TESTING LSST DITHER STRATEGIES FOR SURVEY UNIFORMITY AND LARGE-SCALE STRUCTURE SYSTEMATICS

In order to calculate the coadded depth, we use the modified 5σ limiting magnitude data from OpSim, where the limiting magnitude is "modified" in order to represent a real point source detection depth (Ivezic et al. 2008). Assuming that the signal-to-noise ratio adds in quadrature, as it should for optimal weighting of individual images (see, e.g., Gawiser et al. 2006), we calculate the coadded depth, $5{\sigma }_{\mathrm{stack}}$, in each HEALPix pixel from the modified 5σ limiting magnitude summed over individual observations, $5{\sigma }_{\mathrm{mod},i}$:

$$\begin{eqnarray}&&5{\sigma }_{\mathrm{stack}}=1.25\ {\mathrm{log}}_{10}\left(\displaystyle \sum _{i}{10}^{0.8\times 5{\sigma }_{\mathrm{mod},i}}\right).\end{eqnarray} \tag{ 1 }$$

We find that dithered surveys lead to shallower depths near the borders of the survey region, adding significant noise to the corresponding angular power spectra. In order to clean the spectra, we develop a border masking algorithm to discount pixels at the edges of the survey region, comprising nearly 15% of the survey area. See the appendix for details of the masking algorithm.

Figure 2 shows two projections for the r-band coadded 5σ depth for the various dither strategies after the shallow border has been masked. The first row shows the Mollweide projection of the coadded depth for NoDither and PentagonDitherPerSeason, while the second row shows the corresponding Cartesian projection, zoomed on the LSST WFD survey area ($-180^\circ$ < R.A. < $180^\circ ,-70^\circ$ < decl. < 10°). To conserve space, we show only the latter projection for the rest of the dither strategies. We observe that the survey pointings without any dithering lead to deeper overlapping regions between the fields, and consequently a strong honeycomb pattern in the coadded depth. In contrast, the dithered skymaps have comparatively more uniform depth across the survey region, with smaller-scale variations among the dither strategies.

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Here we note that although dithering in general weakens the honeycomb pattern seen in the undithered survey, we observe horizontal striping from SequentialHexDitherFieldPerNight; in contrast, SequentialHexDitherPerNight and SequentialHexDitherFieldPerVisit show no such behavior. This is an example where a specific dither strategy's behavior is highly dependent on the timescale on which it is implemented: for PerNight timescale, a new dither is assigned to all fields every night, implying that the 217-point lattice is traversed multiple times during the ∼3650-night survey. Similarly, for the PerVisit timescale, although a new dither is assigned to each field every time it is visited, the lattice is traversed multiple times given that every field is visited ∼150 times in the r-band throughout the survey. In contrast, for the FieldPerNight timescale, a new dither point is assigned to each field only when it is observed on a new night. Since a given field is only visited on ∼50 nights in a given filter, only the lower part of the lattice is traversed (as the lattice is traversed starting from bottom left), leading to horizontal striping. We verified this conclusion by rotating the hexagonal lattice by 90°, and observing vertical striping for the FieldPerNight timescale.

In Figure 3, we show a histogram of the r-band coadded depth. We see that the undithered survey leads to a bimodal distribution, with the overlapped regions observed much deeper than the rest of the survey. On the other hand, all the dithered surveys lead to unimodal distributions, as dithering leads to observing the data more uniformly, in agreement with Carroll et al. (2014).

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In order to quantify the angular characteristics reflected in the skymaps, we measure the power spectrum associated with each of the skymaps. Figure 4 shows the power spectra for the coadded depth from each of the dither strategies considered here; we have removed the monopole and dipole using the HEALPix routine remove_dipole. We note that the spectrum corresponding to the undithered survey has a very large peak around ℓ ∼ 150, resulting from the strong honeycomb pattern. In comparison, we see over 10 times less power in the dithered surveys; the ℓ ∼ 150 peak in these surveys is much more comparable to the rest of the spectrum. More specifically, we find that the FieldPerVisit timescale is the most effective at reducing the power for a given dither geometry, while the Random and RepulsiveRandom dithers perform well on all three timescales. Also, we confirm the origins of the ℓ ∼ 150 peak by creating a pure honeycomb, and observing a power spectrum similar to that from the undithered survey.

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Furthermore, we see that the horizontal striping in the SequentialHexDitherFieldPerNight skymap generates a large peak around ℓ ∼ 150, while the rest of the dithered spectra do not exhibit such a strong peak. Curiously, the PentagonDitherPerSeason strategy leads to two large peaks around ℓ ∼ 270—a characteristic different from the rest of the dither strategies' but similar to NoDither, with much less power.

To understand the origins of the characteristic patterns in the skymaps, we consider the a_ℓm coefficients of their spherical harmonic transforms. This allows us to produce the skymaps corresponding to specific ranges of the angular scale ℓ. We show our results in Figure 5 for NoDither, PentagonDitherPerSeason, and SequentialHexDitherFieldPerNight strategies. The top row includes the full power spectrum for each strategy, and the second row shows the corresponding Cartesian projection for 0° < R.A. < $50^\circ ,-45^\circ$ < Decl. < $-5^\circ$. The third and fourth rows show the partial skymaps arising from each of the colored peaks shown in the power spectra in the top row. We observe that for the undithered survey, the ℓ ∼ 150 peak arises from the strong honeycomb pattern, while the second peak arises from structure on the small angular scales. For PentagonDitherPerSeason, we see a milder honeycomb for the ℓ ∼ 150 peak, while the 240 < ℓ < 300 peak arises from structure similar to the corresponding one in the undithered survey. Finally, for SequentialHexDitherFieldPerNight, we can see the source of the strong ℓ ∼ 150 peak: the horizontal striping. For higher-ℓ peaks, we note the weaker structure as compared to the other two strategies. We also performed this a_ℓm analysis individually for the two peaks in 240 < ℓ < 300 and found the underlying structures to be very similar.

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Given our knowledge of the characteristics induced in the coadded depth due to the observing strategy (OS), we now consider the effects of these artifacts on BAO studies. We model the artificial fluctuations in galaxy counts, accounting for photometric calibration errors, dust extinction, and galaxy catalog magnitude cuts. Since BAO studies are redshift dependent, we consider five redshift bins: 0.15 < z < 0.37, 0.37 < z < 0.66, 0.66 < z < 1.0, 1.0 < z < 1.5, and 1.5 < z < 2.0.

We first estimate the number of galaxies in specific redshift bins detected in each pixel at a particular depth using a mock LSST catalog, which is constructed using the outputs of the SAG semi-analytic model for galaxy formation (Cora 2006; Lagos et al. 2008; Tecce et al. 2010; Orsi et al. 2014; Gargiulo et al. 2015; Muñoz Arancibia et al. 2015). The model incorporates differential equations for gas cooling, quiescent star formation, energetic and chemical supernova feedback, the growth of a supermassive black hole, the associated AGN feedback, bursty star formation in mergers, and disk instabilities, all coupled to the merger trees extracted from a dark matter simulation run with GADGET2 (Springel 2005), assuming the standard ΛCDM model (Jarosik et al. 2011). The subhalo populations of merger trees are found using SUBFIND (Springel et al. 2001) after the DM halos were identified using a friends-of-friends algorithm.

We normalize the total r-band galaxy counts to the empirical cumulative galaxy count estimates for LSST (see Abell et al. 2009, Section 3.7.2 for details) at a magnitude cut of r < 25.9 (corresponding to the CFHTLS Deep survey completeness limit of i < 25.5; see Hoekstra et al. 2006; Gwyn 2008 for details). In contrast with Carroll et al. (2014), where Fleming's function (Fleming et al. 1995) was used to account for the incompleteness near the 5σ limit, we use an erfc function. When multiplied by power-law number counts, Fleming's function causes completeness to drop to 20% of its peak at r ∼ 30 before rising again, while the erfc incompleteness function correctly damps down for higher magnitudes. We calculate the number of galaxies, N_gal, in each HEALPix pixel in a given redshift bin as

$$\begin{eqnarray}&&{N}_{\mathrm{gal}}=0.5{\int }_{-\infty }^{{m}_{\max }}\mathrm{erfc}[a(m-5{\sigma }_{\mathrm{stack}})]{10}^{{c}_{1}m+{c}_{2}}{dm},\end{eqnarray} \tag{ 2 }$$

where a is the rollover speed and is chosen to be 1, $5{\sigma }_{\mathrm{stack}}$ is the coadded magnitude depth in the given HEALPix pixel, m_max is the magnitude cut, and c₁ and c₂ are the power-law constants determined from the mock catalogs for specific redshift bins. Here, we assume galaxies to have average colors, i.e., $u-g=g-r=r-i=0.4$, and take this into account by modifying ${c}_{2}$ and m_max in Equation (2) for $u,g,i$ versus r. Given the sharp decline of the erfc function at high magnitudes and the consequent decline in the differential galaxy counts, we consider a magnitude limit of r = 32.0 as no magnitude limit.

Using the number of galaxies in each pixel, we calculate the fluctuations in the galaxy counts ΔN/$\overline{N}$ as (${N}_{\mathrm{gal}}/{N}_{\mathrm{avg}})-1$, where N_avg is the average number of galaxies per pixel across the survey area. Within MAF, this procedure amounts to using a metric to calculate the number of galaxies and then post-processing the galaxy counts to find ΔN/$\overline{N}$.

Here we note that artificial fluctuations in galaxy counts induced by the observing strategy scale the fluctuations arising due to actual LSS. In our calculations, we assume that LSS affects the local normalization of the galaxy luminosity function in a given redshift bin, not its shape. This assumption is valid as long as LSS does not alter the shape of the faint end of the luminosity function, which dominates the galaxy number counts. More precisely, in the ith pixel,

$$\begin{eqnarray}&&\ {\left(\displaystyle \frac{{N}_{\mathrm{gal}}}{{N}_{\mathrm{avg}}}\right)}_{\mathrm{observed},i}={\left(\displaystyle \frac{{N}_{\mathrm{gal}}}{{N}_{\mathrm{avg}}}\right)}_{\mathrm{OS},i}{\left(\displaystyle \frac{{N}_{\mathrm{gal}}}{{N}_{\mathrm{avg}}}\right)}_{\mathrm{LSS},i}.\end{eqnarray} \tag{ 3 }$$

Defining ${\delta }_{i}$ = ΔN${}_{i}$/$\overline{N}$ = (${N}_{\mathrm{gal},i}/{N}_{\mathrm{avg}})-1$, we have

$$\begin{eqnarray}&&\ (1+{\delta }_{\mathrm{observed},i})=(1+{\delta }_{\mathrm{OS},i})(1+{\delta }_{\mathrm{LSS},i}).\end{eqnarray} \tag{ 4 }$$

Since the ensemble average of LSS is zero, we have

$$\begin{eqnarray}\ \langle {\delta }_{\mathrm{observed},i}\rangle & = & \langle {\delta }_{\mathrm{OS},i}\rangle +\langle {\delta }_{\mathrm{LSS},i}\rangle +\langle {\delta }_{\mathrm{OS},i}{\delta }_{\mathrm{LSS},i}\rangle \\ & = & {\delta }_{\mathrm{OS},i}+\langle {\delta }_{\mathrm{OS},i}{\delta }_{\mathrm{LSS},i}\rangle \end{eqnarray} \tag{ 5 }$$

where the angular brackets $\langle ..\rangle$ indicate an ensemble average, defined as an average over many realizations of the universe with one LSST survey. Hence, we have $\langle {\delta }_{{\rm{OS,}}i}\rangle ={\delta }_{{\rm{OS,}}i}$, as the OS-induced structure represents a fixed pattern on the sky for a given LSST observing strategy and OpSim run. Since there is generally no correlation between the OS-induced structure and LSS, the cross-term $\langle {\delta }_{\mathrm{OS},i}{\delta }_{\mathrm{LSS},i}\rangle$ should be negligible; we check and confirm this for a typical dither pattern. Also, we note that this assumption about the correlation between OS-induced structure and LSS breaks down if the survey strategy is correlated with LSS, e.g., Deep Drilling Fields focused on galaxy clusters, as then $\langle {\delta }_{\mathrm{OS},i}{\delta }_{\mathrm{LSS},i}\rangle \ne 0$.

Using Equations (4)–(5), we calculate the power in ${\delta }_{\mathrm{observed},i}$ :

$$\begin{eqnarray}\langle {\delta }_{\mathrm{observed},i}^{2}\rangle & = & \langle {\delta }_{\mathrm{OS},i}^{2}\rangle +\langle {\delta }_{\mathrm{LSS},i}^{2}\rangle \\ & & +2\langle {\delta }_{\mathrm{OS},i}{\delta }_{\mathrm{LSS},i}\rangle +2\langle {\delta }_{\mathrm{OS},i}^{2}{\delta }_{\mathrm{LSS},i}\rangle \\ & & +2\langle {\delta }_{\mathrm{OS},i}{\delta }_{\mathrm{LSS},i}^{2}\rangle +\langle {\delta }_{\mathrm{OS},i}^{2}{\delta }_{\mathrm{LSS},i}^{2}\rangle .\end{eqnarray} \tag{ 6 }$$

As mentioned earlier, $\langle {\delta }_{\mathrm{OS},i}{\delta }_{\mathrm{LSS},i}\rangle$ is negligible since OS-induced structure and LSS are generally not correlated . To check how higher order terms like $\langle {\delta }_{\mathrm{OS},i}^{2}{\delta }_{\mathrm{LSS},i}^{2}\rangle$ compare with $\langle {\delta }_{\mathrm{OS},i}{\delta }_{\mathrm{LSS},i}\rangle$ , we calculate the cross-spectra for a typical dither pattern. We find that $\langle {\delta }_{\mathrm{OS},i}{\delta }_{\mathrm{LSS},i}\rangle$ is dominant over $\langle {\delta }_{\mathrm{OS},i}^{2}{\delta }_{\mathrm{LSS},i}^{2}\rangle$ and therefore these higher order terms are also negligible. Therefore,

$$\begin{eqnarray}&&\langle {\delta }_{\mathrm{observed},i}^{2}\rangle \approx \langle {\delta }_{\mathrm{OS},i}^{2}\rangle +\langle {\delta }_{\mathrm{LSS},i}^{2}\rangle ={\delta }_{\mathrm{OS},i}^{2}+\langle {\delta }_{\mathrm{LSS},i}^{2}\rangle ,\end{eqnarray} \tag{ 7 }$$

implying that the OS and LSS contribute independently to the observed power. ${\delta }_{\mathrm{OS},i}^{2}$ thus represents a bias in our measurement of LSS.

To consider realistic behavior for the observing strategies, we account for the uncertainties arising from photometric calibrations. Given that related systematic errors correlate with seeing (Leistedt et al. 2015) and are expected to decrease with the number of observations, we model the calibration uncertainty ${{\rm{\Delta }}}_{i}$ in the ith HEALPix pixel as

$$\begin{eqnarray}&&{{\rm{\Delta }}}_{i}=\displaystyle \frac{k{\rm{\Delta }}{s}_{i}}{\sqrt{{N}_{\mathrm{obs},{i}}}}\end{eqnarray} \tag{ 8 }$$

where ${\rm{\Delta }}{s}_{i}$ is the difference between the average seeing in the ith HEALPix pixel and the average seeing across the map, ${N}_{\mathrm{obs},i}$ is the number of observations in the ith pixel, and k is a constant such that the variance ${\sigma }_{{{\rm{\Delta }}}_{i}}^{2}={0.01}^{2}$, ensuring the expected 1% errors in photometric calibration (Abell et al. 2009). Figure 6 shows skymaps for these simulated uncertainties for example dither strategies. We note that while dithering does not alter the amplitudes of the photometric calibration uncertainties in our model, it helps mitigate the sharp hexagonal pattern seen in the uncertainties in the undithered survey.

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In order to account for the fluctuations in the galaxy counts arising due to the photometric calibration uncertainties, we modify the upper limit on the magnitude in Equation (2) to be ${m}_{\max }+{{\rm{\Delta }}}_{i}$ for the ith pixel. Since the calibration uncertainties are small, the skymaps for the fluctuations in the galaxy counts after accounting for the calibration uncertainties are indistinguishable from those without. These are shown in the top row in Figure 7.

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Furthermore, we include dust extinction by using the Schlegel-Finkbeiner-Davis dust map (Schlegel et al. 1998) when calculating the coadded depth as well as Poisson noise in the galaxy counts after accounting for both dust extinction and photometric calibration . The bottom row in Figure 7 shows the skymaps for the artificial fluctuations for 0.66 < z < 1.0 after accounting for photometric calibration uncertainties, dust extinction, and the Poisson noise. We find that dust extinction dominates both photometric calibration uncertainties and Poisson noise; it induces power on large angular scales, but it does not wash out the honeycomb pattern in the undithered survey or its low-level residual in the dithered surveys. These trends remain consistent across the five redshift bins.

Finally, in order to account for the spurious power introduced by the depth variations, we consider the relationship between the measured power spectrum and the true one, for a perfectly uniform survey:

$$\begin{eqnarray}&&\langle {P}_{\mathrm{measured}}({\boldsymbol{k}})\rangle =\int d{\boldsymbol{k}}^{\prime} \ {P}_{\mathrm{true}}(k^{\prime} )\ | W({\boldsymbol{k}}-{\boldsymbol{k}}^{\prime} ){| }^{2},\end{eqnarray} \tag{ 9 }$$

where $W({\boldsymbol{k}}-{\boldsymbol{k}}^{\prime} )$ is the survey window function, accounting for the effective survey geometry (Feldman et al. 1994; Sato et al. 2013). Projecting the 3D ${\boldsymbol{k}}$-space onto the 2D ℓ-space, we have

$$\begin{eqnarray}&&{C}_{{\ell },\mathrm{measured}}=\displaystyle \sum _{{\ell }^{\prime} }| {W}_{{\ell }-{\ell }^{\prime} }{| }^{2}\langle {C}_{{\ell }^{\prime} }\rangle +\delta {C}_{{\ell }},\end{eqnarray} \tag{ 10 }$$

where $\langle {C}_{{\ell }}\rangle$ is the expected power spectrum on the full sky, and $\delta {C}_{{\ell }}$ is an error term whose minimum variance is given by (see Dodelson 2003, Chapter 8 for details)

$$\begin{eqnarray}&&\ {({\rm{\Delta }}{C}_{{\ell }})}^{2}=\displaystyle \frac{2}{{f}_{\mathrm{sky}}(2{\ell }+1)}{\langle {C}_{{\ell }}\rangle }^{2},\end{eqnarray} \tag{ 11 }$$

where ${f}_{\mathrm{sky}}$ is the fraction of the sky observed, accounting for the reduction in observed power due to incomplete sky coverage. Since we consider only the WFD survey with masked shallow borders, f${}_{\mathrm{sky}}$ ≈ 37%–39% for the dithered surveys, while ${f}_{\mathrm{sky}}\approx 36 \%$ for the undithered survey. The expected power spectrum can be defined as

$$\begin{eqnarray}&&\langle {C}_{{\ell }}\rangle ={C}_{{\ell },\mathrm{LSS}}+\displaystyle \frac{1}{\bar{\eta }},\end{eqnarray} \tag{ 12 }$$

where $\bar{\eta }$ is the surface number density in steradians⁻¹; see Fall (1978), Huterer et al. (2001), and Jing (2005) for details. The first term in Equation (12) is the LSS contribution to the expected power spectrum, while the second term is the shot noise contribution arising from discrete signal sampling.

With no LSS and negligible shot noise, $\langle {C}_{{\ell }}\rangle \to 0$. However, as shown in Equation (7) , the observing strategy induces a bias in the measured power spectrum, leading to non-zero power even when $\langle {C}_{{\ell }}\rangle \to 0$. The uncertainty in this bias caused by imperfect knowledge of the survey performance limits our ability to correct for the OS-induced artificial structure. More quantitatively, we have

$$\begin{eqnarray}&&{({\sigma }_{{C}_{{\ell },\mathrm{measured}}})}^{2}={({\rm{\Delta }}{C}_{{\ell }})}^{2}+{({\sigma }_{{C}_{{\ell },\mathrm{OS}}})}^{2},\end{eqnarray} \tag{ 13 }$$

where the first term on the right is the minimum statistical uncertainty defined in Equation (11), while the second term corresponds to the contribution from the uncertainty in the bias induced by the observing strategy. Since the "statistical floor" ${\rm{\Delta }}{C}_{{\ell }}$ assumes no bias in ${C}_{{\ell }}$ measurements caused by the observing strategy, the OS-induced uncertainty ${\sigma }_{{C}_{{\ell },\mathrm{OS}}}$ must be subdominant to the statistical floor for an optimal measurement of BAO at a given redshift, i.e.,

$$\begin{eqnarray}&&{\sigma }_{{C}_{{\ell },\mathrm{OS}}}\ll {\rm{\Delta }}{C}_{{\ell }}=\sqrt{\displaystyle \frac{2}{{f}_{\mathrm{sky}}(2{\ell }+1)}}\left({C}_{{\ell },\mathrm{LSS}}+\displaystyle \frac{1}{\bar{\eta }}\right).\end{eqnarray} \tag{ 14 }$$

Here we note that the right side in Equation (14) is formally derived in Shafer & Huterer (2015); also see Huterer et al. (2013). These papers offer a detailed theoretical treatment of artificial structure induced by calibration errors, and while our approach is similar to theirs, we incorporate the additional effects of dust extinction, variations in survey depth, and incompleteness in galaxy detection.

Considering the case where ${C}_{{\ell },\mathrm{LSS}}=0$, we find ${C}_{{\ell },\mathrm{measured}}$, giving us ${C}_{{\ell },\mathrm{OS}}$ for each band and magnitude cut. Since ugri bands are the deepest and appear to have the greatest influence on photometric redshifts (A. Prakash 2016, private communication), we model the overall bias as the mean ${C}_{{\ell },\mathrm{OS}}$ across the four bands. We calculate ${\sigma }_{{C}_{{\ell },\mathrm{OS}}}$ as the standard deviation of ${C}_{{\ell },\mathrm{OS}}$ across the ugri bands, modeling uncertainties due to detecting galaxy catalogs in different bands. Therefore, ${\sigma }_{{C}_{{\ell },\mathrm{OS}}}$ should provide a conservative upper limit on the true uncertainty in ${C}_{{\ell },\mathrm{OS}}$.

The left panel in Figure 8 shows the full-sky galaxy power spectrum with the BAO signal for three galaxy catalogs, r < 24.0, 25.6, 27.5, for the five redshift bins: 0.15 < z < 0.37, 0.37 < z < 0.66, 0.66 < z < 1.0, 1.0 < z < 1.5, and 1.5 < z < 2.0. These spectra are pixelized in order to account for the finite angular resolution of our survey simulations, especially when comparing the uncertainties in ${C}_{{\ell },\mathrm{OS}}$ with the minimum statistical error in measuring BAO. Assuming that all the HEALPix pixels are identical, the pixelized power spectra can be approximated by multiplying the galaxy power spectra with the pixel window function.¹³

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The galaxy power spectra are calculated using the code from Zhan (2006), with modifications to account for BAO signal damping due to nonlinear evolution (Eisenstein et al. 2007). Using the galaxy redshift distribution from Abell et al. (2009), galaxies are assigned to the five redshift bins according to their photometric redshifts, with a time-varying but scale-independent galaxy bias of $b(z)=1+0.84z$ over scales of interest and a simple photometric redshift error model, ${\sigma }_{z}=0.05(1+z)$. Here we assume the cosmology with ${w}_{0}=-1,{w}_{a}=0,{{\rm{\Omega }}}_{m}=0.127,{{\rm{\Omega }}}_{b}=0.0223$, ${{\rm{\Omega }}}_{k}=0$, spectral index of the primordial scalar perturbation power spectrum n_s = 0.951, and primordial curvature power spectrum at k = 0.05 Mpc⁻¹, ${{\rm{\Delta }}}_{R}^{2}=2\times {10}^{-9}$.

The right panel in Figure 8 shows the minimum statistical uncertainty for the five redshift bins for all three galaxy catalogs; the uncertainties are calculated using ${{f}}_{{\rm{sky}}}$ from the undithered survey. We observe that while shallower galaxy catalogs lead to larger ${C}_{{\ell }}$ and ${\rm{\Delta }}{C}_{{\ell }}$, the difference is small and decreases with increasing redshift. For the lowest z-bin, 0.15 < z < 0.37, there is only about an 8% increase in ${C}_{{\ell }}$ and ${\rm{\Delta }}{C}_{{\ell }}$ when comparing the r < 25.6 catalog with r < 24.0.

First we calculate ${C}_{{\ell },\mathrm{OS}}$ and its uncertainties for 0.66 < z < 1.0 after only one year of survey in order to explore the quality of BAO study the first data release will allow. Figure 9 shows the ${C}_{{\ell },\mathrm{OS}}$ uncertainties as well as the minimum statistical error for 0.66 < z < 1.0 for various observing strategies, for r < 24.0 and r < 25.7 (corresponding to the gold sample, i < 25.3). We observe that the undithered survey leads to ${C}_{{\ell },\mathrm{OS}}$ uncertainties that are 1–3× the minimum statistical uncertainty for the gold sample at ${\ell }$ > 100, and only a few dither strategies are effective at reducing the difference. In particular, Random and RepulsiveRandom dithers are the most effective, reducing ${\sigma }_{{C}_{{\ell },\mathrm{OS}}}$ to nearly 1–2× the statistical floor. We note that FermatSpiral and SequentialHex dithers perform nearly as poorly as NoDither when implemented on FieldPerVisit and FieldPerNight timescales, while the PerNight timescale is more effective. On the other hand, we see that FieldPerVisit and FieldPerNight lead to smaller uncertainties for Random and RepulsiveRandom geometries. As expected, we see that a shallower sample r < 24.0 reduces the ${C}_{{\ell },\mathrm{OS}}$ uncertainties; the undithered survey still leads to ${\sigma }_{{C}_{{\ell },\mathrm{OS}}}$ about 3× the statistical floor, while Random and RepulsiveRandom dithers lead to uncertainties comparable to the statistical floor on some timescales. Here we note that since we do not mask borders when considering the one-year data, ${f}_{\mathrm{sky}}$ ≈ 42%–45% for the one-year survey, depending on the dither strategy.

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We then extend the calculation of the OS-induced power to the full 10-year survey. Figure 10 shows ${\sigma }_{{C}_{{\ell },\mathrm{OS}}}$ as well as ${\rm{\Delta }}{C}_{{\ell }}$ for 0.66 < z < 1.0 for three different magnitude cuts: r < 24.0, r < 25.7 and r < 27.5. We find that the undithered survey leads to ${\sigma }_{{C}_{{\ell },\mathrm{OS}}}$ 0.2–4 times the minimum statistical floor for r < 25.7 and r < 27.5; at ${\ell }$ > 100, only a very strict cut of r < 24.0 brings ${\sigma }_{{C}_{{\ell },\mathrm{OS}}}$ below ${\rm{\Delta }}{C}_{{\ell }}$. However, most dither strategies reduce the uncertainties below the statistical floor for galaxy catalogs as deep as r < 27.5, with exceptions of SequentialHex dithers on FieldPerVisit and FieldPerNight timescales. We note here that systematics correction methods such as template subtraction and mode projection can be applied to further reduce the contribution of ${C}_{{\ell },\mathrm{OS}}$ to the total ${C}_{{\ell }}$ uncertainties; e.g., see Elsner et al. (2016), Holmes et al. (2012). Such application appears necessary for the one-year survey as optimizing the observing strategy alone does not reduce the uncertainties in ${C}_{{\ell },\mathrm{OS}}$ below ${\rm{\Delta }}{C}_{{\ell }}$. However, the correction methods may not lead to significant improvements for a dithered 10-year survey, as optimizing the observing strategy is effective in reducing the uncertainties in ${C}_{{\ell },\mathrm{OS}}$ well below the statistical floor.

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To further our understanding, we repeat the 1-year and 10-year analysis for 1.5 < z < 2.0. We find similar qualitative results as those from 0.66 < z < 1.0 analysis: for the 1-year survey, Random and RepulsiveRandom perform well alongside FermatSpiral and SequentialHex on the PerNight timescale, while most dither strategies are effective for the 10-year survey, with the exception of SequentialHex on FieldPerVisit and FieldPerNight timescales.

The effect of magnitude cuts is further illustrated in Table 1, which includes the estimated number of galaxies for 0.15 < z < 2.0 from the r-band coadded depth for the 10-year survey after accounting for photometric calibration errors, dust extinction, and Poisson noise . We see that each magnitude cut eliminates a substantial number of galaxies. Also, as in Carroll et al. (2014), we see that dithering increases the estimated number of galaxies when compared to the undithered survey; the fractional difference in the number of galaxies from dithered to undithered surveys decreases with shallower surveys.