An Empirical Study On Imaging Sensitivity Of SKA1-LOW Based on Point Source Sensitivity

The Square Kilometre Array (SKA) radio telescope will achieve unparalleled sensitivity and angular resolution, substantially progressing our research into the formation and development of the early universe. Sensitivity is one of the most critical telescope metrics, often used to determine the observing mode and observing time of a radio source. It is also used to determine imaging parameters to balance imaging sensitivity and spatial resolution during data processing. In this paper, to meet the needs of continuum imaging pipeline data processing for the SKA1-LOW telescope, we derive equations for point-source-sensitivity (PSS) calculations under multichannel, multipolarization, discrete frequency sampling, discrete time sampling, and different weighting schemes. After validating the correctness of the equation, we improved and refined the sensitivity calculation application in the Radio Astronomy Simulation, Calibration, and Imaging Library, and further calculated the variation of the PSS versus angular resolution for the full-scale SKA1-LOW with different weighting schemes, and provided SKA1-LOW broadband image performance as functions of angular scale for four frequencies. These results can provide not only a theoretical basis for the construction and commissioning of the SKA1-LOW telescope but also guidance for imaging processing in future scientific data processing.


Introduction
The Square Kilometre Array (SKA) is a collaborative effort undertaken by multiple countries and organizations to construct the largest and most sensitive radio telescope in the world (Dewdney et al. 2009).A wide range of questions in astrophysics, fundamental physics, cosmology, and particle astrophysics can be studied with the SKA, extending the observable universe (Braun et al. 2015a).
SKA1-LOW, an integral component of the first phase of the SKA, serves as the low-frequency subarray of the SKA.It will consist of 131,072 log-period dipole antennas, grouped into 512 stations, each containing 256 antennas, covering 50-350 MHz with baselines as long as 65 km (Braun et al. 2019).One half of these stations form a compact core about 1 km in diameter, while the other half sprawls across three spiral arms.The High-Priority Science Objectives for SKA1-LOW include the study of the redshifted 21 cm line of atomic hydrogen in emission and absorption during the Epoch of Reionization (EoR) and Cosmic Dawn (CD) below redshifts of z < 27, characterization of the EoR/CD galactic and extragalactic foregrounds, pulsar (search) surveys, and timing observations, for both deep and wide-field continuum surveys (Braun et al. 2015a;Caiazzo et al. 2017;Labate et al. 2017).
Sensitivity is one of the most significant metrics to consider for a telescope.It is used to measure the weakest signal that can be detected (Wrobel & Walker 1999).In general, the sensitivity of an antenna in an array is the ratio of the antenna's effective area A eff to the system temperature T sys (Ivashina et al. 2008;de Lera Acedo et al. 2015;Sutinjo et al. 2015).
For the construction and commissioning of SKA1, the sensitivity is a significant reference.The requirements of SKA1 sensitivity had been proposed by the design baseline (Dewdney et al. 2013(Dewdney et al. , 2016)).Then, a series of studies Braun et al. (2015bBraun et al. ( , 2019)), Braun & McPherson (2017) analyzed the station, array sensitivity, and expected performance of SKA1-LOW and provided a comprehensive assessment of the sensitivity of SKA1-LOW.Sokolowski et al. (2022) presented a new software tool that enables astronomers and engineers to rapidly predict the anticipated sensitivity of SKA1-LOW prototype stations for arbitrary frequencies, times, and pointings.
Along with the theoretical analysis, prototype arrays were used to further investigate the sensitivity of SKA1-LOW.Van Es et al. (2020) and Bolli et al. (2022) performed electromagnetic (EM) simulations to model the station sensitivity of the SKA1-LOW prototype station Aperture Array Verification System 2 (AAVS2) in isolated pattern and embedded element pattern scenarios.The study demonstrated that sensitivity performance surpasses the requirements, and mutual coupling between the antennas does not adversely affect it.Sokolowski et al. (2021) both modeled and measured the fluctuation in station sensitivity of AAVS2 over five frequencies while tracking local sidereal time.Average measurements of XX and YY polarizations of AAVS2 array sensitivity were provided in conjunction with EM simulations conducted by Macario et al. (2022).Wayth et al. (2021) measured the Engineering Development Array 2 station sensitivity at five frequencies ranging from 70 to 320 MHz.
For astronomers who will use SKA1-LOW to conduct research work in the future, they are more concerned about imaging sensitivity.Astronomers could determine the observing time required for radio sources in science objectives according to their desired imaging sensitivity requirements (Cortes et al. 2023).Several studies have focused on imaging sensitivity, which is usually expressed in terms of point-source sensitivity (PSS).A lower value of PSS implies higher imaging sensitivity and is therefore used as a measure of the limit of weak signals that can be detected under given observing conditions (without confusion noise constraints), as well as for evaluating the root mean square (rms) value of the theoretical thermal noise in an image (Thompson et al. 1986;Crane & Napier 1989;Wrobel & Walker 1999;Ghara et al. 2016;Mort et al. 2016).The advantage of PSS is that the thermal noise level can be estimated without imaging calculation and cleaning.Meanwhile, others have used different approaches to estimate imaging sensitivity.For example, Sinclair et al. (2014) compared and analyzed the image rms deviation of SKA1-LOW with gain or phase errors, utilizing the correlator output sensitivity, while Braun & McPherson (2017) and Braun et al. (2019) utilized the correlator output sensitivity as the imaging sensitivity in order to investigate the imaging performance response of SKA at different angular scales.Rosero (2019) presented taper curves for the Next Generation Very Large Array utilizing various antenna configurations and weighting schemes.
The continuous imaging pipeline (CIP) is used to convert the calibrated continuum visibility into an image (Scaife 2015).The CIP uses a multifrequency, multiscale deconvolution algorithm (MS-MFS; Rau & Cornwell 2011) to clean the dirty image, which needs to be specified with the appropriate parameters including the number of time steps of the output data, the correlator time-averaged duration, the polarization, the number of channels, the weighting scheme, and so on.In imaging processing, astronomers need to choose the most reasonable data processing methods and parameters according to scientific objectives to obtain their desired sensitivity and resolution.However, how to select these parameters is often a problem for astronomers.Despite all the previous studies, astronomers still need sufficient expertize in interferometer data processing when working with scientific data.This is very difficult for astronomers.The study of imaging sensitivity, especially in multifrequency, multiscale, and different weighting methods, can guide astronomers on what parameters should be selected under what circumstances.However, the basic PSS calculation method does not consider discrete-time and frequency sampling, multiple polarizations, and subdivide weighting schemes.
We conducted an empirical study of two critical CIP pipeline parameters, sensitivity and resolution.In Section 2, based on previous work, we propose improved equations for calculating point-source sensitivity in multifrequency, multiscale, and different weighting schemes and verify their accuracy.Section 3 demonstrates the trend of point-source sensitivity with angular resolution for full-scale SKA1-LOW under different weighting schemes after Gaussian tapering.In Section 4, we analyze the results and suggest directions for further improvement.Finally, in Section 5, we summarize.

Point-source Sensitivity
According to Thompson et al. (1986) and Wrobel & Walker (1999), the PSS indicates the capacity of a radio interferometry system to detect weak radio signals, which can be calculated by where k is the Boltzmann constant, T sys is the system temperature, A eff is the effective collection area of a antenna, η is the efficiency factor, n a is the number of antennas, Δν IF is the intermediate frequency bandwidth, τ 0 is the whole observation covers a time interval, and w rms and w mean represent the rms weighting factor and mean weighting factor.
In Equation (1), σ represents the sensitivity of the correlator output in the weak source limit: The number of independent data points in the uv plane n d is expressed as where τ a is the correlator time-average duration (integration time).Equation (1) measures the rms value of image thermal noise in natural weighting when w rms /w mean = 1, and measures the rms value of image thermal noise in uniform weighting, robust weighting, or tapering function when w rms /w mean ≠ 1.However, utilizing Equation (1) directly to analyze SKA continuum imaging sensitivity is not sufficient.The MS-MFS is applied to improve imaging performance in the SKA CIP.Hence, sensitivity calculations should take account of a broader range of scenarios.It is imperative to include multiple channels, polarizations, discrete frequency sampling, and discrete-time sampling in our imaging sensitivity calculations.In addition, to accommodate visibility for gridding operations, we should focus more on assigning weights to the data within each grid.Therefore, instead of simply using w, we need to further subdivide the weights to better characterize the impact and effect of different weighting schemes.We further extended the Equation (1) by referencing the studies of Thompson et al. (2017).We define the number of baselines in the interferometric array as ( ) number of frequency channels as n chan , and number of polarizations as n pol .The total observation time is τ 0 , which includes multiple discrete-time samples.The time interval between two adjacent discrete-time sampling is τ interval .Then, the following equation can be used to calculate the total number of independent data points in the ( ) u v , plane, also known as the number of visibility data where τ 0 /τ interval is the number of discrete-time samplings.If τ interval is equal to τ a , it means that the number of discrete-time sampling during the observation is one.Equation ( 4) is brought into Equation (1) to replace the original In Equation (5), the weight term w i is further partitioned into three components: a tapering function term T k , a density weight term D k , and a weight term reflecting the signal-to-noise ratio of the data points w k .Finally, we can get the optimized PSS calculation as: where s im represents the rms of thermal noise in the image in Jy and A eff is the effective collecting area of an SKA1-LOW station.In Equation (6), Δν is described differently from Δν IF in Equations ( 1) and (2), which represent the bandwidth of each channel.An expression n chan Δν is useful to describe multichannel and discrete frequency sampling.
It should be noted that the right-hand side of the Equation (6) first term is influenced by the chosen observation parameters, such as the interferometer configuration and observation conditions.It represents the fundamental noise level determined by actual observation conditions.The equation's second term includes the rms and mean of the weighting factor, which reflect the relative contribution of the data points to the imaging.This term can be manually adjusted to meet specific research needs or optimize observations later.Different weighting schemes will change the PSS, but the second term reaches a minimum value of W rms /W mean = 1 in natural weighting (because the sensitivity is the highest).Since natural weighting assigns the same weight to each visibility data point, it can be inferred that the weighting term in Equation ( 6) is necessarily equal to 1, resulting in the maximum performance of the natural weighting PSS.A second factor value greater than 1 is produced by uniform weighting, in contrast.This is due to the fact that to achieve equal weighting for each grid, the visibility data within each grid are given fractional weights.Since the number of data points within different grids varies, this leads to differences in fractional weights."Grids" here are these uv grids of visibilities for different weightings or taperings.For this reason, the weighting term in Equation ( 6) is significantly greater than 1.When using other weighting schemes, weighting terms usually fall between natural and uniform weighting.
Compared with the original equations, Equation (6) considers discrete frequency and time sampling and better describes the different weightings.It can flexibly calculate predictions of point-source sensitivity by performing product operations on different observing parameters.

Implementation
The Radio Astronomy Simulation, Calibration, and Imaging Library (RASCIL) expresses radio interferometer calibration and imaging algorithms in Python and Numpy.Starting with version 1.0.0,RASCIL has changed to RASCIL-MAIN, which contains mainly high-level workflows and pipelines.At the same time, the data model and a large number of processing components (functions) have been migrated to the projects "ska-sdp-datamodels" and "ska-sdp-func-python." In order to facilitate algorithmic research, we modified and expanded RASCIL 0.9.6 and created an open-source project named RASCIL2. 3e implemented the PSS function defined in Equation (6) and updated the point-source sensitivity application (ras-cil_sensitivity) in RASCIL2 for SKA1 telescope analysis.In practice, the user realizes the calculation of PSS through the command line interface by inputting the appropriate arguments such as imaging pixels, cell size, weighting scheme, Gaussian tapering parameters, observation time and frequency ranges, and so on.Table 1 lists several common arguments.

Algorithm Validations
To validate the correctness of the algorithm in the presence of thermal noise, we designed the following three experiments using rascil_sensitivity: (1) to validate the correctness of the natural weighting computation; (2) to validate the correctness of the computation for common weighting schemes (natural, uniform, robust); and (3) to validate the correctness of the computation at different angular resolutions (by varying the tapering).These validations are described in detail below.

Validation Based on Imaging Sensitivity Calculator
We compared the predicted values of the rascil_sensitivity with the output of the imaging sensitivity calculator of Sokolowski et al. (2022) under the same observing conditions. 4The parameters are expressed as follows: phase center (0°, − 27°), hour angle range 1 hr, integration time 60 s, number of stations 512, efficiency factor 1, polarization number 2, and total bandwidth 30 MHz.The intermediate frequency starts at 50 MHz and increases by 10 MHz each time up to 350 MHz.In addition, the rascil_sensitivity requires manual input of the A eff /T sys values that refer to the station sensitivity values for the corresponding intermediate frequencies in Table 9 of Braun et al. (2019).The experimental results are shown in Figure 1.
Based on the results in Figure 1, it can be observed that at most of the frequencies, the predicted values from the rascil_sensitivity agree very well with the calculated values from the imaging sensitivity calculator, and the difference between the two is usually less than 10%-20%.However, for data points with frequencies below 160 MHz, there is a large gap, mainly due to the difference between the A eff /T sys calculated by the calculator and the A eff /T sys in Table 9 of Braun et al. (2019).We suspect that this is most likely due to T sys .In Braun et al. (2019) the sky is isotropic, and in the calculator by Sokolowski et al. (2022) they used a Haslam map scaled in frequency with ν −2.5 (Mozdzen et al. 2018).According to Experiment 1, the results of rascil_sensitivity are in general agreement with those given by the calculator when using natural weighting.

Validation Based on Different Weighting Schemes
In radio interferometry, natural, uniform, robust weighting, and tapering functions are four broadly utilized weighting schemes that assign weights to the measured visibilities of the interferometric array.According to the basic principles of radio interference imaging, natural weighting provides the highest sensitivity, while uniform weighting aims for high resolution (The CASA Team et al. 2022).Robust weighting achieves a smooth transition between uniform and natural weighting by adjusting the robustness parameter (R) from −2 to 2. As an efficient visibility weighting scheme, Robust weighting can generate images with significantly improved thermal noise properties at a minimal cost in resolution (Briggs 1995).
Therefore, we evaluated the difference between the predictions of the rascil_sensitivity, the rms of images containing only thermal noise, and the rms of the residual image after WSCLEAN (Offringa et al. 2014) imaging (with sources and thermal noise in the sky model).The reason for this validation approach is that if the CLEAN algorithm converges correctly, the brightness level of the residual image generated should be consistent with the noise model (Bhatnagar & Cornwell 2004).Therefore, comparing whether the predicted value of PSS is consistent with the rms of the residual images in the presence of thermal noise helps to verify the rationality of rascil_sensitivity.We ran the simulation over a field of view of about 4 square degrees.We added 17 sources from the GLEAM catalog with flux densities ranging from 0.01 to 0.1 Jy.
We accounted for thermal noise at each visibility data point, as determined by the Equation (2).The specific observation parameters include a η of 1, a total of 512 stations, Stokes I polarization, a center frequency of 160 MHz, the A eff /T sys of 1.159 m 2 /K for 160 MHz, 30 channels, and a bandwidth of 16 kHz per channel.The hour angle range is 1 hr, with an integration time of 60 s and a time interval of 60 s.Based on the above parameter settings, Figure 2 illustrates the comparison between the PSS predictions, the rms of the thermal noise-only image, and the rms of the residual image after WSCLEAN imaging under different weightings.The left panel displays their respective levels, while the right panel presents the difference between them.The graph confirms the soundness of our rascil_sensitivity in predicting the thermal noise level in natural, uniform, and robust weighting, as it shows near agreement between the three above.

Validation Based On Angular Resolutions
After validating the change in sensitivity for various weighting cases, we further validate the rms of the images after introducing tapering to change the angular resolution.Tapering is a method for correcting weight distribution in the uv plane (Boone 2013) that is commonly implemented using Gaussian tapering to decrease data weights at the outer edges of the uv coverage.This method is effective in suppressing smallscale sidelobes and extending the beamwidth, which can then be used to size the angular resolution.
We repeated the process of the previous validation but added tapering with degrees of 1″, 8″, 16″, 32″, 64″, and 128″, respectively.We plotted the beam size of the tapering processed point-spread function (PSF) as the horizontal axis, the PSS value and the residual image rms as the vertical axis in Figure 3.The original PSF beam size changed according to the degree of processing by tapering at different weightings.For example, in this experiment, the uniform, robust (R = −0.5),and natural weighting PSF beam sizes are 4 66, 7 90, and 85 02, respectively; after processing with 8″ of tapering, these values become 8 43, 12 16, and 124 27, respectively.
Figure 3 shows that the rms values of the residuals generated by imaging the sky model with sources and thermal noise are very close to the predicted values of our rascil_sensitivity.However, the robust (R = −0.5)and natural weighting PSS predictions do not agree well with the residual rms values when subjected to 128″ tapering.This may be due to the fact that the flux density undulations in the residuals obtained from imaging at larger PSF beam sizes may not follow a Gaussian distribution.In contrast, the PSS is a measure of Gaussian thermal noise.This results in a large discrepancy between PSS predictions and actual measurements.Overall, the PSS remains a valid measure of thermal noise level under the expected weighting.
In addition, as seen in Figure 3, after undergoing progressively larger tapering treatments, the sensitivity of natural weighting slowly decreases.Meanwhile, the sensitivity of uniform and (R = −0.5)robust weighting varies little over an angular resolution range of approximately 10″-200″.The presented effect aligns with expectations.In the case of natural weighting with tapering, since natural weighting has the highest sensitivity, progressively larger tapering progressively reduces the weighting of visibility data at the outer edges of the uv coverage, thus progressively reducing the sensitivity.In the other two cases, as tapering increases, the visibility weight of the outer edges of the uv coverage will also decrease.Tapering is similar to a low-pass filter in that it filters out the high spatial frequency portion of an image, thereby smoothing the image somewhat and reducing the appearance of noise.

Imaging Sensitivity of Full-scale SKA1-LOW
After validating the algorithm and codes, we further investigated the taperability with different conditions using rascil_sensitivity.Clark & Brisken (2015) introduced the concept of taperability to measure an array's sensitivity for different resolutions.Asayama (2021) has previously investigated the taperability of imaging sensitivity for SKA1-MID using Common Astronomy Software Applications software, which provides a valuable reference for telescope construction, as well as for astronomers to conduct follow-up studies.
Referring to the study of Asayama (2021), as well as considering the future requirements of SKA1-LOW construction and subsequent scientific studies (e.g., EoR imaging), we explored the relationship between PSS variations and angular resolution with different weighting schemes.
Table 2 lists some important experimental arguments.We chose a fractional bandwidth of 30% and 30 spectral channels with a width of 1% each, as recommended by Braun et al. (2019).In all experiments, we ensured that the value of imaging_oversampling was always 3. And, the integration time of 300 s was chosen to increase the speed of the simulation.
Using the observing parameters, we calculated PSS predictions with uniform, robust, and natural weighting at four typical frequencies after varying degrees of tapering.For each frequency, we performed four sets of experiments to investigate the PSS upon different weighting schemes and tapering.The experimental results are shown in Figure 4, which demonstrates the imaging sensitivity of full-scale SKA1-LOW, presented as relative sensitivity as a function of angular scale.The relative sensitivity is the ratio of the PSS after tapering in uniform, robust, or natural weighting to the PSS in natural weighting, denoted as s s _ im, any wt im, natural .We use relative sensitivity because it allows us to cancel out the first term on the righthand side of Equation (6), leaving only the influence of the weighting factors W rms /W mean .This method allows for a straightforward assessment of sensitivity loss (Braun et al. 2019;Rosero 2019).With values starting at −2 and increasing by 0.5 each time, we can obtain 9 robustnesses (−2, −1.5, −1.0, −0.5, 0.0, 0.5, 1.0, 1.5, 2.0) with the corresponding beam sizes (3.86, 4.47, 5.78, 8.17, 14.82, 22.71, 28.76, 31.91, 32.52) arcsec.The PSS is calculated for each robustness (see black points in Figure 4).
In Figure 4, the horizontal axis represents the PSF beam size after tapering, which is calculated from the geometric mean of its major and minor axes.The vertical axis represents relative sensitivity.Lower values indicate higher imaging sensitivity..25, 0.5, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 1280 consistent imaging sensitivity.This is further evidence of the conclusion stated by Braun et al. (2019) in their sensitivity study using uniform weighting that the SKA1-LOW array configuration designed with a logarithmic spatial distribution has superior taperability.Additionally, it can also be seen from the experimental results that the radio interferometer imaging processor must choose an appropriate weighting scheme.Different weighting modes and tapering will change sensitivity.Astronomers need to balance resolution and sensitivity in imaging processing to satisfy scientific research needs.

Analysis
Taking imaging sensitivity at 160 MHz as an example, for natural weighting with uv-taper, the loss of relative sensitivity is smaller than uniform weighting with uv-taper and robust weighting with uv-taper in the angular resolution range of 30″-400″.This means that in imaging situations where tapering processing is required, the imaged PSF beam size can first be compared to the natural weighting case.If the beam size exceeds that of the natural weighting case, then opting for natural weighting imaging may yield better imaging results.For uniform weighting, the relative sensitivity value is very high, which means poor sensitivity.However, by increasing the degree of tapering, the value of relative sensitivity decreases rapidly.Ideally, a relative sensitivity of about 2 can be achieved (2 is approximately a local minimum value).This suggests that a significant increase in sensitivity can be achieved while moderately reducing resolution by tapering, opening up the possibility of highresolution, high-sensitivity observations.For robust weighting (R = −0.5), the relative sensitivity is less than 2, and by increasing the tapering, the relative sensitivity in the angular resolution range from 8″ to 300″ does not exceed 2. This implies that robust weighting (R = −0.5) is capable of both highresolution, high-sensitivity, and low-resolution observations.
Overall, the effect of tapering on the change in sensitivity in the case of uniform and robust weighting is deeply analyzed in this study.More detailed results at various frequencies are given, which can provide more references for telescope debugging, especially for future scientific research.

Limitations
Although the PSS can be used as a basic benchmark, in actual imaging situations, there may be discrepancies between rms and PSS values due to real-life complexities.Factors such as calibration errors, systematic errors, and instrumental defects may increase the overall rms.Therefore, it is worthwhile to consider various potential sources of error when interpreting image quality and accuracy.For example, PSS calculation must take into account the system temperature setting.The system temperature is influenced by the sky temperature, which is affected by the galactic foreground (Braun et al. 2019).As the foreground is unevenly distributed across the sky, sky temperatures vary in different directions (Sokolowski et al. 2022).In the rascil_sensitivity, the system temperature is a fixed value manually entered.It does not automatically adjust based on the specified phase center and frequency.
In addition, we do not consider confusing noise throughout the study.It is very worthwhile to research how confusion noise affects the PSS.Overall, there is still a lot of work to be done on how to calculate a more accurate PSS that is more in line with practical applications.

Conclusions
SKA1-LOW is under construction.Conducting PSS-related studies is of significant significance for the telescope construction, system commissioning, and scientific data processing of SKA1-LOW.
In this study, we investigated how to compute the PSS in multifrequency, multiscale and multipolarization scenarios by deriving the corresponding computational formulas.We optimized the rascil_sensitivity application in RASCIL2.PSS calculations can be performed easily and efficiently.By specifying the desired array configuration and the corresponding observation parameters, a PSS value can be obtained.
Under the same conditions, we compared the predictions from the rascil_sensitivity with the calculated values from the imaging sensitivity calculator developed by Sokolowski et al. (2022).Moreover, we also compared the rms values of the PSS predictions, the rms values of the images containing only thermal noise, and the rms values of the residual images generated after imaging a sky model containing both thermal noise and sources under different weighting scenarios.Overall, the experimental results show that they are all in reasonable agreement, validating the computational soundness of the rascil_sensitivity.On this basis, we thoroughly investigated how varying weighting schemes (natural, uniform, robust, and tapering) affect PSS and resolution of full-scale SKA1-LOW under different scenarios, and provide a schematic representation of the PSS as a function of PSF beam size at four typical frequencies.
In summary, this study further deepens the PSS calculation and provides a more general PSS calculation method suitable for the SKA1-LOW telescope under discrete sampling, multiscales, and multipolarizations.The results can be directly applied to the commissioning of SKA1-LOW, and can also help astronomers to determine the final imaging parameters and achieve the desired resolution and sensitivity.

Figure 2 .
Figure 2. The left panel presents a comparison of σ PSS , noise-only σ RMS , and WSCLEAN CLEANed σ RMS in different weightings at specific observing parameters.The right panel displays the difference.The "uniform" and "natural" on the horizontal axis represent uniform and natural weighting, respectively.The numbers on the horizontal axis indicate the value of the robustness R in robust weighting.

Figure 3 .
Figure 3.The first row of panels illustrates the alterations in σ PSS , noise-only σ RMS , and WSCLEAN CLEANed σ RMS for uniform, robust (R = −0.5),and natural weightings after tapering.The residuals are shown in the second row of panels, corresponding to the first row.
1. Experiment 1-natural weighting with uv-taper: we analyzed the variation of PSS for different tapering in natural weighting.The tapering took values starting from 0 25, with the latter value being twice the previous value, and we end up with 13 tapering (0 25, 0 5, 1 0, 2 0, ... , 512″, 1024″).The calculated PSS of each tapering are shown in red points in Figure 4. 2. Experiment 2-uniform weighting with uv-taper: we analyzed PSS variation for different tapering in uniform weighting.Tapering values were taken as in Experiment 1.The results are shown in blue points in Figure 4. 3. Experiment 3-robust weighting with uv-taper: the experiment is to analyze the impact of different tapering in robust weighting with robustness R = −0.5 for the PSS.The values of tapering are the same as in the previous experiment.Green points in Figure 4 show the results.4. Experiment 4-robust weighting without uv-taper: we analyzed PSS variation for different robustness in robust weighting.

Figure 4
Figure 4 presents the PSS obtained under different weighting schemes and angular resolutions.The trend of the curves reveals the trend of thermal noise in the images after different weighting and tapering treatments.It is noteworthy that the curves in the middle part of the plots show smooth transitions, indicating that different angular scales provide nearly

Figure 4 .
Figure 4. Relative sensitivity as a function of PSF beam size changed by tapering.The blue line indicates uniform weighting with uv-taper, the green line indicates robust weighting (R = −0.5)with uv-taper, the red line indicates natural weighting with uv-taper, and the black line indicates robust weighting without uv-taper.

Table 1
The Arguments Supported by the Application Figure 1.The comparison of the predicted values of the rascil_sensitivity with the values calculated by the calculator developed by Sokolowski et al. (2022) under natural weighting.

Table 2
The Arguments for the Imaging Sensitivity Analysis