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Since the start of the gravitational wave observation era, no joint high energy neutrino and gravitational wave event has been found. These non-detections could be used for setting an upper bound on the neutrino emission properties for gravitational wave events individually or for a set of them. Although in the previous joint high energy neutrino and gravitational wave event searches upper limits have been found, there is a lack of consistent method for the calculation. The problem addressed in this paper is finding those limits for astrophysical events which are localized poorly in the sky where the sensitivities of the neutrino detectors change significantly and can also emit neutrinos, for example the gravitational wave detections. Here we describe methods for assigning limits for expected neutrino count, emission fluence and isotropically equivalent emission based on maximum likelihood estimators. Then we apply described methods on the three GW detections from aLIGO's first observing run (O1) and find upper limits for them. 

Abstract. The IceCube Neutrino Observatory instruments about 1 km3 of deep, glacial ice at the geographic South Pole. It uses 5160 photomultipliers to detect Cherenkov light emitted by charged relativistic particles. An unexpected light propagation effect observed by the experiment is an anisotropic attenuation, which is aligned with the local flow direction of the ice. We examine birefringent light propagation through the polycrystalline ice microstructure as a possible explanation for this effect. The predictions of a first-principles model developed for this purpose, in particular curved light trajectories resulting from asymmetric diffusion, provide a qualitatively good match to the main features of the data. This in turn allows us to deduce ice crystal properties. Since the wavelength of the detected light is short compared to the crystal size, these crystal properties include not only the crystal orientation fabric, but also the average crystal size and shape, as a function of depth. By adding small empirical corrections to this first-principles model, a quantitatively accurate description of the optical properties of the IceCube glacial ice is obtained. In this paper, we present the experimental signature of ice optical anisotropy observed in IceCube light-emitting diode (LED) calibration data, the theory and parameterization of the birefringence effect, the fitting procedures of these parameterizations to experimental data, and the inferred crystal properties.