of differentiable functions of three variables. The derivative establishes a linear functional defined over this space. Note that in the practical case when the field of electron density is approximated by a table nijkover a discrete three-dimensional grid, the functional derivative turns into a vector of partial derivatives:δP2/δn [RIGHTWARDS ARROW] ∂P2/∂nijk, where i, j, and k are the integers numbering nodes of the spatial grid.

[15] We desire to express GPR for backscatter propagation in terms of components of a more general linear raypath response (RPR) operator. This operator characterizes a two-way raypath associated with a localized scatterer placed at a fixed location (xyz):

  • display math

[16] Here PTx is the group path from transmitter to the scatterer, θTx and βTx are elevation and steer angles at the transmitter, PRx is the group path from the receiver to the scatterer, and θRx and βRx are elevation and steer angles at the receiver. Algorithms for estimating RPR were created earlier [Knepp et al., 2010]. Fundamentals of evaluating RPR from extended ray-tracing equations are outlined inAppendix A.

[17] In view of (3) and taking into account that P2 = PTx + PRx one can obtain

  • display math

[18] Here inline image and inline image represent derivatives of the group path to the backscattering surface with respect to components of the ray launch direction. We will henceforth assume that the backscattering surface is defined by the equation z = 0. Such derivatives are considered here as known quantities because they are easily expressible in terms of components of the state vector of extended ray-tracing equations.

[19] The functional derivatives inline image and inline image should be calculated keeping in mind that we are dealing with backscattering from the Earth's surface. This means that the exit direction from the transmitter is determined by the requirement that the landing point of the transmit ray tracks the landing point of the ray traced from the receiver in the direction θRx, βRx. In other words, the above mentioned functional derivatives may be expressed from the following equation:

  • display math

where x and y are two coordinates specifying position of ray landing point on the ground.

[20] Finally, by utilizing the identities

  • display math

(these expressions are applicable to rays traced from the transmitter as well as to rays traced from the receiver) it is possible to transform (5) to the desired form:

  • display math

[21] This equation expresses GPR in terms of components of RPR and components of the state vector of extended ray-tracing equations.

[22] Expression (8) allows us to employ previously created algorithms for the present task of estimating the matrix LLE which represents the linearized measurement operator for leading edge data samples. This matrix is subsequently incorporated into the linearized measurement operator L employed for reconstructing the ionosphere within GPSII framework [Fridman et al., 2006]. We will call the new inversion algorithm GPSII-CREDO.

2.2. GPSII-CREDO

[23] GPSII-CREDO is an inversion algorithm, which combines the major capabilities of our earlier algorithms GPSII and CREDO. This algorithm is able to fit simultaneously various types of ionospheric data: GPS TEC, vertical sounding or QVI data, and one-hop leading edges of backscatter ionograms. For computational reasons the extended ray tracing equations employed by the version of GPSII-CREDO presented here are not fully magnetoionic. Plausible magnetoionic corrections are introduced by applying a small shift to wave frequency. The amount of shift is estimated from the Appleton formula in the high-frequency approximation [Davies, 1990]. Direct comparison with fully magnetoionic ray tracing has shown that resulting errors of leading edge estimation are small (several kilometers) for the whole frequency band of conventional BIs; for comparison, uncertainties in automatic detection of the leading edge itself are about 30 km.

[24] Figure 2 illustrates one example of the solution sequence. This inversion started at 1203UT and continued with a step of 12 min. Each step typically involved fitting about 100 new TEC data samples, one vertical profile (from the QVI sounder), and about 40 leading edge samples (from 8 sectors of the BI sounder). Figure 2presents a map of the critical frequency resulting from the inversion compared to the output of the climatological model (IRI 2007) at 1503UT (approximately 1003LT). Dotted lines are isolines of critical frequency. Locations of the contributing TEC receivers are marked by triangles and crosses indicate instantaneous ionospheric pierce points at the altitude of 400 km for all contributing satellite-receiver paths. The coverage sector of the BI sounder is indicated by the black dashed line. The QVI vertical profile is measured near the apex of this sector. Solar and geomagnetic environments were quiet on this day (SSN = 10, Kp = 3). Nevertheless the assimilated measurements induce considerable modification of the climatological model. In this example GPSII-CREDO predicts consistently lower critical frequencies over the whole region. The maximum deviation from IRI reaches about 30% within this frame. This deviation appears to be within typical error of IRI predictions obtained by comparing model predictions with data of vertical sounding [see, e.g.,Bertoni et al., 2006].

image

Figure 2. Maps of the critical frequency. (top) The climatological model compared to (bottom) the GPSII/BI inversion. See explanation of details in the text.

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3. Performance Evaluation

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Merging of Backscatter Soundings With GPSII
  5. 3. Performance Evaluation
  6. 4. Conclusions
  7. Appendix A:: Derivation of RPR From Ray-Tracing Equations
  8. Acknowledgment
  9. References

[25] The performance of the ionospheric reconstruction algorithms is evaluated using real OTH radar measurements of targets or beacons with ground truth. In order to evaluate the consistency of our ionospheric models with OTHR observations of beacons we compare OTHR-measured slant coordinates (that is, the slant range and steer) with the slant coordinates obtained via ray tracing through our ionospheric models. In other words, given the predicted ionosphere we can find theoretical rays that connect the OTHR transmitter and receiver with the transponder and to obtain theoretical values for slant range and steer. Discrepancies between theoretical and measured slant coordinates provide good estimates for expected CR errors.

[26] In order to accomplish the above plan we created a stand-alone code for calculating slant coordinates of targets positioned at known ground coordinates. This code accomplishes ray homing to a specified ground location from both receiver and transmitter sites and plausibly takes into account magnetoionic effects in the high-frequency approximation as it was outlined in 2.2. This approximation is well justified at the frequencies of interest (around 16 MHz); direct comparison with fully magnetoionic ray tracing showed errors within 1 km. The ray tracing is performed in three dimensions through the ionospheric model produced by GPSII-CREDO. The output consists of values of slant coordinates (slant range and steer angle at the receiver) for all two-way propagation modes connecting transmitter, target, and receiver.

[27] We tested the algorithm against observations of a transponder in Jamaica illuminated by an OTHR in Virginia. The observations took place on 25 September 2006 from 2051UT to 2400UT. Figure 36below illustrate the performance of the algorithm with different combinations of ionospheric measurements. The slant space validation ray tracing was performed at every time step of the GPSII-CREDO inversion (that is every 12 min, consistent with the OTHR cycle of ionospheric soundings).

image

Figure 3. Slant coordinates for detections of the Jamaica transponder by an OTHR in Virginia (crosses) compared to predictions (circles) obtained using ray tracing through the uncorrected climatological model of the ionosphere (IRI-2007). The slant azimuth is shown in terms of the sine of the radar steer angle as measured from the radar boresight.

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image

Figure 4. Same as Figure 3but predictions (circles) are obtained using ray tracing through the GPSII inversion which incorporated data from a dozen ground-based GPS-TEC receivers and a Virginia QVI sounder.

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image

Figure 5. Same as Figure 3but predictions (circles) are obtained using ray tracing through the GPSII-CREDO inversion which incorporated time series of data from the Virginia QVI and BI sounders.

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image

Figure 6. Same as Figure 3but predictions (circles) are obtained using ray tracing through the GPSII-CREDO inversion which incorporated time series of data from the Virginia QVI and BI sounders and a dozen ground-based GPS-TEC receivers.

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[28] Figure 3is the baseline example obtained using the IRI-2007 climatological model without any other ionospheric data. Slant azimuth data are shown in terms of the sine of the radar steer angle as measured from the boresight plane of the linear antenna array. Note that the climatological model predicts that there is no propagation to the transponder at 23:15 and later. The RMS errors between measurements and predictions are indicated in each plot. In evaluating the RMS error we select theoretical modes which are closest to the observations.

[29] Figure 4shows that incorporation of the data from a dozen of GPS-TEC receivers and the QVI sounder improves propagation prediction in comparison with the purely climatological model (Figure 3). Indeed, the corrected model predicts existence of propagation for a longer interval of time. The RMS error for predicted slant range improves only slightly.

[30] Figure 5shows that the GPSII-CREDO algorithm driven by BI and QVI data demonstrates further improvement over both the purely climatological model (Figure 3) and the TEC-driven algorithm (Figure 4). Indeed, the corrected model predicts existence of propagation for the whole time interval of interest. The RMS error for predicted slant range improves noticeably.

[31] Finally, results for the GPSII-CREDO algorithm driven by all available ionospheric data (BI/QVI/TEC) are presented inFigure 6. We observe improvement over the purely climatological (Figure 3) or TEC/QVI-driven predictions (Figure 4), but there is deterioration in slant range error (from 9 km to 13 km) in comparison with the BI/QVI driven algorithm (Figure 5).

[32] We would like to stress that the above results represent performance of the new algorithm (GPSII-CREDO) which is considerably more advanced than CREDO. One distinction is that CREDO can utilize only BI/QVI data while GPSII-CREDO can as well assimilate data from TEC receivers. Another important advance in GPSII-CREDO is that it utilizes time sequences of ionospheric data, exploiting the notion of continuity of the ionosphere in time. Thus the algorithms are substantially different even when driven by the same BI/QVI data. As a point of comparison, for example, the original CREDO code yielded a slant range misfit for the Jamaica transponder of 28.1 km for the point inFigure 5at 2203UT, which is improved in the GPSII-CREDO case to 10.5 km. Improvement in the slant azimuth was not as dramatic: from 0.0028 to 0.0024 sine units (approximately from 5.6 to 4.8 km in cross range). This observation suggests that the GPSII-CREDO algorithm has potential to considerably improve CR over the original CREDO algorithm.

4. Conclusions

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Merging of Backscatter Soundings With GPSII
  5. 3. Performance Evaluation
  6. 4. Conclusions
  7. Appendix A:: Derivation of RPR From Ray-Tracing Equations
  8. Acknowledgment
  9. References

[33] We have developed an ionospheric inversion algorithm that combines capabilities of our earlier algorithms, GPSII and CREDO. This algorithm is able to fit simultaneously various types of ionospheric data: TEC, vertical sounding data, and one-hop leading edges of backscatter ionograms. The resulting three-dimensional ionospheric model is suitable for performing accurate numerical ray tracing and for assisting in the OTHR CR process.

[34] We performed initial validation tests of the ionospheric models produced by the new inversion algorithm. These tests appear to indicate that in terms of OTHR CR accuracy the new algorithm promises substantial improvement in comparison with its predecessors.

Appendix A:: Derivation of RPR From Ray-Tracing Equations

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Merging of Backscatter Soundings With GPSII
  5. 3. Performance Evaluation
  6. 4. Conclusions
  7. Appendix A:: Derivation of RPR From Ray-Tracing Equations
  8. Acknowledgment
  9. References

[35] The RPR operator (4) is composed of contributions from two propagation legs (from transmitter to the scatterer and from receiver to the scatterer). So it is sufficient to describe equations for just one leg of the path.

[36] The ray-tracing equations [Haselgrove, 1957] have the following general form:

  • display math

[37] Here X = [x1x2x3, … xN]T, N ≥ 6 is the ray state vector, and t is the independent variable of the raypath (propagation time, for example). We indicated in (A1)that the right-hand sideF = [f1f2, … fN]T is generally a functional of the electron density n(r). For convenience we will adopt that x1, x2, and x3 specify spatial position of the ray, x4 and x5 specify direction of the wave normal, and x6 is the group path. Within this section we will use capital letters for vectors and matrices, and corresponding small letters for elements of these matrices.

[38] The extended ray tracing equations are formed by augmenting (A1) with the following linear equation:

  • display math

where R(t) and B are N × N matrices, bij = ∂ fi/∂ xj, and R(t) is initialized as the identity matrix: R(0) = I. Note that expression (A2) consists of N replicas of the linearized version of equation (A1) (one set of linearized equations per column of R).

[39] Equations (A1) and (A2) can be solved simultaneously for a set of exit angles so that the exit directions that home the landing point to a desired location in space can be determined. Let tL be the value of t at the landing point for the homed solution of interest. Given such solution, we can construct the Green function V(tt′) of the initial value problem for the linearized version of equation (A1):

  • display math

[40] Then the Green function for the boundary value problem (subject to the constraint that the ray end points are fixed in space) is

  • display math

where [r1:N,4:5(t)] is a submatrix of the matrix R(t), and [d1:2,1:N(t′)] is a submatrix of the following 3 × N matrix:

  • display math

[41] Now functional derivatives of both components of the ray exit direction as well as the functional derivative of the group path to the landing location can be expressed in terms of known quantities:

  • display math
  • display math

where the second term in the square brackets of (A4) accounts for variation of tL.

[42] The following two properties of functional derivatives may be helpful for evaluating δF/δn in (A3) and (A4). Suppose that a functional φ[n] amounts to a simple function (φ[n] ≡ φ(n(r))). Then inline image, where inline imageis the delta function in three-dimensional space, andφ′(n) = dφ(n)/dn. The following property is useful for handling terms in the ray-tracing equations that contain spatial derivatives of electron density: inline image.

References

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Merging of Backscatter Soundings With GPSII
  5. 3. Performance Evaluation
  6. 4. Conclusions
  7. Appendix A:: Derivation of RPR From Ray-Tracing Equations
  8. Acknowledgment
  9. References