The Electron Density Assimilative Model (EDAM) has been developed to provide real-time characterizations of the ionosphere by assimilating diverse data sets into a background model. Techniques have been developed to assimilate virtual height ionogram traces rather than relying on true height inversions. A test assimilation has been conducted using both GPS and ionosonde data as input. Postassimilation analysis shows that foF2 residuals can be degraded when only GPS data are assimilated. It has also been demonstrated that by using both data types it is possible to have low total electron content and foF2 residuals and that this is achieved by modifying the ionospheric slab thickness.
 The Electron Density Assimilative Model (EDAM) has been developed to provide real-time characterizations of the ionosphere. It comprises a suite of programs that manages disparate ionospheric data sets and assimilates them into a background ionospheric model. EDAM exploits optimal data assimilation techniques that have been developed in the meteorological community over the past few decades. The philosophy has been to design a system that will operate on a single personal computer (PC); that will continue to provide physical results with very sparse data; and from which products can be derived for a range of radio frequency (RF) systems (e.g., radars, communication systems).
 EDAM can assimilate a variety of different types of data: (1) ground-based slant total electron content (TEC) derived from global navigation satellite systems (GNSS) data, (2) space-based TEC derived from GNSS data (both in a radio occultation geometry and using navigation antennas), (3) space-based TEC derived from dual-frequency altimeters, (4) vertical electron density profiles from ionosondes and incoherent scatter radars, and (5) electron density data from in situ sensors. Details of EDAM and of its testing are given by Angling , Angling and Khattatov , and Angling et al. .
 This short note describes how EDAM has been modified to assimilate virtual height traces measured by ionosondes. Assimilating the virtual height profiles avoids having to apply a true height inversion algorithm before assimilation. This note also provides details of a short test conducted to demonstrate the use of the new assimilation routines and the behavior of the model when data from both GPS and ionosondes are assimilated together. The testing is limited, in that postassimilation residuals are examined rather than independent truth data. However, the aim is not to examine the absolute accuracy of the model; rather it is to investigate how the different data types interact to modify the vertical structures in the EDAM electron density grid.
2. The Electron Density Assimilative Model
2.1. Model Description
 The Electron Density Assimilative Model (EDAM) has been developed to assimilate measurements into a background ionospheric model. This background model is provided by IRI2007 [Bilitza and Reinisch, 2008] and the majority of the input data are TEC measurements derived from International GNSS Service (IGS) stations [Beutler et al., 1999]. The assimilation is based on a weighted, damped least mean squares estimation. This is a form of minimum variance optimal estimation (also referred to as Best Linear Unbiased Estimation (BLUE)) that provides an expression for an updated estimation of the state (known as the analysis) that is dependent upon an initial estimate of the state (the background model) and the differences between the background model and the observations [Menke, 1989; Twomey, 1977]. The error covariance matrices of the background model and the observations are also used to control the relative contributions of the background and the observations to the analysis:
where xa is the analysis, xb is the background model, K is the weight matrix, y is the observation vector, B is the background error covariance matrix, and R is the error covariance matrix of the observations [Rodgers, 2000]. H is the nonlinear observation operator that relates the measurements to the state:
where ɛ is the observation error. The observation operator is nonlinear because, in EDAM, the background model is comprised of the log of the ionospheric electron density. H is the Jacobian, whose elements are given by the partial differentials of the observation operator evaluated at the background model; i.e.,
The assimilation is conducted using a tilted dipole magnetic coordinate system that remains fixed in space with respect to the sun. An assimilation time step of 15 min has been used and the electron density differences between the voxels of the analysis and the background model are propagated from one time step to the next by assuming persistence combined with an exponential decay. The time constant for this decay is set at four hours. Thus if the data feed is interrupted, the analysis will decay back to the background model.
2.2. Virtual Height Assimilation
 In the meteorological data assimilation community it is generally accepted that data should be assimilated in a form that requires the least amount of preprocessing. In keeping with this philosophy, new assimilation routines have been developed for EDAM that allow the assimilation of virtual height traces from ionograms. The direct assimilation of virtual height traces avoids having to apply a true height inversion algorithm such as POLAN [Titheridge, 1988] or NHPC [Reinisch et al., 2005]. Indeed, studies have shown systematic differences between these inversion techniques [Sauli et al., 2007]. The true height profile differences arise from differences in the underlying ionospheric models that are used to constrain the inversion in areas that are not sampled by the ionosonde (i.e., the E-F valley region). By assimilating the virtual height profile directly, the resultant true height profiles remain consistent with the EDAM model which may, of course, have been influenced by other data sources.
 The assimilation of virtual height ionogram traces proceeds stepwise through each frequency in the measurement. The assimilation relies on the fact that each frequency penetrates slightly deeper into the ionosphere than the last (Figure 1). This is due to reflection occurring at the height where the transmission frequency is equal to the plasma frequency. Furthermore, as the assimilation progresses, the variances (i.e., diagonal terms of the background error covariance matrix) associated with the lower layers of the background model are progressively reduced so that the largest modifications of the model are constrained to be close to the reflection point.
 In order to incorporate the assimilation of virtual height ionograms into EDAM both an observation operator (i.e., a way of synthesizing the measurement from an ionospheric model) and the Jacobian (the partial derivative of the observation operator) are required.
2.2.1. Observation Operator
 A vertical ionogram consists of the virtual height of reflection plotted as a function of frequency. Therefore, the observation operator must act on the EDAM electron density grid to produce an estimate of virtual height for a particular frequency. The virtual height can be expressed as the integral of the group refractive index between the ground and the reflection point [Budden, 1985]:
where h′ is the virtual height, n′ is the group refractive index, f is the probe frequency, fn is the plasma frequency and z0 is the reflection height. Neglecting collisions and the magnetic field allows the group refractive index to be expressed in terms of electron density. Therefore, equation (5) (and hence the observation operator) can also be expressed in terms of the electron density. Neglecting the magnetic field can produce significant absolute errors in the vertical profile. For instance, the difference between the o and x mode critical frequencies (foF2 and fxF2) can approach 1 MHz for this station. However, for the purposes of this test where we are interested in the residuals between the measurements and the models and in demonstrating how the vertical structure in the model can be modified, neglecting the magnetic field is acceptable.
 The Jacobian is the partial differential of the observation operator with respect to the background model. It consists of two terms: the first due to direct modification of the electron density; and the second due to variations in the reflection height. If the plasma frequency is assumed to vary linearly between voxel centers and that reflection occurs where the plasma frequency is equal to the transmission frequency, then increasing the plasma frequency around the reflection point will act to reduce the reflection height, while decreasing the plasma frequency will increase the reflection height (Figure 2). Given that the plasma frequency is known as a function of height and that the reflection height occurs when the frequency is equal to the plasma frequency, it is possible to write expressions for the rate of change of the reflection height with respect to the plasma frequency.
3. Test Assimilation
 The new assimilation routines have been tested using a network of ten GPS receivers and one ionosonde (EB040) located in Spain (Figure 3). The ionosonde data have been manually scaled by the Ebre Observatory (http://www.obsebre.es). Assimilations have been conducted using just GPS input data, just ionosonde virtual height data, and with both data types. The assimilations have been run for 14 February 2007. In each case, EDAM is cold started at 00:00 UT; i.e., no prior assimilations are conducted to allow time for convergence.
 Postassimilation foF2 and TEC residuals are shown in Figures 4 and 5 respectively for the Ebre ionosonde (EB040) and the Ebre IGS station (EBRE). Results for IRI2007 have been included for reference (Figures 4a and 5a; note that this is the background model used by EDAM). Figures 4 and 5 also show the results when only GPS data are assimilated (Figures 4b and 5b); when only the ionosonde data are assimilated (Figures 4c and 5c); and when both GPS and ionosonde data are assimilated (Figures 4d and 5d). The root-mean-square errors of the residuals for each model run are given in Table 1.
Table 1. RMS Errors of the TEC and foF2 Residualsa
RMS TEC Residual (TECu)
RMS foF2 Residual (MHz)
Note that assimilating both GPS and ionosonde data provides the best overall performance.
GPS + ionosonde
 True height and virtual height vertical profiles have also been extracted from EDAM above the ionosonde. Sample results for 03:00 UT, 06:00 UT, and 11:00 UT are shown for IRI (Figure 6); EDAM when only GPS data are assimilated (Figure 7); EDAM when only the ionosonde data are assimilated (Figure 8); and EDAM when both GPS and ionosonde data are assimilated (Figure 9).
 For the day under test, IRI is generally negatively biased in TEC at night and positively biased in TEC by day. Furthermore, the IRI foF2 is also positively biased during the day, though the nighttime results are more consistent with the observations.
 As expected, the residual TEC exhibited by EDAM is low when GPS data are assimilated. However, when GPS data alone are assimilated, the result can be to degrade the foF2 performance. For example, at 03:00 UT, the postassimilation TEC residual is low for EDAM, but the foF2 residual has increased in comparison to IRI. Across the day, the RMS error of the foF2 residual increases from 0.6 to 0.7 MHz if only GPS data are assimilated. This foF2 degradation can also be seen in Figure 7 where, although the vertical profiles remain physically plausible, the F region peak has been grossly overestimated.
 When only ionosonde data are assimilated, the EDAM residual foF2 is low (Figure 4c), but the residual TEC is largely unchanged from that of IRI (Figure 5c). The benefits of using both data types become apparent when both are assimilated: both the residual TEC and foF2 are simultaneously low (Figures 4d and 5d). The individual RMS errors (Table 1) are not the lowest when both data types are used, but the combination of data provides overall performance. This result is achieved by a modification of the thickness of the F2 layer; that is, the ionospheric slab thickness (the ratio of the TEC to NmF2) has been changed. This modification is produced naturally by the interaction of the two data types and the example vertical profiles in Figure 9 shows that this is done without any major discontinuities in the electron density profile.
 Routines to assimilate virtual height traces from ionosondes in to EDAM have been developed. The assimilation has been tested using a network of GPS stations and a single ionosonde in Spain. Results indicate that postassimilation foF2 residuals can be degraded when only GPS data are assimilated. However, by assimilating both types of data, EDAM can effectively modify the ionospheric slab thickness resulting in simultaneously low TEC and foF2 residuals.
 EDAM has been developed under funding from the United Kingdom Ministry of Defense Science and Technology program. GPS data were obtained from the SOPAC Data Centre. Differential code biases were obtained from the Centre for Orbit Determination in Europe. The vertical ionosonde data were manually scaled by staff at the Ebre Observatory, Spain, and obtained from the University of Massachusetts Lowell DIDBase (http://umlcar.uml.edu/DIDBase/).