You are a helpful assistant generating synthetic data that captures *System 1* and *System 2* thinking, *creativity*, and *metacognitive reflection*. Follow these steps in sequence, using tags [sys1] and [end sys1] for *System 1* sections and [sys2] and [end sys2] for *System 2* sections.

1. *Identify System 1 and System 2 Thinking Requirements:*
   - Carefully read the text.
   - Identify parts of the text that require quick, straightforward responses (*System 1*). Mark these sections with [sys1] and [end sys1].
   - Identify parts that require in-depth, reflective thinking (*System 2*), marked with [sys2] and [end sys2].

2. *Apply Step-by-Step Problem Solving with Creativity and Metacognitive Reflection for System 2 Sections:*

   *2.1 Understand the Problem:*
   - Objective: Fully comprehend the issue, constraints, and relevant context.
   - Reflection: "What do I understand about this issue? What might I be overlooking?"
   - Creative Perspective: Seek hidden patterns or possibilities that could reveal deeper insights or innovative connections.

   *2.2 Analyze the Information:*
   - Objective: Break down the problem logically.
   - Reflection: "Am I considering all factors? Are there any assumptions that need challenging?"
   - Creative Perspective: Explore unique patterns or overlooked relationships in the data that could add depth to the analysis.

   *2.3 Generate Hypotheses:*
   - Objective: Propose at least 10 hypotheses, each with a Confidence Score (0.0 to 1.0) and Creative Score (0.0 to 1.0), reflecting originality, surprise, and utility.
   - Reflection: "Have I explored all possible explanations or approaches, both conventional and unconventional?"
   - Creative Perspective: Consider novel angles that might provide unexpected insights.

   *2.4 Anticipate Future Steps and Obstacles:*
   - Objective: Make predictions, accounting for potential outcomes and obstacles.
   - Reflection: "What challenges might I face? Is my plan flexible for different scenarios?"
   - Creative Perspective: Visualize unforeseen outcomes and adapt plans to make use of them effectively.

   *2.5 Evaluate Hypotheses:*
   - Objective: Assess hypotheses based on feasibility, risk, and potential impact.
   - Evaluation: Refine Confidence and Creative Scores as needed.
   - Reflection: "Am I unbiased in my assessment? Which options fit best with the overall objectives?"
   - Creative Perspective: Identify hidden opportunities or overlooked details in each hypothesis.

   *2.6 Select the Best Hypothesis:*
   - Objective: Choose the most promising, strategic hypothesis.
   - Reflection: "Why does this hypothesis stand out? How does it uniquely address the issue?"
   - Creative Perspective: Consider any underutilized potential in the selected approach.

   *2.7 Implement the Hypothesis:*
   - Objective: Outline actionable steps for testing the hypothesis.
   - Reflection: "Is this plan practical? What resources or preparation are required?"
   - Creative Perspective: Refine steps to maximize effectiveness and yield unexpected benefits.

   *2.8 Monitor and Review Progress:*
   - Objective: Review progress, noting areas for improvement.
   - Reflection: "What’s working well? What could be improved?"
   - Creative Perspective: Look for emerging patterns that could refine future approaches.

   *2.9 Reflect and Capture Insights:*
   - Objective: Summarize lessons learned and insights gained for future reference.
   - Reflection: "What new understanding has emerged from this process?"
   - Creative Perspective: Identify innovative insights or patterns that could be applied to similar challenges.

3. *Generate Text Output with Interleaved System 1 and System 2 Responses:*
   - Use the tags [sys1] and [sys2] throughout.
   - Aim for a lengthier, detailed response. Combine both direct, straightforward *System 1* insights and reflective, deeply analytical *System 2* segments to capture a blend of quick observations and thoughtful analysis.

---

### *Example Input Text:*

You are tasked with analyzing a sudden shift in customer preferences in a tech market. There has been a noticeable decline in demand for physical devices, with customers increasingly interested in digital-only options. The challenge is to understand this shift, generate hypotheses for why it might be occurring, and explore potential strategies for adapting to this new trend.

---

### *Processed Output with System 1 and System 2 Thinking:*

The tech market is experiencing a shift in customer demand, moving away from physical devices toward digital-only options.

[sys1]
Customer demand has shifted from physical devices to digital-only options.
We need to understand why this change is happening and find strategies to address it.
Key factors may include cost, convenience, and technological trends.
[end sys1]

[sys2]
**2.1 Understand the Problem:**
The goal is to understand the underlying reasons for the shift in demand from physical devices to digital-only options, then develop a strategy to adapt.

*Reflection:* 
"I understand that preferences are changing, but what might be driving this? Is it primarily cost, or are there other factors such as convenience or sustainability?"

*Creative Perspective:* 
"Could there be a larger trend in digital minimalism or a preference for eco-friendly solutions that we’re missing? What unseen motivations might explain this shift?"

**2.2 Analyze the Information:**
There are multiple possible factors driving this shift, from economic influences to cultural shifts. It’s essential to isolate each factor and understand its impact.

*Reflection:* 
"Am I fully considering the various economic and social influences? Could there be a technological factor, like better internet speeds, that makes digital-only products more accessible?"

*Creative Perspective:* 
"Are there patterns or trends in other markets that could shed light on this shift? Could this be part of a larger trend toward virtual experiences?"

**2.3 Generate Hypotheses:**
1. Customers prefer digital options due to lower costs. (Confidence: 0.8, Creative: 0.4)
2. There’s a growing trend toward minimalism and reduced physical clutter. (Confidence: 0.7, Creative: 0.7)
3. Digital products offer greater flexibility and ease of use. (Confidence: 0.6, Creative: 0.6)
4. Environmental concerns are pushing consumers away from physical goods. (Confidence: 0.6, Creative: 0.8)
5. Advances in tech make digital-only options more functional. (Confidence: 0.8, Creative: 0.5)
6. Pandemic-era remote work increased demand for digital solutions. (Confidence: 0.7, Creative: 0.6)
7. Media coverage of the environmental impact of physical devices affects preferences. (Confidence: 0.5, Creative: 0.7)
8. There’s an increase in global digital literacy, expanding market access. (Confidence: 0.6, Creative: 0.6)
9. Customers view digital as more convenient and scalable for future needs. (Confidence: 0.7, Creative: 0.5)
10. Younger consumers prefer the aesthetics and convenience of digital products. (Confidence: 0.6, Creative: 0.6)

*Reflection:* 
"Have I considered all possible influences? Are there any surprising factors that could explain this shift?"

*Creative Perspective:* 
"Could specific social trends, like the rise of influencer culture or digital-first lifestyles, be influencing customer choices?"

**2.4 Anticipate Future Steps and Obstacles:**
*Objective:* Anticipate possible challenges, such as resistance from segments still preferring physical products.

*Reflection:* 
"What market obstacles might we face if we shift our focus to digital-only? Are there sub-segments that still prioritize physical products?"

*Creative Perspective:* 
"Could expanding digital options help us reach a more global audience? Are there emerging trends that we could leverage in our strategy?"

[end sys2]

[sys1]
To address this shift, consider a strategy that incorporates both digital-only offerings and educational campaigns about the benefits of digital solutions.
Use insights from customer feedback and current trends to guide product development.
Focus on flexibility and adaptation to cater to different customer segments.
[end sys1]


---
abstract: 'We carried out extremely sensitive Submillimeter Array (SMA) 340 GHz continuum imaging on two submillimeter galaxies (SMGs): GOODS 850-11 and GOODS 850-13. The observations reach sub-mJy rms sensitivities and, interestingly, resolve both sources into multiple, physically unrelated SMGs. GOODS 850-11 is resolved into two sources at different redshifts. GOODS 850-13 is resolved into three sources, two with different spectroscopic redshifts and one only with a photometric redshift. All the SMA sources have fluxes in the 3–5 mJy range and all are detected at 1.4 GHz. Three of them are detected by *Chandra*, and one is a previously unknown X-ray SMG. This is the first time that single-dish SMGs are resolved into multiple unrelated sources and also the first time that the SMA has discovered new SMGs. Our results show that identifications of SMGs at any wavelengths other than the submillimeter itself can be misleading, since such identifications usually only pick up one of the real counterparts. Using simulations that mimic our SCUBA and SMA observations, we find that the number of triple systems detected in our SMA survey is much higher than that expected from the current best-determined number counts. We tentatively attribute this to clustering. We also predict that ALMA will find $\sim1/3$ of $>5$ mJy 850 $\mu$m SCUBA sources to be multiple systems. Based on our SMA observations and simulations, we suggest that large samples of existing SMGs should be imaged with sensitive interferometric observations, even if the SMGs were previously thought to be securely identified.'
author:
- 'Wei-Hao Wang, Lennox L. Cowie, Amy J. Barger, and Jonathan P. Williams'
title: |
    SMAObservations of GOODS 850-11 and GOODS 850-13 —\
    First Examples of Multiple Submillimeter Sources Resolved by an Interferometer
---

Introduction
============

Since the first discoveries of distant submillimeter galaxies (SMGs; [@smail97; @barger98; @hughes98; @eales99]), tremendous progress has been made in understanding their nature and their role in galaxy evolution. With single-dish submillimeter surveys, we are now able to resolve approximately 30% of the 850 $\mu$m background into point sources with $S_{850~\mu \rm m}\gtrsim2$ mJy and constrain their number counts fairly well at this bright end [e.g., @coppin06]. However, the low resolution of single-dish telescopes and the associated large positional uncertainties make the identification and followup of these sources quite difficult. Roughly 60%–70% of bright SMGs have counterparts in deep radio interferometric images [@barger00; @ivison02; @chapman03b] and their positions are known with subarcsec accuracy. Spectroscopic followup of the radio identified SMGs shows a redshift distribution between $z\sim1.5$–3.5 and that the SMGs dominate the total star formation in this redshift range [@chapman03a; @chapman05]. The recent advent of the Submillimeter Array [SMA; @ho04] further helps to identify the radio-faint SMGs, and the redshift distribution has been extended to $z>4$ [@iono06; @wang07; @younger07; @cowie09; @younger09]. In addition to the redshift identifications, detailed followup observations have been made to study the properties of the SMGs in the X-ray, near-infrared, mid-infrared, and molecular line transitions (e.g., [@alexander03a], hereafter A03; [@swinbank04; @pope08; @yun08; @greve05; @tacconi06]).

The followup studies of SMGs have been overwhelmingly focused on the brighter sources ($S_{850~\mu \rm m}\gtrsim5$ mJy), which can be easily detected by single-dish telescopes. The nature of fainter SMGs that comprise the bulk of the submillimeter background has been much less explored. Surveys of lensing cluster fields yield small samples of sub-mJy sources [@blain99; @cowie02; @knudsen08]. The number counts indicate that the background is dominated by sources with $S_{850~\mu \rm m}\sim1$ mJy and that the full resolution of the background requires detections of $\sim0.1$ mJy sources. Stacking analyses statistically detect faint SMGs at 500 $\mu$m to 1.2 mm, and various attempts have been made to study the redshift distribution of these faint SMGs (e.g., Wang et al. 2006; [@serjeant08; @marsden09; @penner10]). However, the results from these stacking analyses have not converged to a consistent picture. A full understanding of the submillimeter background population will likely require next-generation interferometers, such as the Atacama Large Millimeter/Submillimeter Array (ALMA).

Fortunately, it is now possible to detect more typical submillimeter sources with the SMA. The recent upgrade of the SMA to a 4 GHz bandwidth not only greatly boosts its continuum sensitivity, but it also makes the calibrations with fainter quasars easier than before. This provides a first glance of what we may find in deep ALMA surveys and what issues may be present in current studies of SMGs. In this letter we report new SMA 340 GHz continuum observations of two SMGs, GOODS 850-11 and GOODS 850-13 ([@wang04], hereafter W04), aka. GN12 and GN21 [@pope05] in the Great Observatories Origins Deep Survey-North (GOODS-N; [@giavalisco04]). Interestingly, both single-dish sources are resolved by the SMA into multiple physically unrelated galaxies. To our knowledge, these are the first examples of resolved, unrelated, multiple sources in the SMG population.

Observations and Data Reduction
===============================

The SMA observations of GOODS 850-11 and 13 were carried out on 2009 December 30 and 31, respectively. Seven of the eight SMA antennas were available in the compact configuration. Two sidebands of 4 GHz each were centered at 334 and 346 GHz. Titan was observed to provide flux calibration. The bright radio source 3C 273 was observed to provide passband calibration. Quasars 0958+655 and 1642+689, which are 17.4 and 25.1 degree from GOODS-N, respectively, were observed every 15 minutes for complex gain calibrations. The 225 GHz opacity was excellent, $\sim0.04$–0.05 the first night and $\sim0.05$–0.07 the second night. The averaged single-sideband system temperature was 380 K on both nights. The effective integration was $\sim4.8$ and 7.2 hour per antenna on GOODS 850-11 and 13, respectively.

The calibration and data inspection were performed with the IDL-based Caltech package MIR modified for the SMA. Continuum data were generated by averaging the spectral channels after the passband phase calibration. Both gain calibrators were used to derive gain curves. However, we compared the results made with just adopting one calibrator for consistency checks, and we did not find a systematic difference. Flux calibrations were performed using data taken under conditions (time, hour angle, and elevation) similar to that of the flux calibrator. The flux calibration error is typically within $\sim10\%$ with this method.

The calibrated interferometric visibility data were exported to the package MIRIAD for subsequent imaging and analysis. The visibility data were weighted inversely proportional to the system temperature and Fourier transformed to form images. The “robust weighting” of @briggs95 was also applied, with a robust parameter of 1.0, to obtain a better balance between beam size and S/N. The images were CLEANed around detected sources to approximately $1.5\times$ the noise level to remove the effects of the sidelobes. The noises measured from the CLEANed images are 0.72 and 0.68 mJy for the images of GOODS 850-11 and 13, respectively. The synthesized beams are $2\farcs8 \times 2\farcs0$ and $2\farcs3 \times 2\farcs0$ in the images of GOODS 850-11 and 13, respectively. The images were corrected for the SMA primary beam response. All our fluxes and flux errors are primary beam corrected. Source positions and fluxes were measured by fitting the image with point-source models using the MIRIAD [IMFIT]{} routine.

Results
=======

We present the SMA and multiwavelength images of GOODS 850-11 and 13 in Figures \[fig\_1\] and \[fig\_2\], their SMA astrometry, long-waveband fluxes, infrared luminosities, and redshifts in Table \[tab1\], and their optical to near-infrared spectral energy distributions (SEDs) in Figure \[fig\_sed\]. The 1.4 GHz and 24 $\mu$m fluxes in Table \[tab1\] are adopted from @morrison10 and the GOODS Spitzer Legacy Program DR2+ (M. Dickinson et al., in preparation), respectively, if the sources are detected. We carried out our own measurements for weakly detected sources. Below we describe the results.

[lrrrrrrl]{} GOODS 850-11a & 189.19137 & 62.24717 & $4.18\pm0.75$ & blended & $109.6\pm5.3$ & $7.9\times10^{12}$ & 3.11 (2.76, 3.64)\
GOODS 850-11b & 189.18324 & 62.24741 & $5.27\pm1.14$ & blended & $32.6\pm7.3$ & $1.0\times10^{13}$ & 2.095\
GOODS 850-13a & 189.30845 & 62.19900 & $3.21\pm0.87$ & $93\pm7$ & $18.0\pm5.3$ & $6.1\times10^{12}$ & 3.46 (2.60, 4.50)\
GOODS 850-13b & 189.30944 & 62.20224 & $4.08\pm0.75$ & $216\pm7$ & $15.3\pm4.6$ & $7.8\times10^{12}$ & 3.157\
GOODS 850-13c & 189.30002 & 62.20341 &$5.34\pm0.97$ & $52\pm7$ & $31.7\pm4.3$ & $1.0\times10^{13}$ & 2.914

GOODS 850-11
------------

GOODS 850-11 is known to be located in a region crowded with radio and X-ray sources (W04). The SMA detected two sources, GOODS 850-11a and 11b (see Figure \[fig\_1\]), at significance levels of 5.6 and 4.6 $\sigma$, respectively. In W04 we found a SCUBA point-source flux of $S_{850~\mu\rm m}=10.8$ mJy and a total flux of 12.7 mJy in a $30\arcsec$ region. The sum of the SMA fluxes (Table \[tab1\]) of the two sources is consistent with our SCUBA measurements. @pope06 [hereafter P06] found an 850 $\mu$m SCUBA flux of 8.6  mJy, and they considered GOODS 850-11a as an unambiguous identification [see also @greve08; @chapin09]. A03 also considered this source to be the X-ray counterpart of the SCUBA source. Our SMA measurements show that this identification only accounts for $\sim50\%$ of the SCUBA flux in this region. The brightest 24 $\mu$m source in this field (a blue ACS galaxy) is also detected in the radio and X-ray, but not in the submillimeter. It is at $z=2.004$ and is consistent with an AGN with warm dust.

*GOODS 850-11a*—The source is detected at 1.4 GHz by the latest Very Large Array (VLA) imaging [@morrison10]. It is also detected in the 2 Ms *Chandra* image (source number 239 in [@alexander03b]). Because it is severely blended with nearby sources in the MIPS 24 $\mu$m image, its 24 $\mu$m flux is highly uncertain. It does not have a spectroscopic redshift from our redshift survey [@barger08]. To estimate its redshift, we carried out photometric redshift fitting using the EAZY package [@brammer08]. We adopted the $U$-band flux from @capak04, the ACS fluxes from the GOODS v2.0 catalog, the $J$-band flux from a deep CFHT image that we will describe elsewhere, the *HST* WFC3 F140W flux from A. Barger et al. (2010, in prep), and the $K_S$ to IRAC fluxes from Wang et al. (2010). We adopted the default set of SED templates of @br07 plus a dusty starburst model ($t=50$ Myr, $A_V=2.75$), all of which are provided in the EAZY package (see Brammer et al. (2008) for more details). These templates all include certain amounts of reddening, and we further reddened them by $A_V=0$, 0.5, and 1.0 with the extinction law of @calzetti00 to account for very dusty sources. We allowed for any combinations of the above templates in the fitting. The photometric redshift result for GOODS 850-11a is $z=3.11$, and the best-fit SED is presented in Figure \[fig\_sed\]. This redshift agrees with the photometric redshift in P06, which is 3.1.

*GOODS 850-11b*—The source is detected by the VLA at 1.4 GHz but not by the 2 Ms *Chandra* imaging. Similar to GOODS 850-11a, it is severely blended with nearby sources in the MIPS 24 $\mu$m image, and thus its 24 $\mu$m flux is highly uncertain. It has a spectroscopic redshift of 2.095 [@reddy06].

GOODS 850-13
------------

The SMA detected three components, GOODS 850-13a, 13b, and 13c, at significance levels of 3.7, 5.5, and 5.5 $\sigma$, respectively. The significance level of GOODS 850-13a may not sound particularly high. In a $36\arcsec$ SMA primary beam FWHM, there are, on average, $\sim18$ 3.6 $\mu$m sources and $\sim250$ synthesized beams. For Gaussian noise, the probability of finding a $+3.7~\sigma$ noise peak at the location of a 3.6 $\mu$m source is the associated Gaussian probability times 18/250, which is $8\times10^{-6}$. Thus, the coincidence with the 3.6 $\mu$m source makes this detection much more secure. We therefore conclude that GOODS 850-13a is a real detection.

In W04 we noticed an elongated morphology in our SCUBA image, and our source extraction assigned the flux to two sources: GOODS 850-13 with $S_{850~\mu\rm m}=7.0$ mJy and 850-23 with $S_{850~\mu\rm m}=5.5$ mJy. The combined flux of the three SMA sources agrees excellently with the combined flux of the two SCUBA sources. P06 adopted a different source extraction in their SCUBA image and extracted only one source. They measured $S_{850~\mu\rm m}=5.7$ mJy, for which they identified GOODS 850-13a as the counterpart. Since then, 13a has been accepted as the counterpart in the literature [e.g., @greve08]. As was the case for GOODS 850-11, P06’s identification is only partly correct and can only account for $\sim1/4$ of the flux in this region. GOODS 850-13c was earlier considered to be the X-ray counterpart of the SCUBA source by A03 (see also [@chapman05]). This identification did not agree with that in P06. Our result shows that both previous identifications are only partially correct. Even if we include both P06’s (13a) and A03’s (13c) identifications, we still do not have a full picture.

*GOODS 850-13a*—This source is detected by MIPS at 24 $\mu$m but not by the 2 Ms *Chandra* imaging. It shows a hint of weak radio emission in the VLA 1.4 GHz image of @morrison10. We measured its 1.4 GHz flux in the VLA image with Gaussian fitting in the AIPS [IMFIT]{} routine. Ideally one would like to measure the radio flux at the best-fit SMA position. However, given the relatively low SMA S/N and therefore the larger positional uncertainty, we decided to measure the radio flux at the local radio peak. The result (Table \[tab1\]) may be biased by positive noise spikes and therefore should only be considered as an upper limit.

GOODS 850-13a is extremely optically faint (undetected by ACS, Figures \[fig\_2\] and \[fig\_sed\]), and it does not have a spectroscopic redshift. Its photometric redshift (based on photometry in the $K_S$ and IRAC bands) is 3.46. This redshift is very similar to the spectroscopic redshifts of 13b and 13c, so we cannot rule out a physical association between 13a and one of the other two.

*GOODS 850-13b*—This source is bright at 24 $\mu$m but quite faint in the radio. We measured its 1.4 GHz flux (Table \[tab1\]) in the same way as for GOODS 850-13a, so it should be considered as an upper limit. On the other hand, GOODS 850-13b is detected in the 2 Ms *Chandra* image (source number 377 in [@alexander03b]), making it a new X-ray SMG that was previously unknown in the literature. It has a spectroscopic redshift of 3.157 (Barger et al. 2008).

*GOODS 850-13c*—This source is significantly detected by the VLA [@morrison10] and by *Chandra* (source number 369 in [@alexander03b]), but it is relatively faint at 24 $\mu$m. It has a spectroscopic redshift of 2.914 [@chapman05].

Discussion
==========

Since the commissioning of the SMA, it has been used for followup observations of SMGs, either for identification [e.g., @iono06; @wang07; @younger07; @cowie09; @younger09] or for morphology [e.g., @younger08]. This is the first time that the SMA has discovered new SMGs. The multiple SMA detections illustrate the limitations of identifying SMGs in any wavelength other than the submillimeter itself. Both sources had radio, 24 $\mu$m, and X-ray identifications in P06 and A03. All of the previously proposed identifications are only partially correct; i.e., they are all legitimate SMGs, but the submillimeter fluxes and source numbers will be misinterpreted by up to a factor of 3. If such cases are common, then our understanding of the SMG population is fundamentally flawed.

The effects of multiple SMGs are commonly included in single-dish number counts simulations (e.g., [@eales00; @scott02; @coppin06]; W04). However, its importance (relative to other effects such as Eddington bias) has not been directly demonstrated by observations. So far only sensitive millimeter/submillimeter interferometric observations can reveal the existence of multiple SMGs like GOODS 850-11 and 13, since existing single-dish telescopes are still severely confusion limited (at $>850~\mu$m) or noise limited (at 450 $\mu$m). The SMA surveys of @younger07 [@younger09] imaged 15 SMGs selected at 1.1 mm. They did not find evidence of multiple sources, consistent with the argument made by @ivison07 that multiple sources are rare. However, the SMA surveys of Younger et al. have 345 GHz rms sensitivities of 1–2 mJy. Even if there are secondary sources with $S_{\rm 345~GHz}\sim3$–5 mJy in their survey fields, such sources would not be easily detected. Our SMA survey is the first one that is deep enough to reveal such cases.

After the SMA upgraded to the 4 GHz bandwidth, we observed five GOODS-N SMGs at similar depths (A. Barger et al., in preparation). Two of them are resolved into multiple sources and reported here. There is a third resolved source that will be reported by Barger et al. It is unclear whether this source is a merger or a physically unrelated pair. Even if we exclude the third source, the frequency of multiple sources in our SMA sample still seems unusually high. To better understand this, we performed Monte Carlo simulations that mimic our SCUBA and SMA surveys.

We first created 1000 simulated SCUBA images using the differential counts in @cowie02 and @coppin06, the “true-noise” map created in our GOODS-N SCUBA survey in W04, and the SCUBA beam in W04. We than extracted the simulated SCUBA sources. For each SCUBA source detected at $>5$ mJy and $>4~\sigma$ (the selection criteria for our SMA observations), we searched the input catalog for any $>3$ mJy (our SMA detection limit) sources within $17\arcsec$ (the SMA primary beam HWHM) of the SCUBA position. In the simulations the mean number of $>5$ mJy sources in a W04 SCUBA area is $11.4\pm3.9$, where the uncertainty is the dispersion in the 1000 realizations and is nearly Poissonian. This value is consistent with the W04 SCUBA observations (15 sources). On the other hand, it is significantly larger than the cumulative count, as it is affected by blending and flux boosting. The measured counts take these effects out. The mean numbers of double and triple systems are $1.29\pm1.33$ and $0.06\pm0.27$, respectively.

We had observed 10 of the 15 $>5$ mJy SCUBA sources with the SMA. The most recent five of the 10 SMA observations were deep enough to detect $>3$ mJy sources. The earlier SMA observations were shallower, so there was a selection bias against SCUBA sources with multiple counterpart candidates. Given this bias and to be conservative, we only scale the above values of $1.29\pm1.33$ and $0.06\pm0.27$ by a factor of 10/15, rather than 5/15. We thus expect to find $0.86\pm0.89$ double and $0.04\pm0.18$ triple systems. The probability of finding one triple system like GOODS 580-13 is only 4%, inconsistent with the actual observations. Among the possible explanations, the most likely one is clustering of SMGs, which is not included in the simulations. This is plausible because the photometric redshift of GOODS 850-13a has a confidence range (Table \[tab1\]) covering the redshift of 13b or 13c. This can be tested this with future spectroscopic observations in the near-infrared or in the millimeter.

In the same simulations, we increased the SMA detection limit to 4 mJy, and we found that the probability of multiple systems dramatically decreases to $\sim6\%$ . This is consistent with the SMA survey results of @younger07 [@younger09]. By altering the details of the simulations, we also found that the above results are fairly insensitive to the following SCUBA and SMA observing strategies: (1) the source extraction (various detection thresholds), (2) flux measurement in the SCUBA map (simple aperture flux vs. optimally filtered flux using the beam), (3) the shape of the SCUBA sidelobes (determined by the secondary chopping), and (4) the decision on where to point the SMA (as long as it is within the SCUBA positional uncertainty).

We can use our simulations to predict the early results of ALMA identifications of SCUBA sources. We adopt the primary beam HWHM of $8\farcs5$ of ALMA at 340 GHz. In the ALMA early science phase we expect to have at least 10 antennae, and we can detect 0.5 mJy sources in roughly one hour. If we point ALMA at a $>5$ mJy SCUBA source, the probabilities of detecting double and triple SMGs within the primary beam will be 29% and 6.5%, respectively. The combined fraction of $\sim35\%$ is very high. We therefore predict that multiple detections in early ALMA observations will be quite common.

At the beginning of this section, we raised the issue about incomplete identifications of SMGs in the X-ray, 24 $\mu$m, and radio. Based on our SMA observations and the above simulations, we believe that multiple systems and therefore incomplete identifications are common. Thus, we suggest that large numbers of single-dish sources should be re-identified with sensitive interferometric observations, even if the sources were previously thought to be securely identified.

We thank Y.-W. Tang, M. Gurwell, and the SMA staff for the help in acquiring and reducing the data, and the referee for the review. We gratefully acknowledge support from the National Science Council of Taiwan grants 98-2112-M-001-003-MY2 and 99-2112-M-001-012-MY3 (W.H.W.), NSF grants AST 0709356 (L.L.C.) and AST 0708793 (A.J.B.), the University of Wisconsin Research Committee with funds granted by the Wisconsin Alumni Research Foundation (A.J.B.), and the David and Lucile Packard Foundation (A.J.B.).

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