  The Hodge numerical invariants of a variation of Hodge structure over a
smooth quas--projective variety are a measure of complexity for the global
twisting of the limit mixed Hodge structure when it degenerates. These
invariants appear in inequalities which they may have correction terms, called
Arakelov inequalities. One may investigate the correction term to make them
into equalities, also called Arakelov equalities. We investigate numerical
Arakelov type (in)equilities for a family of surfaces fibered by curves. Our
method uses Arakelov identities in a weight 1 and also in a weight 2 variations
of Hodge structure (cf. \cite{GGK}), in a commutative triangle of fibrations.
We have proposed to relate the degrees of the Hodge bundles in the two
families. We also compare the Fujita decomposition of Hodge bundles in these
fibrations. We examine various identities and relations between Hodge numbers
and degrees of the Hodge bundles in different levels.
