Hydrology Modeling - EnKF and Monte Carlo Simulation

Ensemble Kalman Filter
Source: http://en.wikipedia.org/wiki/Ensemble_Kalman_filter
The ensemble Kalman filter (EnKF) is a recursive filter suitable for problems with a large number of variables, such as discretizations of partial differential equations in geophysical models. EnKF makes the assumption that all probability distributions involved are Gaussian; when it is applicable, it is much more efficient than the particle filter.
EnKFs represent the distribution of the system state using a collection of state vectors, called an ensemble, and replace the covariance matrix by the sample covariance computed from the ensemble. The ensemble is operated with as if it were a random sample, but the ensemble members are really not independent - the EnKF ties them together. One advantage of EnKFs is that advancing the pdf in time is achieved by simply advancing each member of the ensemble.

Ensemble
Source: http://en.wikipedia.org/wiki/Numerical_weather_prediction#Ensembles
It is impossible to definitively predict the state of the atmosphere, owing to the chaotic nature of the fluid dynamics equations involved. Furthermore, existing observation networks have limited spatial and temporal resolution, especially over large bodies of water such as the Pacific Ocean, which introduces uncertainty into the true initial state of the atmosphere. To account for this uncertainty, stochastic or "ensemble" forecasting is used, involving multiple forecasts created with different model systems, different physical parametrizations, or varying initial conditions. The ensemble forecast is usually evaluated in terms of the ensemble mean of a forecast variable, and the ensemble spread, which represents the degree of agreement between various forecasts in the ensemble system, known as ensemble members.

Monte Carlo Simulation
Source: http://en.wikipedia.org/wiki/Monte_Carlo_method
The name "Monte Carlo" was popularized by physics researchers Stanislaw Ulam, Enrico Fermi, John von Neumann, and Nicholas Metropolis, among others; the name is a reference to the Monte Carlo Casino in Monaco where Ulam's uncle would borrow money to gamble.
Monte Carlo simulation methods are especially useful in studying systems with a large number of coupled degrees of freedom, such as fluids, disordered materials, strongly coupled solids, and cellular structures. More broadly, Monte Carlo methods are useful for modeling phenomena with significant uncertainty in inputs, such as the calculation of risk in business, hydrological modeling.
Monte Carlo methods are a class of computational algorithms that rely on repeated random sampling to compute their results. Monte Carlo methods are often used when simulating physical and mathematical systems. Because of their reliance on repeated computation and random or pseudo-random numbers, Monte Carlo methods are most suited to calculation by a computer. Monte Carlo methods tend to be used when it is infeasible or impossible to compute an exact result with a deterministic algorithm.