## 1 Introduction  
Intact, low-pressure ultrafiltration membranes can completely remove particulates and pathogens and thus produce drinking water that meets the stringent regulations on water quality [1]. For this, accurate and efficient integrity monitoring of the membrane system is fundamental to guarantee the quality of filtered products and detect the presence of oversized pores or defects that can compromise the retention capability of the filter [2]. Integrity tests should be sensitive to defects as small as 3 \(\mu\)m, which is based on the lower size range of _Cryptosporidium_ oocysts, so that the tests can make sure any defect that is large enough for oocysts to pass will prompt a response from the integrity test being used [1,3,4]. In such a context, the detection method should be highly sensitive, quick and easy, have a signal that can be interpreted by programmable logic controller (PCL), and should be able to be carried out as frequently as possible. Generally, conventional membrane integrity monitoring techniques, including direct and indirect methods, have been reported [2]. Among the various integrity monitoring methods, the pressure decay test (PDT), a direct method, appears to be a reliable and common method. The generic protocol for a pressure decay test described in the United States Environmental Protection Agency (USEPA) guidelines is as follows: (1) drain the water from one side of the membrane, (2) pressurize the drained side of the fully wetted membrane, and (3) isolate the pressure source and monitor the decay for a designated period of time. This method is very sensitive in detecting leaks and integrity breaches and takes advantage of existing equipment, such as pressure and flow meters, to test the integrity over a large number of membranes. However, the method still has some weaknesses, including the inability to continuously monitor integrity, the inability to measure the pressurized air volume for calculation, the potential false positive results if the membrane is not fully wetted, and the difficulty in detecting horizontally oriented membranes due to potential draining and air venting problems [2; 5; 6]. The log removal value (LRV) can be calculated to express the membrane integrity, but the effectiveness of PDT in detecting membrane integrity changes is affected by the characteristics of the membrane surface. Correct LRV calculations require accurate inputs obtained under representative conditions, such as filtrate flow capacity, air-liquid conversion ratio, volumetric concentration factor, smallest verifiable decay rate, and volume of pressurized air in the system during the test. Moreover, the timing of the PDT has an impact on the calculated LRV, which can be counterintuitive in certain circumstances [2; 6]. The calculation of the sensitivity of the test method requires an accurate measurement of the volume of pressurized air in the system, and an important concern associated with the pressure decay test is the potential for larger decay rates, even within the upper control limit, to affect the accuracy of the test. For example, if the total pressure decay over the duration of the test reduces the applied pressure on the membrane to a level below what is sufficient to meet the resolution criterion, the test would not comply with the requirements of the rule [1]. Therefore, changes in the pressure and the pressure decay rate (PDR) in the membrane gas chamber needs to be analyzed. [PERSON] et al. [7] developed a decision-aid tool to comply with the USEPA rule that requires a resolution of 3 \(\upmu\)m when the pressure is higher than 500 mbar, although the model needs to be calibrated on each full-scale membrane system to refine the prediction of LRV. [PERSON] et al. [8] conducted a sensitivity analysis of pressure-based integrity test and solved the defect size resolution using Monte Carlo and probability bounds analysis for five commercial membrane designs.