The construction of a general parameterization for inverse analysis of welding processes using incomplete or minimal experimental data, although demonstrated in principle, has yet to be firmly established. The structure of such a parameterization follows from the concepts of model and data spaces that imply the existence of an optimal parametric representation for a given class of inverse problems. Accordingly, the corresponding optimal parametric representation lies in the model space and is determined by the characteristics of the available data sets spanning the associated data space and the nature of the data sampling for purposes of parameter determination via the appropriate optimization methods. Data adopted for inverse heat-transfer analysis of welds includes solidification cross-sections, thermocouple measurements and microstructural changes. We discuss here the sensitivity of temperature histories, obtained by inverse analysis, with respect to shape characteristics of longitudinal solidification cross-sections, along the symmetry plane of the weld, which are strongly correlated with convective heat transfer, but are difficult to determine experimentally. Accordingly, we consider the sensitivity of temperature histories with respect to assumptions concerning the shapes of longitudinal solidification cross-sections. An understanding of the nature of this sensitivity is significant for establishing the feasibility of a general parameterization of particular types of welding processes.