## 3 Uncertainties, Limitations, and Sources of Error in SEM Measurements
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Analytical Methods/Techniques** & **Year** & **Minerals/Materials** & **References** \\ \hline SEM, CT, Raman Spectroscopy & 2021 & Saturated brine, wellbore cement & [146] \\ \hline SEM–EDS, XRD, IRS, XRF & 2023 & Gabbro-anorthosite, seawater, and mafic rock & [147] \\ \hline SEM–EDS, AM-SEM, FE-SEM, CT & 2023 & Mineralogical analysis of petroleum geology & [148] \\ \hline SEM-FIB & 2020 & Mineralized scale patterns on the cell periphery & [149] \\ \hline SEM, EMP, Raman Spectroscopy, BSE & 2019 & Petrified wood, Mn-oxide minerals & [150] \\ \hline SEM–EDS, Monte Carlo Simulations & 2021 & Glass fiber-reinforced cement & [151] \\ \hline SEM, XRD, XRF, FTIR & 2022 & High ash coal, fluidized bed gasifier & [152] \\ \hline SEM–EDS, XRD & 2023 & Mineral-forming bacteria & [153] \\ \hline SEM–EDS, XRD, TGA & 2023 & Self-healing cement-based minerals & [154] \\ \hline SEM, XRD, XPS, FTIR & 2023 & Biomimetic mineralized cement & [155] \\ \hline SEM, TEM, PLM, EBSD, SAED & 2023 & Talc, amphiboles, and biopyriboles & [156] \\ \hline SEM = Scanning Electron Microscopy, SDD = Silicon Drift Detector, EDS = Energy-Dispersive X-ray Spectroscopy, EPMA = Electron Probe X-ray Microanalyzer, WDS = Wavelength-Dispersive Method, XRD = X-ray Diffraction, TEM = Transmission Electron Microscopy, BSE = Backscattered Emission, TMR = Transverse Micro-Radiography, SMI = Surface Micro-Hardness, \(\omega\)RF = Micro X-ray Fluorescence, LWIR = Long Wave Infrared, SAM = Spectral Angle Mapper, EDX = Energy-Dispersive X-ray Spectroscopy, XPS = X-ray Photoelectron Spectroscopy, FT-IR = Fourier-Transform-Infrared Spectroscopy, UV = Ultraviolet Visible, TGA = Thermogravimetric Analysis, TRKP = Total Reflection X-ray Fluorescence, FIB = Focus In Beam, \(\omega\)CT = X-ray Micro-Computed Tomography, XLI = X-Lined Hypophosphatemia, IRS = Infrared Spectrometry, AM = Automated Mineralogy, FE = Field Emission, EMP = Electron Microprobe, PLM = Polarized Light Microscopy, EBSD = Electron Backscatter Diffraction, SAED = Selected Area Electron Diffraction. \\ \hline \hline \end{tabular}
\end{table} 
Table 2: _Cont._The accepted statistical methods, which use the point counting method on polished sections and assess errors in mineral grades, agree well with the bootstrap method [170; 171; 172]. This method has the advantage of being assumption-free and can be applied to a wide range of particle characteristics [162]. It does not assume a biomedical distribution. These methods imply that the standard deviation of mineral grades is proportional to the square root of the number of particles measured or the total area of particles measured. The relative standard deviation of measurements for any mineral grade can be estimated as follows [173]:  
\[RSD=ax^{-0.5} \tag{6}\]  
where \(RSD\) is the relative standard deviation, \(a\) is a coefficient, and \(x\) is the mineral grade. The bootstrap method can also provide information about the measurement of how much total area (grains) is needed to reach a given uncertainty. In addition to the uncertainty, SEM also has some drawbacks, including (but not limited to) limited depth of penetration that primarily provides surface information; and low accelerating voltages that provide low-resolution images, while increasing the voltage starts damaging the surface of the sample.