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$p,q,r,t$ and $s$ are measured variables and are used by the function $f$ give as: $$f=\left(\frac{-\left(\sqrt{\frac{p^2\left(q^2-r^2\right)}{t^2}}\right)+\sqrt{\frac{p^2\left(q^2-r^2\right)}{t^2}-4\left(\frac{p^2}{t^2}\right)}}{2}\right)\cdot s$$

I am trying to do a global sensitivity analysis of this mathematical model in order to determine the sensitivity of each measured variable (uncertainty) on the output (uncertainty). I am using a low discrepancy estimator like sobol sequences in which the indices are calculated using Monte Carlo method and the initial steps to create the sample matrix requires the probability distribution of the input variables. The input variables are physically measured quantities (example: distance between two points). I do not have the data for these measurements (Input variables) as it is just a model so, I cannot even have a rough estimate of how the distribution would look like.

What would be the best approach?. Which probability density function must be applied to each input variable and what should be their range?.

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    Raw answer. The more thorough approach to "sensitivity" of an expression with respect to a variable is its **derivative**. If one has to agregate these derivatives in order to have a global estimator, it is possible to take a weighted average of the derivatives wrt the different variables. – 2017-01-08
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    I am trying to use the low discrepancy estimator like sobol sequences. In which the indices are calculated using Monte Carlo method and the initial steps to create the sample matrix requires the probability distribution of the input variables. How do I define those distributions?. – 2017-01-08
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    Don't you think that this information should be included into your question ? – 2017-01-08

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there many methods st : Indices based on correlation and linear regression. Indices based on the variance (Sobol indices)

for the Sobol indices we have $ S_i= { Var(E(Y|X_i)) \over Var(Y) } $

where the numerator represents the partial variance and the denominator represents the total variance.

for more details consult this reference : 

I. M. Sobol. Sensitivity estimates for nonlinear mathematical models.
Math. Modeling Comput. Experiment, 1(4) :407–414 (1995), 1993.