[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).

[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

where *n*_{0}(**r**, *t*) is a background model of the ionosphere, and *u*(**r**, *t*) 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 of*u*(**r**, *t*) 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 vector*U* is related to the vector of available measured quantities *Y*:

[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* = 〈*ηη*^{T}〉 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 *M*_{LE}[*U*]. We also need to augment *L* with *L*_{LE} = *δM*_{LE}[*U*]/*δu* which is the linearization of *M*_{LE}. 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 *L*_{LE}can 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

where *P*_{2}denotes the group path for two-way propagation from the transmitter to ground surface and back to the receiver,*n*denotes the three-dimensional distribution of electron density,*θ*^{Rx} and *β*^{Rx} are the elevation and steer angles at the receiver. In view of (1) there is a straightforward relation between GPR and *L*_{LE}. 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 *n*_{ijk}over a discrete three-dimensional grid, the functional derivative turns into a vector of partial derivatives:*δP*_{2}/*δn* ∂*P*_{2}/∂*n*_{ijk}, 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 (*x*, *y*, *z*):

[16] Here *P*^{Tx} is the group path from transmitter to the scatterer, *θ*^{Tx} and *β*^{Tx} are elevation and steer angles at the transmitter, *P*^{Rx} is the group path from the receiver to the scatterer, and *θ*^{Rx} and *β*^{Rx} 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 *P*_{2} = *P*^{Tx} + *P*^{Rx} one can obtain

[18] Here and 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 and 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 *θ*^{Rx}, *β*^{Rx}. In other words, the above mentioned functional derivatives may be expressed from the following equation:

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

[20] Finally, by utilizing the identities

(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:

[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 *L*_{LE} 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].