Determine the coefficient of the exceptional divisor 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 $k_j$?
 A: 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.
