[9] The MHD tomography analysis is based on the concept of CAT analysis developed by Kojima et al. [1998]. However, to introduce MHD code into the CAT analysis, some additional procedures and assumptions are required.
[10] The region to be treated is outside of a sphere with a radius of 50 Rs. This choice of the inner boundary distance is made for two reasons. One is that the IPS measurements at the STEL radio frequency (327Mhz) has sensitivity for r ≥ 50 Rs and the other is that the MHD simulation code becomes very simple when only the superAlfvénic region is treated.
[11] The MHD tomography analysis is an iterative procedure modifying the solar wind velocity distribution on the inner boundary sphere. The modification of the boundary velocity is made in three steps. In the first step, MHD simulation of solar wind is carried out using a provisional distribution on the inner boundary at 50 Rs, and the steady state of the threedimensional MHDbased solar wind at r ≥ 50 Rs is calculated (described in section 2.1). In the second step (described in section 2.2), the IPS velocity observations are simulated in this numerical solar wind, and the discrepancies between the velocities from the IPS simulations and the actual IPS observations are calculated for each LOS. In the last step (described in section 2.3), the velocity distribution on the inner boundary surface is modified such that the discrepancies between numerical and actual IPS velocities are reduced.
[12] The iterations can be started from any arbitrary boundary distribution. We use a uniform solar wind speed distribution of 500 km/s. The process of modification is repeated until the IPS velocity discrepancies become sufficiently small. The threedimensional numerical solar wind obtained is MHDbased and best matches IPS observations.
2.2. Simulation of IPS Velocity Measurement in Numerical Solar Wind
[28] The IPS velocity measurement is simulated in the numerical threedimensional solar wind obtained by the MHD simulation, and the discrepancy between the actual and simulated IPS velocities is evaluated. If the discrepancy is not small enough, the discrepancy amount is used to modify the flow speed distribution on the inner boundary surface.
[29] The IPS simulation is made with the integration [Kojima et al., 1998],
where is the velocity vector of the solar wind plasma flow observed in the rest frame, is the position vector of a point on the LOS at the distance z from Earth, and w′ is the normalized weight. The square of electron density fluctuation (δn_{e})^{2} is the weighting factor of radio diffraction, and the other weighting factor w(z) is made by using the Born approximation and taking the Fresnel propagation filter and the size of the radio sources into account [Young, 1971]:
where q is a wavenumber, Λ is the wavelength of observation (92 cm at STEL IPS observation, corresponding to 327 MHz), θ_{0} is the angular size of radio source assumed to be 0.1 arcsecond, and a cubic powerlaw spectrum q^{−3} for the fluctuations is assumed.
[30] To complete equation (17), the electron density fluctuation must be determined. Although the density fluctuation in reality may depend on not only the local solar wind conditions but also its origin in the solar corona, we utilize the powerlaw dependence of the density fluctuation on the density and flow speed found by Asai et al. [1998]. Although Asai et al. classified the data into three flow speed classes and derived powerlaw dependences for each class, the present analysis assumes the power index is constant and equal to the average of Asai et al.'s powerlaw indices,
The LOS integration of equation (17) is made for the regions within 1 AU, because the weighting factors at r ≥ 1 AU are sufficiently small to be negligible.
2.3. Modification of Boundary Map of Solar Wind Flow Speed
[31] The modification of the flow speed on the inner boundary, V_{r}(50 Rs,θ,ϕ), is made such that the discrepancy of IPS velocity, ΔV ≡ V_{sim} − V_{obs}, will be reduced. To assign the position and guess the amount of modification of boundary velocity, the discrepancy ΔV and the normalized weight w′(z) on LOS are traced back to the inner boundary sphere along the MHDbased streamline with RungeKutta method. Thus projected ΔV and w′(z) are restored in the numerical lattice around the foot point of the streamline together with other weights w_{spread} and w_{err}.
[32] The weight w_{spread} is a Gaussian spread function around the source position,
where η is the angular distance from the foot point of the streamline to the concerned numerical lattice. The typical angular size η_{0} is determined from the transverse travel distance of an acoustic wave. Assuming the typical distance from LOS to the inner boundary ΔR_{typical} to be 80 solar radii, the typical sound speed c_{typical} to be 50 km/s, and typical solar wind speed V_{typical} to be 400 km/s, the angular size of domain of influence on the inner boundary sphere η_{d.i.} is approximated as
Since the numerator and denominator of c_{typical}/V_{typical} are in positive correlation, η_{d.i.} is rather insensitive to the conditions. We use this value as the constant halfwidth of the spread function. This angular width is approximately twice as large as the angular interval of the computational grid and the spread function helps to avoid numerical instability in the MHD simulation.
[33] Another weight w_{err} is a function of observational error V_{err} that is evaluated at the time when the observational IPS velocity is calculated from the time lags of IPS signals among four STEL IPS sites. The typical V_{err} is 10 ∼ 20 km/s for a typical solar wind of 500 km/s. In this study this weight is defined as a function of the ratio of V_{err} to the observed IPS velocity V_{obs},
It should be mentioned here that IPS velocity data with V_{err}/V_{obs} ≥ 0.3 are not used in the analysis.
[34] This procedure is done for all actual IPS observations during the time period concerned, and the amount of velocity modification at each lattice (j, k) is calculated as
where the integer m and l denote the digit position on LOS and the IPS observation number, respectively.
[35] Finally, the flow speed at the lattice (j, k) is modified by
The positive factor α(≤1) prevents overshoot of the solution. We set α = 0.8 in the present analysis.
[36] To examine the degree of agreement between the derived and observed IPS data, the average of the discrepancy
and the standard deviation
are checked during the analysis. Table 1 tabulates the values of and σ(ΔV) obtained in the analysis for Carrington rotation (CR) 1844, as an example. It is observed that rapidly decreases and approaches zero and σ(ΔV) decreases to 10% or better level of the average IPS velocity _{obs}. Since the discrepancies evaluated in the final iteration are generally sufficiently small, we can say that the derived threedimensional solar wind structure matches the IPS observations well.
Table 1. Average and Standard Deviation of the Discrepancy Between the Actual and Synthetic IPS Velocities^{a}Trial Time  1  2  3  4  10  20 


 16.65  0.96  −3.00  −1.12  −0.04  −0.04 
σ(ΔV)  100.07  56.05  48.73  45.40  39.06  37.96 
$\ \overline{V_{\rm obs}}$  438.55  438.55  438.55  438.55  438.55  438.55 
[37] The standard deviation σ(ΔV) is also used in the data selection. Because IPS observation detects not only the steady component of solar wind but also transient components like propagating disturbances and because the actual solar wind varies during one Carrington rotation, some of IPS data are not suitable for the present purpose of reconstructing the quiet component of solar wind. First, we carry out a provisional analysis with five iterations and then discard the IPS velocity data whose discrepancy is greater than 1.5sigma,
After this data selection, the iterative process is repeated.