A control survey technique using a laser tracker and a digital level was introduced to the KEK e^{−}/e^{+} injector linac in 2020. Control surveys are continuously demonstrated during the machine’s downtimes every summer. Analysis of the two-year data reproduces their trends in terms of the fiducial points on the beam line. In our paper, we report on systematic coordinates and their error distributions evaluated by a control survey, compare them with a numerical survey simulation, and discuss newly encountered issues.

## I. INTRODUCTION

The operation of the KEK e^{−}/e^{+} injector linac (645 m long in total) on the Tsukuba campus started in 1989 and achieved 200 000 operation hours in 2020. It consists of a 120 m straight section, a 36 m arc section, which reverses the beam’s advancing direction, and a 492 m straight section. The latter was divided into 82 m (sector C) and 410 m (sectors 1–5) sections. Only the 410 m-straight section was utilized as the injector during the TRISTAN (until 1995) and Photon Factory (PF) operations. The former 120 m-straight section (sectors A and B), the arc section (J-arc), and sector C were constructed for the previous KEKB operation. The KEK injector linac is currently divided into 59 units.

Such standard accelerator components as accelerator tubes, magnets, vacuum systems, and diagnostic systems were mounted on a unit girder and installed in each unit. Based on their own beam energies, the KEK injector linac simultaneously distributes e^{−} or e^{+} beams to four different ring accelerators: PF (2.5 GeV e^{−}, 450 mA), PF-Advanced Ring (PF-AR, 6.5 GeV e^{−}, 50 mA), SuperKEKB High Energy Ring (HER, 7 GeV e^{−}, 2600 mA), or Low Energy Ring (LER, 4 GeV e^{+}, 3500 mA) .

PF is the Japanese dedicated synchrotron radiation light source accelerator in the x-ray region, which successfully emitted the first synchrotron radiation in 1982 and continues to operate today with several upgrades. PF is now the driving force behind both structural analysis and the development of matter and materials in a wide range of fields.

PF-AR was modified from TRISTAN’s front accelerator, where top quark research was demonstrated in the early 1980s. PF-AR is currently operating as a high-intensity pulsed synchrotron radiation light source for chronological experiments on molecules, crystals, and geoscience research.

On the other hand, SuperKEKB has been widely upgraded, both in the KEKB accelerator complex and the Belle-II detector. As a result, the detection performance of the decay points has been drastically improved, and such improvements are critical for experiments that indicate the time dependent CP violation of B mesons. Since the first successful collision of e^{−} and e^{+} beams in April 2018, the Belle-II experiment has steadily stored its experimental data, and in June 2022, the SuperKEKB recorded a peak luminosity of 4.65 × 10^{34} cm^{−2} s^{−1} and an integrated luminosity of 424 fb^{−1}.

^{35}cm

^{−2}s

^{−1}. For this purpose, the following normalized emittances for injection beams in both horizontal and vertical components are required:

To achieve these emittances and provide high quality, stable beams to each storage ring, the alignment tolerances for the magnets on each unit girder and the neighboring two unit girders must be settled at 50 and 100 *μ*m in one sigma, respectively. Note that since these calculated tolerances remain tentative, more realistic ones are now being estimated.

During both the SuperKEKB construction and upgrade stages and the recovery from the massive 2011 earthquake, each unit girder was aligned with a He–Ne laser baseline and quadrant photo-diode (QPD) sensors, which were mounted on both ends of the girder, referring to the laser-pointing coordinates on the sensor.^{1,2} Note that a laser tracker was utilized for isolated-magnet alignment until 2018.

A conventional control survey technique^{3} using a laser tracker and a digital level was introduced to the KEK injector linac in 2020 and has been continuously demonstrated in its downtimes every summer. For a control survey analysis, a geodetic line correction,^{4} which is evaluated using both the laser QPD data and the control survey data, is applied to the level data in addition to a conventional control survey analysis. This paper describes the systematic coordinates and their error distributions and discusses newly encountered issues.

## II. CONTROL SURVEY APPARATUS IN THE KEK INJECTOR LINAC

### A. Three-dimensional coordinates survey

The conventional control survey technique, which is well-established and feasible at a synchrotron light source, SPring-8,^{3} has been introduced to the KEK injector linac in 2020. The three-dimensional coordinates of magnets, monuments on the wall, and both ends of each unit girder (∼800 fiducial points) are surveyed with a laser tracker (Leica AT-401^{5}) and a spherical, 2.5 in. diameter mirror reflector (SMR). All fiducial point names are controlled by QR code labels. The survey work and post-processed analysis are performed by commercial software, Hexagon Spatial Analyzer (SA) Ultimate.^{6} A hand-held device with iOS is utilized for the remote operation of the laser tracker and fiducial point name input by QR codes.

A schematic view of a control survey with a laser tracker is shown in Fig. 1, and the length, number of units, magnets, and monuments for each sector are summarized in Table I. A Cartesian coordinate system is defined with two fiducial points: the center of a pulsed dipole magnet (PX_A1_M) installed in sector A and the center of a DC quadrupole magnet (QD_B7_4) in sector B, which is 111 m apart from PX_A1_M. The origin of the coordinate system is set at the center of PX_A1_M, and the vertical *z* axis is defined as the gravity direction at PX_A1_M. The *y* direction (from north to south) is determined with the above two magnets, and *x* is orthogonal to the *yz*-plane (from east to west), as shown in Fig. 1.

Sector name . | A . | B . | J-arc . | C . | 1 . | 2 . | 3 . | 4 . | 5 . |
---|---|---|---|---|---|---|---|---|---|

Length (m) | 48.0 | 76.8 | 36.1 | 81.6 | 83.3 | 92.9 | 69.9 | 79.5 | 76.8 |

No. of units | 4 | 8 | 1 | 8 | 7 | 7 | 8 | 8 | 8 |

No. of magnets | 65 (63) | 14 | 35 (36) | 18 (16) | 74 | 90 | 20 (10) | 16 (8) | 34 (26) |

No. of monuments | 15 | 19 | 10 | 21 | 32 | 32 | 17 | 19 | 20 |

Sector name . | A . | B . | J-arc . | C . | 1 . | 2 . | 3 . | 4 . | 5 . |
---|---|---|---|---|---|---|---|---|---|

Length (m) | 48.0 | 76.8 | 36.1 | 81.6 | 83.3 | 92.9 | 69.9 | 79.5 | 76.8 |

No. of units | 4 | 8 | 1 | 8 | 7 | 7 | 8 | 8 | 8 |

No. of magnets | 65 (63) | 14 | 35 (36) | 18 (16) | 74 | 90 | 20 (10) | 16 (8) | 34 (26) |

No. of monuments | 15 | 19 | 10 | 21 | 32 | 32 | 17 | 19 | 20 |

At least one station point (corresponding to the location of a survey instrument) is settled in each sector, and a networked control survey for the KEK injector linac is configured with 63 station points. One station point at a bypass line, which connects sectors A and 1, was added in 2021 (Fig. 1). The typical measurement accuracies for both the horizontal and vertical angles and the distance with the laser tracker are summarized in Table II. Air conditioning must be turned off (at least in any surveying area) during a laser tracker survey since the air flow degrades the measurement accuracy, especially the distance measurement. Unfortunately, since the air conditioning was fully operational during this survey, the distance accuracy of the KEK injector linac survey was ∼3 times worse than the SPring-8 survey,^{3} during which the air conditioning was completely stopped.

Leica AT-401 (3D-coordinates) | |

Horizontal angle | 2.7 ± 1.1 μrad |

Vertical angle | 2.5 ± 1.0 μrad |

Distance | $18.9\u221211.0+14.3\mu $m |

Trimble DiNi0.3 | |

Level | 4.7 μm |

Leica AT-401 (3D-coordinates) | |

Horizontal angle | 2.7 ± 1.1 μrad |

Vertical angle | 2.5 ± 1.0 μrad |

Distance | $18.9\u221211.0+14.3\mu $m |

Trimble DiNi0.3 | |

Level | 4.7 μm |

### B. Leveling survey

^{7}) and a bar-code scale. The 19 selected magnets are surveyed by moving the digital level to 36 station points: 18 points from sectors A to 5 and 18 from sectors 5 to A. Thus, it consists of a closed-loop survey. All line of sight (LOS) lengths, which connect station and fiducial points, must be identical to improve measurement accuracy. All of the level data are connected by differences in the elevation (

*h*

_{i},

*i*= 1, 2, …, 36) between these points. Since the leveling is a closed-loop survey, the first and last fiducial points are identical (PF_A1_M). Thus, the height difference (Δ

*H*) is given by

*H*has a finite value due to systematic error, which is known as loop closure error. Therefore, each

*h*

_{i}needs to be corrected by adding correction factor

*δ*

_{i}, and we assume that it increases proportionally to path length

*s*

_{i}at each fiducial point as follows (the loop closed correction):

*H*is evaluated as ∼120

*μ*m in the case of the KEK injector linac survey, which is equivalent to the case of SPring-8. The standard deviation for a 1 km double-run leveling of Trimble DiNi0.3 is 0.3 mm (catalog value), and the average LOS length is 15.8 m. Thus, the level of survey accuracy is estimated to be 4.7

*μ*m, which is also shown in Table II.

### C. Network analysis

Both three-dimensional coordinates and the leveling of the accelerator components are combined, and network analysis is done with them using SA. The network analysis is iterated to optimize the coordinates of the common fiducial points, which are surveyed at a minimum of three station points, by adjusting both the transfer and rotation matrix elements for all the station points by solving the observation equations with the least squares method.

*r*,

*θ*,

*ϕ*), where

*r*is the distance,

*θ*is the horizontal angle, and

*ϕ*is the vertical angle. Here, we discuss and treat the surveyed coordinates in the Cartesian coordinate system,

*F*, actual

*S*, and survey errors

*V*,

*F*as a function of the fiducial points’ nominal coordinates as

*F*′ and correction part Δ

*F*with a form of the Taylor expansion and denoted as a linear function,

*x*′,

*y*′, and

*z*′ are the approximated coordinates of the survey, written with their corrections

*dx*,

*dy*, and

*dz*,

*V*is denoted as

*i*common fiducial points are surveyed by

*k*station points. Here, we define two variables:

*M*= 3

*ik*and

*N*= 3

*k*, where 3 denotes the degree of coordinates

*x*,

*y*, and

*z*surveyed by the laser tracker. The observation equation is now written as

**is the vector of the residual errors, dimension**

*V**N*;

*A*is the Jacobian matrix of

*M*×

*N*;

**is the vector of the unknown corrections to the approximated coordinates, dimension**

*x**N*; and

**is the vector of the observed minus calculated (from the approximated coordinate) value, dimension**

*B**N*.

*W*(=

*C*

^{−1}) is introduced,

*C*is the covariant matrix and $\sigma i2$ is the variance, the coordinate uncertainty of the

*i*-th observation. Note that the weight factors are defined for each station point in SA since there is no correlation among the observations surveyed by different station points. Therefore, only the diagonal matrix elements are valid, and the off-diagonal ones are treated as zero only for the weight definitions. Next, the convergence condition of the least squares solution, Eq. (14), is rewritten as

At least one of the fiducial point’s coordinates is required to be fixed to obtain the converged solution of Eq. (18). Our control survey has 64 station points: 63 for the three-dimensional coordinate survey and one for the leveling survey described in Secs. II A and II B. Actually, the level survey has a total of 36 station points, as discussed above, although the number of leveling station points is treated as one in the network analysis due to the SA specifications. Only the elevation differences for the 19 fiducial points measured by the digital level are treated as fixed points since the digital level’s accuracy is relatively superior to that of the laser tracker. Further details of the numerical treatments or optimization processes in SA, such as the transformation and the rotation of station points, are described elsewhere.^{8,9}

Now, let us review the survey results. Since the conventional control survey was introduced to the KEK injector linac in 2020, we have performed the control surveys every year and completed data analyses for the 2020 and 2021 survey data. Figure 2 shows the evaluated survey data for both the *x* (upper) and *z* (lower) coordinates of the fiducial points on all the magnets in the KEK injector linac along the path length. The *x* coordinate is treated and discussed as the residual *δx* between the surveyed (*obs*) and designed (*des*) ones. The residuals of the coordinates (□) between the 2020 and 2021 survey data for both *x* (Δ*x*) and *z* (Δ*z*) are overlaid on the right axes on each histogram. As a reference, the sectors and locations of the expansion joints among the building blocks are displayed with hatching and vertical dotted lines, respectively. The discontinuities of the distribution of the *x* coordinates at the J-arc entrance and exit and the gradual slopes of the 0.12–0.13 mrad are recognized for both the 2020 and 2021 survey data, shown in Fig. 2 (upper).

The discontinuities at the J-arc and the slope were confirmed at the construction phase of the SuperKEKB upgrades by a theodolite, and the slope was not corrected due to the time constraints before the SuperKEKB’s upcoming commissioning. Δ*x* gradually increased beyond sector 3. We identified a small difference in the slope between the 2020 and 2021 surveys due to its gradual increase; since the cause remains unclear, this phenomenon is not understood yet, whether the analysis error or the mechanical deformation of the building blocks are related. We are closely monitoring the slope variation and analyzing further data, as well as preparing some crack-displacement-measuring sensors at the expansion boundaries to monitor the displacement among the building blocks.

A geodetic line correction, discussed in Sec. II D, was not applied to the *z*-coordinate distribution in Fig. 2. The details of the *z*-coordinate distribution are discussed below with the geodetic line correction.

Coordinate uncertainties for each fiducial point are calculated as diagonal matrix elements of the covariant matrix [Eq. (15)]. The evaluated survey uncertainties for the *x* (*U*_{x}, ○), the *y* (*U*_{y}, △), and the *z* (*U*_{z}, □) coordinates of all the magnets and monuments were compared with the years 2020 (upper) and 2021 (lower) in Fig. 3. Both *U*_{x} and *U*_{y} have characteristic distributions compared to such storage rings as SPring-8, where the distributions are monotonously periodic for the entire circumference.^{3} The variations of *U*_{x} along the path length show remarkable peak structures at the J-arc and sectors 1–3 for both the 2020 and 2021 surveys. The variation of *U*_{y} indicates a step structure at the J-arc, and at sectors A–B, it is larger than sectors C–5 by almost twice for both years. On the other hand, *U*_{z} almost equals the entire path length. Note that the weight factors [Eq. (15)] of the digital level are adjusted as their uncertainties are equivalent to ≃4.7 *μ*m. In this paper, we ignore the *y* coordinate, except for its uncertainty, since it does not provide any significant information compared to the *x* and *z* coordinates. The slope shape of *U*_{x} in sectors 1–2 of the 2021 survey is different from that of the 2020 survey due to the addition of a station point at the bypass line that connects sectors A and 1. The increases in survey uncertainty at sector 5 might be caused by a numerical analysis error that is also well-reproduced by simulations.

### D. Geodetic line correction for level components

We introduce and apply a geodetic line correction to transfer the survey data to the beam analysis and 3D model control in a computer aided design (CAD) environment, which require absolute coordinates with respect to an idealistic single plane. Moreover, since the coordinates of all control points are referred to from the CAD model in simulation, the correction is applied to the survey data to efficiently compare with the calculation.

*r*= 6371 × 10

^{3}m, the elevation difference Δ

*H*for a straight line of

*L*= 500 m is estimated to be

*θ*≃ 39

*μ*rad for

*L*= 500 m. In general, a network survey is performed by moving its station points along the geodetic curve. Therefore, a level network survey for any components on a long straight line, such as linacs over 50 m long, is required to apply the geodetic line correction, where Δ

*H*exceeds a few tens of micrometers, a value equivalent to the typical measurement accuracy of laser trackers.

We used two pieces of elevation data, a laser QPD (*z*_{PD}) and a leveling survey with a digital level (*z*_{DL}), for the geodetic line correction in the KEK injector linac. Before the QPD measurements for both sectors A–B and C–5, tilt adjustments were made for each laser injection point as their pointings were directly located at the centers on both ends of the QPDs. Each QPD housing has an arm structure with its fiducial point, whose level is equivalent to the QPD center. These fiducial points are used in network surveys with the laser tracker and the digital level. Since the QPD measurement system’s operation ceased in 2019 due to a fire, we separately used past datasets for the QPD measurements: sectors A–B measured in 2018 and sectors C–5 measured in 2016.

*z*

_{PD}(□),

*z*

_{DL}, and their residuals (or elevation

*dz*, ○) are related by

*z*

_{DL}distribution is corrected as its start and end elevations correspond to those of

*z*

_{PD}, which are also plotted as $zDL\u2032$ (△). We found that the maximum difference of elevation $(zPD\u2032)$ in sectors C–5 (

*L*≃ 500 m) is estimated to be 6.4 mm, which equals 5 mm in a rough calculation with Eq. (19).

The elevation residuals described in Eq. (21) are individually fitted with third polynomial fitting functions (*dz*_{fit}) for the *y* coordinate of each survey year in both sectors A–B and C–5.

*z*

_{cor}, are finally derived as

The corrected elevations of *z* of all the magnets surveyed in 2020 and 2021 are compared in Fig. 5, and the differences in corrected elevation Δ*z* between the 2020 and 2021 data are plotted. Compared to the case without correction [Fig. 2 (lower)], the maximum elevations of ∼9 mm at the J-arc and the minimum elevations of −4 mm at the boundary between sectors 2 and 3 were reduced, and the deviations of the elevation converged within −1.6–0.5 mm after applying the correction to the elevation for each year.

## III. SIMULATION OF CONTROL NETWORK SURVEY

We simulated a control network survey and performed it using one of the functions integrated into the SA Ultimate: Measurement Plans (SA–MP^{6}) for the following two reasons: (1) to optimize the numbers and the coordinates of both station points and monuments to reduce survey uncertainties as low as possible, and (2) to understand the characteristics and differences between real coordinates and those designed for fiducial points. No environmental effects, such as temperature and relative humidity distributions or air flow, were taken into account for the virtual control survey simulation. Since all the fiducial points are aligned on the designed coordinates, where the beam line level is on a completely flat plane, no geodetic line correction is required for the simulation.

The following is the simulation procedure:

Prepare two coordinate files of (1) station points and (2) magnets, QPD arms, and monuments as ASCII texts. The coordinates of (2) consist of both three-dimensional coordinates and levels that are assumed to be, respectively, surveyed by the laser tracker and the digital levels, where their survey accuracies depend on distance.

Connect each station point (1) to the fiducial points (2) within a maximum distance of 30 m from the station point as a real survey.

script survey works with the SA–MP and virtually constructs control survey datasets.

Execute a network analysis for the above virtual control survey data by applying the weight values for the digital level data.

The contents of the ASCII coordinate files for (1) and (2) are described as follows:

station points: {

*station point name*},*x*_{st},*y*_{st},*z*_{st},fiducial points: {

*station point name*}, {*fiducial point name*},*x*_{fp},*y*_{fp},*z*_{fp}.

The nine sectors, which are divided into 4–8 units for each (Table I), consist of 59 units. In the simulations, two sets for the number of station points and three sets for the number of monuments [i.e., 6 (=2 × 3) cases] are defined and compared, as shown in Table III. We re-define a case ID with *M*–*N* by combining individual indices *M*-: 1, 2 and *N*-: A, B, C (Table III), respectively, to simplify the following discussion. Case 1–A is approximately equivalent to the real control survey configuration. Note that a station point was also defined in the bypass line in the simulation.

Station point configurations . | Case 1 . | Case 2 . | Real control survey . |
---|---|---|---|

No. of station points | 63 | 103 | 61 |

No. of station points/unit | 1 | 2 | 1 |

Intervals in y (m) (averaged) | 10.0 | 5.1 | 10.3 |

Level of sensor (m) (min./max./ave.) | 1.8 (fix) | 1.8 (fix) | 1.2/2.1/1.5 |

Station point configurations . | Case 1 . | Case 2 . | Real control survey . |
---|---|---|---|

No. of station points | 63 | 103 | 61 |

No. of station points/unit | 1 | 2 | 1 |

Intervals in y (m) (averaged) | 10.0 | 5.1 | 10.3 |

Level of sensor (m) (min./max./ave.) | 1.8 (fix) | 1.8 (fix) | 1.2/2.1/1.5 |

Monument point configurations . | Case A . | Case B . | Case C . | Real control survey . |
---|---|---|---|---|

No. of points | 138 | 260 | 383 | 192 |

No. of points/unit | 2 | 4 | 6 | 2–3 |

Intervals in y (m) (averaged) | 4.6 | 2.5 | 1.7 | 4.1 |

Level of monument (m) (min./max./ave.) | 1.4 (fix) | 1.4 (fix) | 1.4 (fix) | 1.2/1.5/1.2 |

Monument point configurations . | Case A . | Case B . | Case C . | Real control survey . |
---|---|---|---|---|

No. of points | 138 | 260 | 383 | 192 |

No. of points/unit | 2 | 4 | 6 | 2–3 |

Intervals in y (m) (averaged) | 4.6 | 2.5 | 1.7 | 4.1 |

Level of monument (m) (min./max./ave.) | 1.4 (fix) | 1.4 (fix) | 1.4 (fix) | 1.2/1.5/1.2 |

The six configuration cases are compared for the evaluated horizontal residuals *δx* between calculated *x* and the designed one [Fig. 6 (upper)] and calculated elevation *z* [Fig. 6 (lower)] for all the magnets along the path length, respectively. All the fiducial points, which were virtually surveyed, are principally oriented on the designed coordinates with realistic survey accuracies depending on each distance, i.e., both long straight sections, where sectors A–B and 1–5 are parallel. However, the evaluated horizontal residuals *δx* varied over the middle of sector 1, although the elevation did not. We propose the following two reasons for these behaviors:

Two long straight sections, sectors A–B and C–5, are mechanically connected to the J-arc and the bypass line (Fig. 1), i.e., sectors from sector A to the middle of sector 1 consist of a loop structure as a storage ring. Therefore, a constraint on the horizontal coordinates for the loop structure is relatively severe compared to a one-path structure between the middle of sector 1 and the end of sector 5.

A packing factor of accelerator components over sector 3 is rather lower than the other area (Table I), denoting that the number of commonly surveyed fiducial points may be inadequate and, thus, such fewer packing factors may cause divergence on residual

*δx*.

On the other hand, Fig. 7 (top), (middle), and (bottom), respectively, show the calculated survey uncertainties, *U*_{x}, *U*_{y}, and *U*_{z}. These results reproduce the uncertainty distributions well in the real control survey (Fig. 3). The calculated survey uncertainty comparison among the six cases (Fig. 7) concludes that Case 2–C achieved the minimum uncertainties for *U*_{x}, *U*_{y}, and *U*_{z}, although only *U*_{x} exceeds the alignment tolerance of 100 *μ*m for unit girders. Further optimization of the number of station points and monuments or drastic improvement of the survey technique is required.

The characteristic shapes of the uncertainty distribution, especially for *U*_{x} and *U*_{y}, remain for each case. Although the characteristic distributions of real control survey uncertainties were initially considered due to environmental factors, we confirmed that at least there was no correlation between the uncertainties and the temperature or the relative humidity distribution because no environmental factors were considered in the simulations. One candidate for this characteristic distribution is that the nonuniformity of the number of common survey points for each station point along the entire path length may generate characteristic trends of survey uncertainty. Some divergences were found at sector 3 for both *U*_{y} and *U*_{z} in Case 1–C, as well as some singular points in the calculation. We also investigated the uncertainty divergences of *U*_{x} and *U*_{z} in sector 5 due to the lack of common survey points.

Currently, our control survey is demonstrated only with the laser tracker and digital level. Introducing other measurement systems, such as a stretched-wire measurement system^{10,11} or an iris diaphragm laser alignment system,^{12} is expected to improve the accuracy of our control survey, especially for the *y*-component, and this is our future plan.

## IV. SUMMARY

We successfully performed a three-dimensional control survey using a conventional technique with a laser tracker and a digital level.

A geodetic line correction for leveling was found to be essential, especially for a survey in a long straight line that compared the survey data measured by a laser QPD and a conventional method.

Horizontal coordinate *x* has discontinuities at the J-arc and a gradual slope of 0.12–0.13 mrad from the J-arc exit to the end of sector 5. It was already confirmed at the construction phase of the SuperKEKB upgrade by theodolite. The slight slope difference between the 2020 and 2021 data increases over sector 3, for which more statistics are required for an investigation. In addition, some crack-displacement-sensors are now prepared at the expansion boundaries among building blocks to monitor their displacements.

A realistic control network survey was simulated by SA–MP through which six cases of different configurations were compared in terms of both station points and monuments to optimize the number and location of both the station and fiducial points to reduce the survey uncertainty and investigate the characteristics and differences between the real coordinates and the designed ones of the fiducial points. We found that the evaluated survey uncertainties of *U*_{x}, *U*_{y}, and *U*_{z} were minimum in Case 2–C, although only *U*_{x} exceeded the targeted alignment tolerance. The characteristic shapes of the survey uncertainty remained in all cases, especially for *U*_{x} and *U*_{y}. To reduce each uncertainty distribution as much as possible, i.e., to remove each characteristic shape of the distribution, increasing the number of common survey points between station points might be more effective than optimizing the configuration in terms of the numbers of station points and monuments in each unit.

Our latest survey results are now ready to be transferred to the beam dynamics codes in the KEK injector linac to conduct a more realistic beam dynamics analysis.

## ACKNOWLEDGMENTS

First, we thank K. Hisazumi, K. Kimura, Y. Mizukawa, K. Suzuki, and S. Ushimoto of the Mitsubishi Electric System & Service Co. Ltd. for their patient control survey work. We thank Dr. S. Matsui of RIKEN Harima, Dr. K. Mishima of PASCO Corporation, and Professor T. Kamitani of KEK for their beneficial discussions and comments. We also appreciate the precise advice provided by Professor N. Iida of KEK and her colleagues based on their perspectives on beam dynamics.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Y. Okayasu**: Conceptualization (lead); Data curation (lead); Formal analysis (equal); Investigation (equal); Methodology (lead); Supervision (lead); Writing – original draft (lead). **T. Suwada**: Conceptualization (equal); Investigation (equal); Methodology (equal); Project administration (equal); Writing – review & editing (equal). **K. Kakihara**: Conceptualization (equal); Investigation (equal); Methodology (equal). **M. Tanaka**: Data curation (equal).

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

## REFERENCES

^{+}/e

^{−}injector linac

*Engineering Surveying Technology*