# Whitney stratification for proper morphisms

Let $$f: X \to \Delta$$ be a flat, projective morphism, smooth over the punctured disc $$\Delta^*:=\Delta \backslash \{0\}$$ and central fiber $$f^{-1}(0)$$ is a reduced, simple normal crossings divisor. Does there exist a Whitney stratification of $$X$$ with one strata consisting of $$f^{-1}(\Delta^*)$$? For example, in the case $$X$$ is of relative dimension $$1$$ (i.e., $$X$$ is a family of curves) and the central fiber is the union of two smooth, irreducible curves $$Y_1, Y_2$$, is $$\{f^{-1}(\Delta^*)\} \coprod \{(Y_1 \cup Y_2)\backslash (Y_1 \cap Y_2)\} \coprod \{Y_1 \cap Y_2\}$$ a Whitney stratification of $$X$$? Any idea/reference will be most welcome.

EDIT: Assume $$X$$ is a smooth, complex manifold.