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14th International Symposium on Particle Image Velocimetry – ISPIV 2021  
August 1-5, 2021 | Chicago, IL USA 

Numerical Uncertainty in Density Estimation for 
Background Oriented Schlieren 

 
Jiacheng Zhang1§, Lalit K. Rajendran2§, Sally P. M. Bane2, Pavlos P. Vlachos1* 

 
1 Purdue University, School of Mechanical Engineering, West Lafayette, USA. 

2 Purdue University, School of Aeronautics and Astronautics, West Lafayette, USA. 

*pvlachos@purdue.edu 

Abstract 
Background Oriented Schlieren (BOS) is an image-based density measurement technique. BOS estimates the 

density gradient from the apparent distortion of a target pattern viewed through a medium with varying density 
using cross-correlation, tracking, or optical flow algorithms. The density gradient can then be numerically integrated 
to yield a spatially resolved estimate of the density [1]. A method was recently proposed to estimate the a-posteriori 
instantaneous and spatially resolved density uncertainty for BOS [2] and showed good agreement between the 
propagated uncertainties and the random error. However, the density uncertainty quantification method could not 
account for the systematic uncertainty in the density field due to the discretization errors introduced during the 
numerical integration, which could be much larger than the displacement random errors [2]. In this work, we 
propose a method to estimate the numerical uncertainty introduced by the density integration in BOS measurements, 
using a Richardson extrapolation framework. A procedure is also introduced to combine this systematic uncertainty 
with the random uncertainty from the previous work to provide an instantaneous, spatially-resolved total uncertainty 
on the density estimates. The method will be tested with synthetic fields and synthetic BOS images.  

With the Richardson extrapolation [3], the discretization error of a numerical estimation can be estimated based 
on the residual between two sets of results with different grid levels as:  
 𝜖!̅ = − "!#""!

$##%
, (1) 

where 𝜖!̅ is the estimated numerical error of the result obtained on a grid with spacing ℎ, 𝑓! and 𝑓$! are the results 
obtained on the grids with spacing ℎ and 𝑟ℎ, respectively, with 𝑟 being the downsampling factor (usually 𝑟 = 2), 
and 𝑝 is the order of accuracy of the discretization scheme. In this study, 𝑝 is 2 since the second-order central 
differencing scheme was used for carrying the numerical integration. The estimated error is then employed as the 
numerical uncertainty (𝑈 = |𝜖!|). With the random uncertainty obtained using the previous method [2] as the 
standard deviation of the random error distribution, and the numerical uncertainty interpreted as the standard 
deviation of the bias error distribution, the standard total uncertainty can be expressed as 
 𝑈&'&()* = 𝑈+,(-* + 𝑈$(./'0* , (2) 
thereby providing a framework for combining the random uncertainty estimates to estimate the overall uncertainty 
in the density integration.  

The proposed uncertainty estimation method was tested using a synthetic sinusoidal scalar field as:  
 𝑓(𝑋, 	𝑌) = 𝑠𝑖𝑛 6*1

2
𝑋7 𝑠𝑖𝑛 *1

2
𝑌, (3) 

where 𝑓 represents the scalar field, and 𝜆 represents the wavelength. A zero-mean Gaussian distributed noise was 
added to the scalar field with prescribed noise levels. One thousand (1000) realizations of the corrupted field were 
generated, and for each realization, the integration was performed with the noisy gradient fields to estimate the 
error. The results are shown in figure 1 for two noise levels: 1% and 10% of the peak value of the scalar field. It is 
seen in both levels that the spatial variation of the total uncertainty matches that of the total error, and the RMS of 
the total uncertainty coincides with the RMS of the error distribution. This validates the framework used to combine 
the bias and random uncertainty estimates. 

 
§ These authors contributed equally to this work. 



The numerical uncertainty estimation was also applied to the synthetic BOS images rendered using a ray-
tracing based image generation methodology [4]. A sinusoidal density field was chosen for the error analysis as 
described by equation (4): 
 𝜌(𝑋, 	𝑌) = 𝜌3 + 𝛥𝜌3 𝑐𝑜𝑠 6

*1
2
𝑋7 𝑐𝑜𝑠 6*1

2
𝑌7, (4) 

where 𝜌3 is the ambient density, Δ𝜌3 is the peak density difference and 𝜆 is the wavelength. A 2D slice of the 
density field is shown in figure 2(a). The rendered images were processed using PRANA with a standard cross-
correlation procedure for two passes in an iterative window deformation framework. A sample instantaneous 
displacement field are shown in figure 2(b). The 
displacement fields were used to calculate the depth 
averaged density gradient field ∇𝜌, which were then 
spatially integrated using the Poisson solver to obtain 
the projected density field. The error of the density 
field was determined as the deviation from the original 
density field used to render the synthetic images, and 
the numerical uncertainty was estimated using the 
Richardson extrapolation method. The results of the 
density error and uncertainty are compared in figure 
2(c) and (d). The numerical uncertainty of the density 
integration was lower than the total density error, 
because the density error was also due to the 
discrepancy between the ray tracing displacements 
and the theoretical displacements because of the linear 
approximation of the non-linear ray trajectory through 
density gradients [4]. 

Efforts are ongoing to apply the proposed method 
to experimental BOS data.  

References 
[1] M. Raffel, “Background-oriented schlieren (BOS) techniques,” Exp. Fluids, vol. 56, no. 3, pp. 1–17, 2015. 
[2] L. K. Rajendran, J. Zhang, S. Bhattacharya, S. P. M. Bane, and P. P. Vlachos, “Uncertainty quantification 

in density estimation from background-oriented Schlieren measurements,” Meas. Sci. Technol., vol. 31, no. 
5, 2020. 

[3] P. J. Roache and P. M. Knupp, “Completed Richardson extrapolation,” Commun. Numer. Methods Eng., 
vol. 9, no. 5, pp. 365–374, 1993. 

[4] L. K. Rajendran, S. P. M. Bane, and P. P. Vlachos, “PIV/BOS synthetic image generation in variable density 
environments for error analysis and experiment design,” Meas. Sci. Technol., vol. 30, no. 8, 2019. 

 

Figure 1 Error and uncertainty statistics for the sinusoidal field for two noise levels. (a) and (b) represent the spatial variation of the statistics 
and probability density functions respectively for a 1% noise level, with (c) and (d) representing the results for the 10% noise level. 
 

Figure 2 Results of the analysis with synthetic BOS images. (a) 2D 
slice of the density field used to render the synthetic BOS images, (b) 
image displacements from cross-correlation analysis, (c) error in the 
density field, and (d) numerical uncertainty