Decomposition of variables in horizontal and vertical modes
Multiplicative decomposition as defined in the post is often called factorization. It is not always possible to achieve an exact factorization. There would be some residual difference that we could denote as eps(x, y, z, t).

U(x, y, z, t) = uv(z) * uh(x,  y,  t) + eps(x, y, z, t)


Such a decomposition is not unique, however you may choose a pair of functions that minimizes the value of eps(x, y, z, t) according to some metric.

For further reading consider the description of sklearn Non-negative matrix factorization function.