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Intrapartum fetal surveillance aims at preventing neonatal morbidity and mortality due to asphyxia. Continuous fetal heart rate monitoring helps in reducing asphyxia yet generates many unnecessary c-sections. In fetal heart rate time series analysis, variability is a key parameter as its decrease is used to identify asphyxia. Multifractal analysis can be envisaged as a new statistical tool enabling to revisit variability analysis. Applied to data collected at an academic hospital, the wavelet Leader based multifractal analysis is shown here to i) permit a rich and relevant characterization of their variability, ii) to achieve a significant discrimination between healthy and non-healthy fetus, iii) to enable this valid discrimination to hold much earlier than during the last 15 min preceding delivery. This hence opens promising tracks to decrease the number of unnecessary c-sections.