Alexahin, Y.I. JACoW Geneva, Switzerland 978-3-95450-180-9 10.18429/JACoW-NAPAC2016-THPOA17 English 1132-1134 THPOA17 ion emittance optics simulation controls Contribution to a conference proceedings 2017 2017-01 http://dx.doi.org/10.18429/JACoW-NAPAC2016-THPOA17 https://jacow.org/napac2016/papers/thpoa17.pdf In a strongly nonlinear system the particle distribution in the phase space may develop long tails which contribution to the covariance (σ) matrix should be suppressed for a correct estimate of the beam emittance. A method is offered based on Gaussian approximation of the original particle distribution in the phase space (Klimontovich distribution) which leads to an equation for the σ matrix which provides efficient suppression of the tails and cannot be obtained by introducing weights. This equation is easily solved by iterations in the multi-dimensional case. It is also shown how the eigen-emittances and coupled optics functions can be retrieved from the σ matrix in a strongly coupled system.