<table><caption>**Table 12**
(Continued)</caption><thead><tr><th>Column</th><th>Format</th><th>Unit</th><th>Description</th></tr></thead><tbody><tr><td>Signif2_Peak</td><td>E</td><td>...</td><td>Source significance in peak interval in σ units</td></tr><tr><td>Flux2_Peak</td><td>E</td><td>cm<sup>-2</sup> s<sup>-1</sup></td><td>Peak integral photon flux from 100 MeV to 100 GeV</td></tr><tr><td>Unc_Flux2_Peak</td><td>E</td><td>cm<sup>-2</sup> s<sup>-1</sup></td><td>1σ error on peak integral photon flux</td></tr><tr><td>Time2_Peak</td><td>D</td><td>s (MET)</td><td>Time of center of interval in which peak flux was measured</td></tr><tr><td>Peak2_Interval</td><td>E</td><td>s</td><td>Length of interval in which peak flux was measured</td></tr><tr><td>Flux2_History</td><td>48E</td><td>cm<sup>-2</sup> s<sup>-1</sup></td><td>Integral photon flux from 100 MeV to 100 GeV in each two-month interval</td></tr><tr><td>Unc_Flux2_History</td><td>2 × 48E</td><td>cm<sup>-2</sup> s<sup>-1</sup></td><td>1σ lower and upper error on Flux2_History<sup>c</sup></td></tr><tr><td>Sqrt_TS2_History</td><td>48E</td><td>...</td><td>Square root of the Test Statistic in each two-month interval</td></tr><tr><td>ASSOC_FGL</td><td>18A</td><td>...</td><td>Most recent correspondence to previous FGL source catalogs, if any</td></tr><tr><td>ASSOC_FHL</td><td>18A</td><td>...</td><td>Most recent correspondence to previous FHL source catalogs, if any</td></tr><tr><td>ASSOC_GAM1</td><td>18A</td><td>...</td><td>Name of likely corresponding 2AGL source, if any</td></tr><tr><td>ASSOC_GAM2</td><td>18A</td><td>...</td><td>Name of likely corresponding 3EG source, if any</td></tr><tr><td>ASSOC_GAM3</td><td>18A</td><td>...</td><td>Name of likely corresponding EGR source, if any</td></tr><tr><td>TEVCAT_FLAGS</td><td>A</td><td>...</td><td>P if positional association with non-extended source in TeVCat<br>E if associated with an extended source in TeVCat, N if no TeV association</td></tr><tr><td>ASSOC_TEV</td><td>24A</td><td>...</td><td>Name of likely corresponding TeV source from TeVCat, if any</td></tr><tr><td>CLASS1</td><td>5A</td><td>...</td><td>Class designation for associated source; see Table 7</td></tr><tr><td>CLASS2</td><td>5A</td><td>...</td><td>Class designation for low-confidence association</td></tr><tr><td>ASSOC1</td><td>28A</td><td>...</td><td>Name of identified or likely associated source</td></tr><tr><td>ASSOC2</td><td>26A</td><td>...</td><td>Name of low-confidence association or of enclosing extended source</td></tr><tr><td>ASSOC_PROB_BAY</td><td>E</td><td>...</td><td>Probability of association according to the Bayesian method<sup>d</sup></td></tr><tr><td>ASSOC_PROB_LR</td><td>E</td><td>...</td><td>Probability of association according to the Likelihood Ratio method<sup>e</sup></td></tr><tr><td>RA_Counterpart</td><td>D</td><td>deg</td><td>R.A. of the counterpart ASSOC1</td></tr><tr><td>DEC_Counterpart</td><td>D</td><td>deg</td><td>decl. of the counterpart ASSOC1</td></tr><tr><td>Unc_Counterpart</td><td>E</td><td>deg</td><td>95% precision of the counterpart localization<sup>f</sup></td></tr><tr><td>Flags</td><td>I</td><td>...</td><td>Source flags (binary coding as in Table 5)<sup>g</sup></td></tr></tbody></table>

**Notes.**

<sup>a</sup> The coordinates are rounded, following the IAU convention. The letter at the end can be c (coincident with interstellar clump), e (extended source), i (for Crab Nebula inverse Compton), or s (for Crab Nebula synchrotron).

<sup>b</sup> From the 95% ellipse, assuming a Gaussian distribution.

<sup>c</sup> Separate 1σ errors are computed from the likelihood profile toward lower and larger fluxes. The lower error is set equal to NULL and the upper error is derived from a Bayesian upper limit if the 1σ interval contains 0 (TS < 1).

<sup>d</sup> NaN in this column when ASSOC1 is defined means that the probability could not be computed, either because the source is extended or because the counterpart is the result of dedicated follow-up.

<sup>e</sup> Probabilities <0.8 are formally set to 0.

<sup>f</sup> For extended counterparts, this reports their extension radius.

<sup>g</sup> Each condition is indicated by one bit among the 16 bits forming Flags. The bit is raised (set to 1) in the dubious case, so that sources without any warning sign have Flags = 0.

<table><caption>**Table 13**
LAT 4FGL FITS Format: Components Extension</caption><thead><tr><th>Column</th><th>Format</th><th>Unit</th><th>Description</th></tr></thead><tbody><tr><td>LowerEnergy</td><td>E</td><td>MeV</td><td>Lower bound of component's energy interval</td></tr><tr><td>UpperEnergy</td><td>E</td><td>MeV</td><td>Upper bound of component's energy interval</td></tr><tr><td>ENumBins</td><td>I</td><td>...</td><td>Number of bins inside energy interval</td></tr><tr><td>EvType</td><td>I</td><td>...</td><td>Event type selection for this component</td></tr><tr><td>ZenithCut</td><td>E</td><td>deg</td><td>Maximum zenith angle for this component</td></tr><tr><td>RingWidth</td><td>E</td><td>deg</td><td>Difference between RoI radius and core radius</td></tr><tr><td>PixelSize</td><td>E</td><td>deg</td><td>Pixel size for this component (of exposure map in unbinned mode)</td></tr><tr><td>BinnedMode</td><td>I</td><td>...</td><td>0 = Unbinned, 1 = Binned</td></tr><tr><td>Weighted</td><td>I</td><td>...</td><td>1 if weights were applied to this component</td></tr></tbody></table>

The extension Hist_Start (format unchanged since 1FGL) contains the definitions of the time intervals used to build the light curves. The new extension Hist2_Start (same format) describes the time intervals used to build the second series of light curves.

**Appendix B**
**Weighted log-likelihood**

In 3FGL, we introduced a first attempt at accounting for systematic errors in the maximum likelihood process itself, at the source detection level. It was not used in the source
characterization, however, for lack of a suitable framework. The standard way to account for systematic errors (for example in XSPEC$^{110}$) is to define them as a fraction $\epsilon$ of the signal and add them to the statistical errors in quadrature, in a $\chi^2$ formalism. This can be adapted to the maximum likelihood framework by introducing weights $w_i < 1$ (Hu & Zidek 2002) as

$$ \log \mathcal{L} = \sum_i w_i (n_i \log M_i - M_i) \quad (12) $$

where $M_i$ and $n_i$ are the model and observed counts in each bin, and the sum runs over all bins in space and energy. The source significance can then be quantified in the same way, via the Test Statistic TS = $2 \log(\mathcal{L}/\mathcal{L}_0)$ in which $\mathcal{L}$ and $\mathcal{L}_0$ are the (weighted) log-likelihood with and without the source of interest, respectively.

Since the statistical variance in Poisson statistics is the signal itself, a first guess for the weights could be

$$ w_i = \frac{M_i}{M_i + (\epsilon M_i)^2} = \frac{1}{1 + \epsilon^2 M_i} \quad (13) $$

However, that formulation has a serious flaw, which is that it is not stable to rebinning. If one splits the bins in half, then $M_i$ is split in half while $\epsilon$ stays the same (it is defined externally). In the limit of very small bins, obviously the weights will all tend to 1 and the $\log \mathcal{L}$ formula will tend to the unweighted one, even though nothing has changed in the underlying data or the model.

The solution we propose, originally presented in Ballet & Fermi LAT Collaboration (2015), is to define a suitable integral over energy ($E$) and space ($\mathbf{r}$) $N(\mathbf{r}, E)$, which does not depend on binning. $M_i$ in the weight formula is then replaced by $N(\mathbf{r}_i, E_i)$ taken at the event's coordinates. For the integral over space, since the catalog mostly deals with point sources, the logical solution is to integrate the background under the PSF, i.e., to convolve the model with the PSF $P(\mathbf{r}, E)$, normalized to 1 at the peak (this is equivalent, for a flat diffuse emission, to multiplying by the PSF solid angle). Note that the model already contains the PSF, so this amounts to applying a double convolution to the sky model.

For the energy integral, the choice is less obvious. The source spectrum is not a narrow line, so convolving with the energy dispersion (similar to what is done for space) is not justified. An integral over the full energy range would give the same weight to all energies, which is clearly not what we want (there is no reason to downplay the few high-energy events). The option we adopt here is to start the integration at the current energy:

$$ w_i = \frac{1}{1 + \epsilon^2 N(\mathbf{r}_i, E_i)} \quad (14) $$

$$ N(\mathbf{r}_i, E_i) = \int_{E_i}^{E_{\max}} S(\mathbf{r}_i, E) dE \quad (15) $$

$$ S(\mathbf{r}, E) = \frac{dM}{dE}(\mathbf{r}, E) * P(\mathbf{r}, E) \quad (16) $$

where $dM/dE$ is the differential model. As energy increases, the spectra (in counts) decrease and the LAT PSF gets narrower so the convolution makes $S$ even steeper than $dM/dE$. As a result, the integral giving $N$ is nearly always dominated by the lowest energies, so the exact upper bound $E_{\max}$ is not critical.

<figure><img src="image_27.png" /><figcaption>**Figure 22.** Data-based log-likelihood weights as a function of latitude across the Galactic Center, at 100 MeV, 300 MeV, 1 GeV, and 3 GeV, assuming all events are used throughout, and the same zenith cut at 105°. The dips at some latitudes are point sources, which are included in the data-based weights. Those weights were not used in 4FGL (which uses separate event types), they are shown here only for illustration.</figcaption></figure>

The only spectral region where it is important is the very lowest energies (<100 MeV) where the effective area rises steeply. In order not to penalize the lowest energies too much, we set $E_{\max} = 2E_i$ in Equation (15).

There are two possibilities to define $dM/dE$. Since the main origin of the systematic error is the diffuse emission, we can restrict $dM/dE$ to the diffuse emission model only (we call the result model-based weights). On the other hand, there are also systematic uncertainties on sources due to PSF calibration and our imperfect spectral representation, so another option is to enter the full model (or the data themselves) into $dM/dE$ (we call the result data-based weights). That second choice limits spurious sources next to bright sources. There is of course no reason why the level of systematics $\epsilon$ should be the same for the diffuse emission model and the sources, but in practice, it is a reasonable approximation.

Another important point, for the procedure to be stable, is that the weights should not change with the model parameters. So $dM/dE$ must be defined beforehand (for example from a previous fit). In this work, we use data-based weights computed from the data themselves, with a common $\epsilon$. The data are not as smooth as the model, but this is not a problem in the regime of large counts where weights play a role.

We assume here that $\epsilon$ is a true constant (it depends neither on space nor on energy). For a given $\epsilon$, the weights are close to 1 at high energy and decrease toward low energy. At a given energy, the weights are smallest where the data is largest (in the Galactic ridge). We illustrate that behavior in Figure 22, merging all event types together (not what we do in 4FGL), for 8 yr and $\epsilon = 3\%$. The width of the trough in the Galactic Ridge gets narrower at high energy, as the PSF improves. At 100 MeV, the weights are everywhere less than 12%. They reach 50% at high latitude at 250 MeV, and 90% at 500 MeV. This justifies our choice of discarding 75% of the events below 100 MeV and 50% below 300 MeV (Table 2). The entire sky is limited by systematic effects below 300 MeV. On average, in the Galactic ridge (a little better than the very center shown in Figure 22), the weights are 0.5% at 100 MeV, 1.5% at 250 MeV, 5% at 500 MeV, 20% at 1 GeV, 60% at 2 GeV, and reach 90% at 4.5 GeV.

<footnote n="110">110 https://heasarc.gsfc.nasa.gov/xanadu/xspec/</footnote>
<figure><img src="image_28.png" /><figcaption>Figure 23. Contribution to TS as a function of energy for a power-law source with Γ = 2.5 at high latitude, with and without weights. This assumes all events are used throughout (and with the same zenith cut at 105°), as in Figure 22.</figcaption></figure>

Another way to illustrate the effect of the weights is shown in Figure 23 (similar to Figure 18 of the 1FGL paper). It shows the contribution to TS of all energies, for a rather soft source at high latitude (the background and exposure are averaged over all latitudes larger than 10°), with and without weights. Energies below 300 MeV contribute very little when the weights are applied. This remains true with the actual data selection used in 4FGL.

A specific difficulty remains, because at a given energy, we split the data into several components, each corresponding to a particular event type (with a different PSF). Since the systematics act in the same way on all components, the weights must be computed globally (i.e., weights must be lower when using PSF2 and PSF3 events than when using PSF3 alone). On the other hand, the resulting uncertainties with two components should be smaller than those with a single component (adding a second one adds information). In this work, we started by computing weights $w_k$ individually for each component $k$ (the dependence on $E$ and $r$ is left implicit). Then, we assumed that the final weights are simply proportional to the original ones, with a factor $\alpha < 1$ ($\alpha$ depends on $E$ and $r$ as well). A reasonable solution is then

$$N_{\min} = \min_{k} N_{k} \qquad (17)$$

$$K_{\text{tot}} = \sum_{k} \left( \frac{N_{\min}}{N_{k}} \right)^{2} \qquad (18)$$

$$\alpha = \frac{1 + \epsilon^2 N_{\min}}{1 + \epsilon^2 N_{\min} K_{\text{tot}}} \qquad (19)$$

$$w_k = \frac{\alpha}{1 + \epsilon^2 N_k} \qquad (20)$$

$K_{\text{tot}}$ and $\alpha$ are 1 if one component dominates over the others, and $K_{\text{tot}}$ is the number of components if they are all similar. The effect of this procedure is depicted in Figure 24 at 1 GeV, the lowest energy at which we use all event types. It illustrates

<figure><img src="image_29.png" /><figcaption>Figure 24. Data-based weights at 1 GeV for ZMax = 105° as a function of latitude (for the interesting [-30°, 30°] region) across the Galactic Center, for different PSF event types, computed according to Equation (20). These weights were actually used in 4FGL. The average (over event types) weight is larger than the weight using all events together at the same 1 GeV energy (blue dashed line in Figure 22). This is because keeping event types separate is more favorable than merging them and losing the event type information.</figcaption></figure>

quantitatively how the PSF0 events are unweighted at low latitudes, compared to better event types.

## ORCID iDs

M. Ajello https://orcid.org/0000-0002-6584-1703

M. Axelsson https://orcid.org/0000-0003-4378-8785

L. Baldini https://orcid.org/0000-0002-9785-7726

J. Ballet https://orcid.org/0000-0002-8784-2977

D. Bastieri https://orcid.org/0000-0002-6954-8862

J. Becerra Gonzalez https://orcid.org/0000-0002-6729-9022

R. Bellazzini https://orcid.org/0000-0002-2469-7063

E. Bissaldi https://orcid.org/0000-0001-9935-8106
E. Charles https://orcid.org/0000-0002-3925-7802

C. C. Cheung https://orcid.org/0000-0002-4377-0174

F. D'Ammando https://orcid.org/0000-0001-7618-7527

A. Desai https://orcid.org/0000-0001-7405-9994

A. Domínguez https://orcid.org/0000-0002-3433-4610

A. Franckowiak https://orcid.org/0000-0002-5605-2219

Y. Fukazawa https://orcid.org/0000-0002-0921-8837

S. Funk https://orcid.org/0000-0002-2012-0080

P. Fusco https://orcid.org/0000-0002-9383-2425

F. Gargano https://orcid.org/0000-0002-5055-6395

D. Gasparrini https://orcid.org/0000-0002-5064-9495

M. Giroletti https://orcid.org/0000-0002-8657-8852

T. Glanzman https://orcid.org/0000-0001-9649-3871

S. Guiriec https://orcid.org/0000-0001-5780-8770

A. K. Harding https://orcid.org/0000-0001-6119-859X

K. Hayashi https://orcid.org/0000-0001-6922-6583

M. Kerr https://orcid.org/0000-0002-0893-4073

M. Kuss https://orcid.org/0000-0003-1212-9998

S. Larsson https://orcid.org/0000-0003-0716-107X

M. Lemoine-Goumand https://orcid.org/0000-0002-4462-3686

J. Li https://orcid.org/0000-0003-1720-9727

I. Liodakis https://orcid.org/0000-0001-9200-4006

F. Loparco https://orcid.org/0000-0002-1173-5673

B. Lott https://orcid.org/0000-0003-2186-9242

P. Lubrano https://orcid.org/0000-0003-0221-4806

S. Maldera https://orcid.org/0000-0002-0698-4421

A. Manfreda https://orcid.org/0000-0002-0998-4953

L. Marcotulli https://orcid.org/0000-0002-8472-3649

M. N. Mazziotta https://orcid.org/0000-0001-9325-4672

M. Meyer https://orcid.org/0000-0002-0738-7581

N. Mirabal https://orcid.org/0000-0002-7021-5838

T. Mizuno https://orcid.org/0000-0001-7263-0296

M. E. Monzani https://orcid.org/0000-0002-8254-5308

A. Morselli https://orcid.org/0000-0002-7704-9553

I. V. Moskalenko https://orcid.org/0000-0001-6141-458X

V. S. Paliya https://orcid.org/0000-0001-7774-5308

M. Pesce-Rollins https://orcid.org/0000-0003-1790-8018

V. Petrosian https://orcid.org/0000-0002-2670-8942

T. A. Porter https://orcid.org/0000-0002-2621-4440

S. Rainò https://orcid.org/0000-0002-9181-0345

R. Rando https://orcid.org/0000-0001-6992-818X

M. Razzano https://orcid.org/0000-0003-4825-1629

S. Razzaque https://orcid.org/0000-0002-0130-2460

O. Reimer https://orcid.org/0000-0001-6953-1385

R. W. Romani https://orcid.org/0000-0001-6711-3286

F. K. Schinzel https://orcid.org/0000-0001-6672-128X

C. Sgrò https://orcid.org/0000-0001-5676-6214

D. A. Smith https://orcid.org/0000-0002-7833-0275

D. J. Thompson https://orcid.org/0000-0001-5217-9135

L. Tibaldo https://orcid.org/0000-0001-7523-570X

D. F. Torres https://orcid.org/0000-0002-1522-9065

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