[25] The method of auxiliary sources is a numerical technique suitable for a variety of scattering problems [*Uberall et al.*, 1987]. The boundary conditions are imposed only on a discrete number of points of the physical boundaries, called collocation points. The scattering fields in each area satisfy the Helmholtz equation because they are written as weighted finite sums of the local Green's functions. These functions correspond to auxiliary sources posed on a set of points. The sources are located outside the area whose field they generate so that the investigated quantities inside it are smooth as a result of scattering. Sometimes it is convenient to group adjacent regions and to employ the modified Green's functions which incorporate the additional boundary conditions [*Leviatan et al.*, 1983]. If the forms of the boundary quantities from both sides of the physical surfaces coincide being converging with increasing number of auxiliary sources, then the approximate field is very close to the unique solution of the problem.

[26] The method of auxiliary sources will be applied by regarding the cylindrical scatterer as one region and the remaining area (stepped discontinuous waveguide with two layers) as another. That is why the expression of the Green's function *G*(*z*, *x*, *z*′, *x*′) determined in sections 2–4 is required. The single magnetic component in the remaining area (superscript *G* in parentheses) is written as:

where (*z*_{Gu}, *x*_{Gu}), *u* = 0, ., (*U* − 1) are the auxiliary points of the sources generating the field of the remaining area. They are not contained in it but are located into the cross section of the cylindrical scatterer (which has been conditionally removed). It is obvious from (35) that the computation of the field outside the scatterer requires repeated implementation (for each auxiliary source) of the spectral integration and the mode matching procedures described previously. In the same way, the single axial magnetic component inside the dielectric obstacle (superscript *g* in parentheses) is given by

where the quantity *g*(*z*, *x*, *z*′, *x*′) represents the Green's function of the bound-free area in case it is filled by the material of the scatterer.

The auxiliary sources generating the field inside the underground formation are posed on the points (*z*_{gu}, *x*_{gu}), *u* = 0, ., (*U*-1) located outside of the cross section of the scatterer. The coefficients *C*_{Gu} and *C*_{gu} can be determined through the enforcement of the boundary conditions.

[27] Owing to the circular shape of the cylinder, the auxiliary surfaces (lines in two dimensions) are chosen to be circles with radii *a*_{g} > *a* for the field inside the formation and *a*_{G} < *a* for the field in the remaining area. The auxiliary points are distributed uniformly on the circles as in Figure 3. For the external circle *z*^{2} + *x*^{2} = *a*_{g}^{2} the points are defined by *z*_{gu} = *a*_{g} cos, *x*_{gu} = *a*_{g} sin. For the internal circle *z*^{2} + *x*^{2} = *a*_{G}^{2} the auxiliary points are given by *z*_{Gu} = *a*_{G} cos, *x*_{Gu} = *a*_{G} sin with *u* = 0, .(*U*-1). The continuity of the tangential field components to the scatterer's bound *z*^{2} + *x*^{2} = *a*^{2} will be imposed on discrete equispaced points *z*_{u} = *a* cos, *x*_{u} = *a* sin. The azimuthal electric field (*E*_{ϕ}) is given in terms of the single magnetic component (*H*_{y}) by:

where ε is the local dielectric constant and ϕ = atan2(*x*, *z*). The coefficients *C*_{Gu} and *C*_{gu} of (36), (35) will be determined by the following 2*U *× 2*U* linear system:

for *v* = 0, ., (*U* − 1). The circles with radii *a*_{G} and *a*_{g} are placed far enough from the actual surface to obtain smoother results and better fitting of the boundary quantities. Simultaneously, *a*_{G}, *a*_{g} are chosen close enough to *a* for the sake of stability of the linear system above. From the convergent formulas satisfying the boundary conditions, one can evaluate the produced field on the observation point at horizontal distance *z*_{o} from *x* axis (on the ground surface *x* = *W*).