Consider the following setting: suppose that $X$ is a smooth variety and let $f:X\rightarrow \Delta$ be a smooth morphism outside the origin $0$. Let the central fiber $X\_0$ be a reduced (Cartier) divisor with simple normal crossings. Here a reduced divisor is a Weil divisor $D=\sum\_{i}n\_i D\_i$ with all $n\_i=1$. Take some successive blowing ups along the singular locus of $X\_0=\sum \_{i\in I}X\_i$ to get a proper modification $\mu:V\rightarrow X$ from a smooth variety $V$ to $X$. We then write the total transform$\mu^\*{X\_0}=\sum \_{i\in I}\mu\_\*^{-1}X\_i+\sum\_{j\in J}k\_j V\_j^\prime$ with $k\_j\geq 1$. Here, we can ensure the strict transform $\sum \_{i\in I}\mu\_\*^{-1}X\_i$ to be smooth.


Here is my question, can we determine the ramification divisor of the morphism $\mu$? Namely, if we write $$K\_V\equiv\mu^\*K\_X+\text{Ram}(\mu),$$
then what is $\text{Ram}(\mu)$, should it be written as $t\_j V^\prime\_j$? If so, what is the relationship between $t\_j$ and $k\_j$?



Determine the coefficient of the exceptional divisor
Ok, here's an expanded version of what I said in the comment.


SNC case
--------


First, I'm going to create a special log resolution of the pair $(X, X\_0)$. We assume $d = \dim X$. We are assuming that $X$ is nonsingular and $X\_0$ is simple normal crossings.


We begin by letting $S\_{n}$ denote the stratum of this pair of dimension $n$. In other words, $S\_0$ is the points that are intersections of components of $X\_0$, $S\_1$ is the curves that are components of $X\_0$, etc. Let $Y^0 \to X$ be th blowup of all the points in $S\_0$. This separates the strict transforms of the components of $S\_1$. We let $S\_{1}^0$ denote the set of those strict transforms (likewise with $S\_{2}^0$, etc). Let $E\_{0,i}^0 \subseteq Y\_0$ denote the set of exceptional divisors.


Now, we can write
$$
K\_{Y^0} = \pi\_0^\* K\_X + (d-1) \sum E\_{0,i}^0
$$
since we are blowing up nonsingular centers of codimension $d$ in a nonsingular variety. We also ahve
$$
\pi\_0^\* X\_0 = (\pi\_0)^{-1}\_\* X\_0 + d \sum E\_{0,i}^0.
$$
since each $E\_{0,i}^0$ maps to a point in $S\_0$ which is the intersection of $d$ components of $X\_0$.


Next, we blowup the curves in $S\_1^0$ (these are curves in $Y^0$). Note, each curve may intersect some $E\_{0,i}^0$, but it cannot contain any, and likewise no $E\_{0,i}^1$ contains any curve in $S\_1^0$.. Let $E\_{1, i}^1$ denote the exceptional divisors of this new blowup and $E\_{0,i}^1$ the strict transforms of the old exceptional divisors. In this case we have
$$
K\_{Y^1} = \pi\_{1,0}^\* K\_{Y^0} + (d-2) \sum {E\_{1,i}^1} = \pi\_1^\* K\_X + (d-1) \sum E\_{0,i}^1 + (d-2) \sum E\_{1,i}^1.
$$
Here we crucially used the fact that no element of the the $S\_1^0$ were contained in $E\_{0,i}^0$.
By a similar consideration, we also have
$$
\pi\_1^\* X\_0 = (\pi\_1)^{-1}\_\* X\_0 + d \sum E\_{0,i}^1 + (d-1) E\_{1,i}^1.
$$


Continuing in this way, we eventually obtain a log resolution that separates the components of $X\_0$, $\pi : Y \to X$ and
$$
K\_{Y} = \pi^\* K\_X + (d-1) \sum E\_{0,i} + \dots + 1 \sum E\_{d-2,i}
$$
where the $E\_{0,i}$ are the strict transforms of the exceptional divisors from $Y^0 \to X$, $E\_{1,i}$ are the strict transforms of the exceptional divisors from $Y^1 \to Y^0$, etc.
and also
$$
\pi^\* X\_0 = (\pi)^{-1}\_\* X\_0 + d \sum E\_{0,i} + (d-1) \sum E\_{1,i} + \dots + 2 \sum E\_{d-2, i}
$$
In other words for each $E$ an excpetional divisor on $Y$ (or an element of the strict transform), the coefficient of $\pi^\* X\_0$ is exactly one bigger than the coefficient in the relative canonical divisor (what you called Ram).


SLC case
--------


Let's now assume that $X\_0$ is semi-log canonical (SLC) but not necessarily SNC. In this case, since $X\_0$ is Gorenstein, this is equivalent to it being Du Bois (see work of Kovács and Doherty). It is also equivalent to the pair $(X, X\_0)$ being log canonical by inversion of adjunction.
Anyways, since $(X, X\_0)$ is log canonical, for any resolution of singularities, $\pi : V \to X$, if we write
$$
K\_V + D = \pi^\* (K\_X + X\_0)
$$
we know that all the coefficients of $D$ are $\leq 1$ (this is the definition of log canonical). In other words, if we write
$$
K\_V = \pi^\* K\_X + \mathrm{Ram}\;\;\;\; \text{ and }\;\;\;\; \pi^\* X\_0 = (\pi)^{-1}\_\* X\_0 + \sum b\_i E\_i
$$
where $E\_i$ are the exceptional divisors (note $\mathrm{Ram} = \sum c\_i E\_i$ for some $c\_i$)
then we have that
$$
D = \pi^\* K\_X + \pi^\* X\_0 - K\_V = \pi^\* K\_X + \pi^\* X\_0 - \pi^\* K\_X - \mathrm{Ram} = (\pi)^{-1}\_\* X\_0 + \sum b\_i E\_i - sum c\_i E\_i.
$$
In other words, for each exceptional divisor $E\_i$, we have that the coefficient of
$$
D = \pi^\* X\_0 - \mathrm{Ram}
$$
is $b\_i - c\_i$ and
$$
b\_i - c\_i \leq 1
$$
or in other words $b\_i \leq c\_i + 1$.


Note, we had equality in the SNC case when I picked a very special resolution of singularities.


