Nazgul Hd

[Ashlyn Robello]

Italways occurred to me how distinct a character the Lieutenant of Barad-dr is; and the power, authority and status he seems to have had in Mordor, considering he was just a man.(If interested, see the two-part article on the Military Structure of Mordor here )

This is an interesting idea. I guess we first has to look at the actual nazgul, how long did they bear their 9 rings before they became wraiths? Did they all become wraiths one at a time, or all at once. If indeed the MoS is on his way to becoming a wraith, by what device or process he he doing so? How long has he been under the guile of Sauron? and how much longer would it have taken? if at all! Great idea though.

Then again, if we decide to sustain this Mouth of Sauron theory, could he have been wearing one of the Nine Rings of Power given to the Men? After becoming Ringwraiths, did they need to wear the Rings or could Sauron have used them for other purposes such as this?

Context. Gamma-ray bursts (GRBs) can be located via arrival time signal triangulation using gamma-ray detectors in orbit throughout the solar system. The classical approach based on cross-correlations of binned light curves ignores the Poisson nature of the time series data, and it is unable to model the full complexity of the problem.

Results. Our novel method can robustly estimate the position of a GRB as verified via simulations. The uncertainties generated by the method are robust and in many cases more precise compared to the classical method. Thus, we have a method that can become a valuable tool for gravitational wave follow-up.

Open Access article, published by EDP Sciences, under the terms of the Creative Commons Attribution License ( ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

To improve upon the cIPN, we propose a new IPN algorithm we call nazgul1, built on a hierarchical, forward modeling approach to likelihood-based inference. We demonstrate through the analysis of mock data sets generated via realistic simulations that the nazgul algorithm produces precise statistical localizations with well-calibrated uncertainties, even in challenging settings where the cIPN algorithm does not. As much of the data necessary for IPN localizations of real observed sources is not (at present) publicly available, we restrict our investigation to the simulated regime and focus on the methodological development. We plan a future study with real data to assess its viability and systematics. However, we explicitly make available all the required software2 to reproduce our results and to assist in further development of the approach with the hope that nazgul can become a valuable tool for the GRB localization community.

Fig. 2.Illustrative sketch of the problem as well as the framework for our forward model. A GRB emits a signal which is detected at two different times in detectors at different locations. These signals are translated into data sampled with different temporal resolutions and different detector effective areas resulting in two light curves which are delayed with respect to each other.

That is, β is formed as a length 2k column vector of elements with independent standard Normal distributions. For reference, the corresponding covariance matrix may be computed as , which is similar to the strategy employed in some X-ray timeseries analyses (Zoghbi et al. 2012). An accessible introduction to the random Fourier feature approximation, albeit from the perspective of spatial Gaussian random fields rather than timeseries, is given by Milton et al. (2019).

In Fig. 3, we illustrate our hierarchical model as a directed acyclic graph. This diagrammatic mode of presentation highlights the connections between the parameters at each level of the hierarchical model and the data. Here, open circles represent latent variables to be inferred while closed circles indicate data. Quantities not enclosed in circles are constant information that do not exhibit measurement uncertainty. Finally, quantities enclosed in diamonds represent deterministic combinations of latent variables and/or fixed quantities.

In order to demonstrate and validate our method, we have created a simulation package4 that allows for the virtual placement of GRB detectors throughout the solar system and creates energy-independent Poisson events sampled from a given light curve model. Using this package, we select a number of representative combinations of satellite configurations and light curve shapes and perform analyses on mock data simulated under each setting. We additionally compute the localization of the simulated GRB with the cIPN method to serve as a comparison. Our procedure for simulating mock datasets is described in Appendix A. In the following we first demonstrate the ability of the nazgul model to fit a single time series data set, and then proceed to fit two and three detector configurations to recover time delays and perform localizations. By repeating the simulation and localization process many times over, we confirm that the associated uncertainty intervals of the nazgul algorithm are well calibrated in a Frequentist (long-run) sense, i.e., that each quoted uncertainty interval contains the true source location with a frequency close to its nominal level.

Here we demonstrate the ability of RFFs to model simulated light curves with Poisson count noise. A GRB is simulated with four overlapping pulses each beginning five seconds after the pulse preceding it. In Fig. 4 we can see the posterior traces of the fit compared with the true light curve. We can check the reconstruction of the light curves via posterior predictive checks (Gabry et al. 2019, PPCs) where counts in each time bins are sampled from the inferred rate integrated over the posterior (see Fig. 5).

We have checked that we are able to recover a variety of pulse shapes and combinations. Thus, even without the aid of a secondary signal from another detector, RFFs can learn the shape of the latent signal.

When the strength of the signal is decreased, we notice that the inferences become less constraining, but the overall ability of the model to reproduce the latent signal is sufficient to find properties of the light curve that can be time delayed.

The simulation for only two detectors allows us to validate the simplest configuration of detectors possible which results in a single annulus on the sky. For this simulation, the configuration consists of one detector in low-Earth orbit and another at an altitude mimicking an orbit in L1. The light curve shape chosen consists of two pulses and the detectors are given effective areas that differ by a factor of two. Each light curve is sampled at a different cadence (100 ms and 200 ms). Details are given in Tables 1 and 2.

As can be seen in Fig. 6, the algorithm correctly finds annuli enclosing the true position of the simulated GRB. The latent signal is recovered as shown in Fig. 6 and the PPCs adequately reproduce the data as seen in Fig. 7. We can examine the correlations between position and time that occur naturally due to our forward model approach in Fig. 8. By using different combinations of temporal binnings, we find that the result is not dependent on this parameter unless the binning is too large to resolve the time delay. Even in this case; however, we find that the result is only marginally affected with larger uncertainties on the time delay (Fig. 9).

Fig. 6.Skymap of the two detector validation. The purple contours indicate the 1, 2, and 3σ credible regions with increasing lightness. The posterior samples are shown in green and the simulated position is indicated by the cross-hairs.

First we examine the method on our example simulation. While our new algorithm is independent of binning and pre/post- source temporal selection, we find that the same is not true the cIPN cross-correlation approach, and different selections can have vastly different results. Thus, we choose a selection, having knowledge of the simulation parameters5 that encompasses the burst interval and has enough lag between the signals to fully map out the so-called reduced χ2-contour as a function of lag. We also chose different combinations temporal binnings for the light curves ranging from 50 to 200 ms; the coarser corresponding to what was used for the nazgul method. The results from two of the fits at different resolution is show in Fig. 13. We can see that the choice of binning immensely influences the confidence intervals. Additionally, the 1σ region misses the true value while the 3σ region is much larger than the nazgul posterior.

To see if these results can be generalized, we perform cross-correlation on each of the 500 simulations. We can repeat the exercise above and compute the fraction of fits that capture the simulated value given a confidence interval6. Figure 14 shows that for different temporal binnings, the credible intervals do not exhibit appropriate coverage. For coarser binnings, the coverage begins to approach expectations. This is likely due to the number of counts in each interval increasing for these binnings and thus satisfying the Gaussian approximation of the Poisson distribution. This poses severe difficulties for detector geometries which would need high-resolution to find sub-second time delays between light curves. We note that there are some combinations of resolution that result in adequate coverage in the 3σ intervals, but in situations involving real data, it would be impossible to know which combinations would be required.

Fig. 14.Confidence intervals of Δt versus the fraction of times the true, simulated value is within that credible intervals. The green line represents a perfect, one-to-one relation. Here, coarser binnings are represented by lighter purple colors.

Moreover, we can examine the relative widths of the uncertainties of the two methods at various confidence intervals. We find that while our algorithm is well behaved (see Fig. 15), the 1σ uncertainties of the cIPN method are smaller and (as shown above) underestimated, while the 3σ uncertainties are appreciably larger and likely over-estimated.

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