Goniometric measurements are essential for the determination of many optical quantities, and quantifying the effects of errors in the rotation axes on these quantities is a complex task. In this paper, we show how a measurement model for a four-axis goniometric system can be developed to allow the effects of alignment and rotation errors to be included in the uncertainty of the measurement. We use three different computational methods to propagate the uncertainties due to several error sources through the model to the rotation angles and then to the measurement of bidirectional reflectance and integrated diffuse reflectance, a task that would otherwise be intractable. While all three methods give the same result, the GTC Python package is the simplest and intrinsically provides a full uncertainty budget, including all correlations between measurement parameters. We then demonstrate how the development of a measurement model and the use of GTC has improved our understanding of the system. As a consequence, taking advantage of negative correlations between measurements in different geometries allows us to minimise the total uncertainty in integrated diffuse reflectance, lowering the standard uncertainty from 0.0029 to 0.0015.

<p><b>We are continuously making judgements based on the appearance of objects around us. The appearance of an object depends on the way in which the object’s surface reflects, transmits, and absorbs the incident light. In recent years, there has been an increasing demand for traceable measurements of ‘appearance’ across various industries. While there has been extensive work carried out in national metrology institutes to develop traceable colour measurements, there are more and more requirements for multidimensional measurements, for example of the bidirectional reflectance and transmittance distribution functions (BRDF, BTDF). Measurements of these quantities require an instrument that allows changes to both the incident and detection angles, such as a goniospectrophotomer. Achieving traceability for measurements of the bidirectional scattering distribution function (BSDF, referring to either the BRDF or BTDF) and integrated BSDF requires an in-depth understanding of the instrument and all of the errors in the measurements.</b></p> <p>This thesis describes the measurement model developed for measurements of BSDF and integrated BSDF using the Measurement Standards Laboratory of New Zealand’s (MSL) goniospectrophotometer. This model considers all of the errors in the alignment of the instrument and the angles set by the instrument and propagates these through to the measured BSDF. A correction factor for the non-uniformity in the response of the detector is calculated and a new design for the detector is proposed to remove the need for this correction factor. The effect of the instrument signature is also considered.</p> <p>The measurement model is applied to various samples to demonstrate the insights that can be gained from a detailed model, which are then used to help lower the measurement uncertainty. The relative standard uncertainty in the BRDF of a white Spectralon sample at 550 nm is less than 0.1% for in-plane measurements, while the relative standard uncertainty in the diffuse reflectance (calculated from integrated BRDF measurements) of a white Spectralon sample at 560 nm is 0.25%.</p> <p>Finally, the model is applied to measurements of BTDF, which is integrated to calculate transmittance haze. The haze values measured using the goniospectrophotometer are compared with measurements of haze made according to the documentary standards using an integrating sphere. It is demonstrated that these two methods of measuring transmittance haze do not agree due to a conflict in the definition of transmittance haze. The BTDF measurements are used to simulate the integrating sphere haze and to calculate the sensitivity of the sphere measurements to various alignment errors.</p>

In recent years, there has been a growing interest in the measurements of the bidirectional reflectance distribution function (BRDF) in industry and research and development. However, there is currently no dedicated key comparison to demonstrate the scale conformity. To date, scale conformity has been proved only for classical in-plane geometries, in comparisons between different national metrology institutes (NMIs) and designated institutes (DIs). This study aims at expanding that with nonclassical geometries, including, for the first time, to the best of our knowledge, two out-of-plane geometries. A total of four NMIs and two DIs participated in a scale comparison of the BRDF measurements of three achromatic samples at 550 nm in five measurement geometries. The realization of the scale of BRDF is a well-understood procedure, as explained in this paper, but the comparison of the measured values presents slight inconsistencies in some geometries, most likely due to the underestimation of measurement uncertainties. This underestimation was revealed and indirectly quantified using the Mandel–Paule method, which provides the interlaboratory uncertainty. The results from the presented comparison allow the present state of the BRDF scale realization to be evaluated, not only for classical in-plane geometries, but also for out-of-plane geometries.

The validity of calibration and measurement capability (CMC) claims by national metrology institutes is supported by the results of international measurement comparisons. Many methods of comparison analysis are described in the literature and some have been recommended by CIPM Consultative Committees. However, the power of various methods to correctly identify biased results is not well understood. In this work, the statistical power and confidence of some methods of interest to the CIPM Consultative Committees were assessed using synthetic data sets with known properties. Our results show that the common mean model with largest consistent subset delivers the highest statistical power under conditions likely to prevail in mature technical fields, where most participants are in agreement and CMC claims can reasonably be supported by the results of the comparison. Our approach to testing methods is easily applicable to other comparison scenarios or analysis methods and will help the metrology community to choose appropriate analysis methods for comparisons in mature technical fields.