Finding Equations of Curves 
I was wondering if it would be possible to find the equations of the curves in the picture, or some sort of approximation. It looks like a square root function to me. I have attempted to plot the points in my calculator and do a regression, but there is no option for a square root regression. Any help would be appreciated.
 A: As suggested in comments, plot the logarithm of travel time $t$ as a function of the logarithm of the epicenter distance $d$. You will observe a very linear trend. So, using linear regression based on $\big(\log(d_i),\log(t_i)\big)$ you  should get the coefficients for $$\log(t)=\alpha +\beta \log(d)$$
I picked eleven points from the  $S$-wave curve (probably not very accurate values) and obtained $$\log(t)=1.44494+0.765722 \log (d)$$ which has a very high degree of determination $(R^2=0.999834)$ that is to say that this simple model fits quite accurately the data. 
Going to exponentials, this would translate to $$t=4.24159~ d^{0.765722}$$ All of these can be done with calculator.
Later in your studies, you will learn that this is just a preliminary step of a rigorous nonlinear regression since what we did here is to minimize $$SSQ_1(\alpha ,\beta)=\sum_{i=1}^n \Big(\alpha +\beta \log(d_i)-\log(t_i)\Big)^2$$ which not the same as minimizing $$SSQ_2(\gamma ,\delta)=\sum_{i=1}^n \Big(\gamma~d_i^{\delta}-t_i\Big)^2$$ which is what has to be done (assuming that the model is valid) since what are measured are $t_i$'s and $d_i$'s and not their logarithms. But this preliminary step, easy to do, provides very good estimates.
Continuing the full procedure with the same data and nonlinear regression, what I obtained is $$t=4.6706 d^{0.711472}$$ you can compare with the previous simple procedure.
