GCR environmental models II: Uncertainty propagation methods for GCR environments

©2014 American Geophysical Union. All rights reserved.

GCR Environmental Models II:

Uncertainty Propagation Methods for GCR Environments

Tony C. Slaba1 and Steve R. Blattnig

1

1 NASA Langley Research Center, Hampton, VA, USA

Corresponding author email: [email protected]

This article has been accepted for publication and undergone full peer review but has not been through the copyediting, typesetting, pagination and proofreading process which may lead to differences between this version and the Version of Record. Please cite this article as doi: 10.1002/2013SW001026


Abstract

In order to assess the astronaut exposure received within vehicles or habitats, accurate

models of the ambient GCR environment are required. Many models have been developed and

compared to measurements, with uncertainty estimates often stated to be within 15%.

However, inter-code comparisons can lead to differences in effective dose exceeding 50%.

This is the second of three papers focused on resolving this discrepancy. The first paper

showed that GCR heavy ions with boundary energies below 500 MeV/n induce less than 5% of

the total effective dose behind shielding. Yet, due to limitations on available data, model

development and validation is heavily influenced by comparisons to measurements taken

below 500 MeV/n. In the current work, the focus is on developing an efficient method for

propagating uncertainties in the ambient GCR environment to effective dose values behind

shielding. A simple approach utilizing sensitivity results from the first paper is described and

shown to be equivalent to a computationally expensive Monte Carlo uncertainty propagation.

The simple approach allows a full uncertainty propagation to be performed once GCR

uncertainty distributions are established. This rapid analysis capability may be integrated into

broader probabilistic radiation shielding analysis and also allows error bars (representing

boundary condition uncertainty) to be placed around point estimates of effective dose.


1. Introduction

High energy galactic cosmic rays (GCR) pose a serious health risks for astronauts that can drive

mission planning and vehicle design [Cucinotta et al. 2013]. In order to quantify the exposure received from

GCR by astronauts behind shielding materials, several computational tools are required. Rigorous quantification

of the uncertainty of each of these tools is important to overall risk assessment and requires careful examination.

A number of models have been developed to describe the ambient spectra of GCR ions relevant for human

shielding applications, and the uncertainty of most of these models has been stated to be less than 15% [NCRP

2006]. However, when the models are evaluated over a common epoch and transported through to effective

dose, relative differences can easily exceed 50% [Mrigakshi et al. 2013; Slaba and Blattnig 2014].

This type of inconsistency indicates the need for a more rigorous validation approach with uncertainty

metrics that are better tied to exposure quantities of interest for space radiation shielding applications. This is the

second of three papers focused on addressing this need. In the first paper [Slaba and Blattnig 2014], a sensitivity

analysis was performed to quantify the extent to which each GCR ion and energy, prior to entering any shielding

material or tissue, contributes to effective dose behind shielding. It was found that GCR ions with Z > 2 and

boundary energy below 500 MeV/n induce less than 5% of the total effective dose behind shielding. This is an

important finding given that most of the GCR models are heavily calibrated and validated against measurements

taken below 500 MeV/n by the Advanced Composition Explorer/Cosmic Ray Isotope Spectrometer

(ACE/CRIS) instrument [Stone et al. 1998]. It is therefore possible for two GCR models to accurately reproduce

the ACE/CRIS data and have similar overall uncertainty statements while inducing very different effective dose

values behind shielding.

In this work, two methods for propagating GCR model uncertainties into effective dose are discussed.

This general procedure, which quantifies the expected variation in effective dose due to uncertainty in the input

GCR boundary condition, is often referred to as uncertainty propagation. This type of analysis requires

knowledge of the distribution of uncertainties being propagated. Even though GCR model uncertainties are

typically expressed and communicated as a single number (e.g. 15%), there is actually a distribution of

uncertainties occurring for each ion and energy region. The uncertainties arise from a lack of measurements and

measurement uncertainty as well as the natural variation in the environment with time. In many cases, these

distributions are not precisely known, and assumptions are made by subject matter experts. Since rigorously

developed GCR model uncertainty distributions have not yet been developed (focus of next paper),


representative distributions will be used with shape parameters based on previously published accuracy

statements [NCRP 2006] or chosen to highlight important points about the analysis.

One method for propagating these uncertainty distributions is through Monte Carlo (MC) sampling. In

this approach, a model uncertainty distribution is sampled and applied as a perturbation to a nominal GCR

boundary condition spectrum. The perturbed boundary condition is then propagated through to the exposure

quantity of interest, and the process is repeated many times. The end-result is a distribution of exposure

quantities. It is shown in this work that this computationally expensive procedure is equivalent to a simpler and

more efficient approach utilizing results derived from the sensitivity analysis [Slaba and Blatting 2013]. The

mathematical formulation of the fast method and computational examples establish equivalence to the MC

approach. The advantage of the simpler approach is that a full uncertainty propagation may be performed

rapidly once the GCR uncertainty distributions are known. This rapid analysis capability would be helpful in

broader radiation analyses utilizing probabilistic methods. Most importantly, it allows error bars, representing

uncertainty in the GCR environmental model, to be easily placed around any point estimate of effective dose.

Effective dose is being used in the present papers due to its widespread use in past and current NASA design

studies. However, the general approach outlined herein could also be applied to other exposure quantities such

as dose or dose equivalent. This would enable environmental model uncertainty to be quantified in transport

code validation comparisons [Zeitlin et al. 2013a, 2013b; Hassler et al. 2014].

The next, and final, paper addressing GCR model validation metrics and uncertainty quantification will

combine results from the previous sensitivity analysis as well as the efficient uncertainty propagation method

presented here. The end-result will be a more meaningful and rigorously developed uncertainty estimate that is

tied closely to effective dose values behind shielding. Further, the combined effort of all three papers will

provide a simple and rigorously developed computational framework for placing error bars around a point

estimate of effective dose behind spherical shielding. The method could be easily extended and applied to other

exposure quantities and geometries as well.

2. Uncertainty Propagation Methods

In this section, two methods are presented for propagating relative uncertainties in the GCR boundary

condition into effective dose values behind shielding. A direct MC sampling approach is described first,

followed by a simpler and more efficient method utilizing the relative contribution values (weights) given by


Slaba and Blattnig [2014]. Computational examples are given to show that the methods are equivalent.

Throughout this section, the nominal GCR boundary condition is generated using the Badhwar-O'Neill

2010 (BON2010) GCR model [O'Neill 2010] for the solar minimum period of October 1976. Only the proton

portion of the GCR spectrum is being considered for the sake of computational simplicity. The computational

procedures developed herein could be applied directly to any ion in the GCR spectrum. Effective dose was

computed using the ICRP 60 quality factor [ICRP 1990] and ICRP 103 tissue weights [ICRP 2007] with the

FAX (Female Adult voXel) human phantom [Kramer et al. 2004] as described by Slaba et al. [2010a]. Transport

was performed with HZETRN-π/EM [Wilson et al. 1991; Slaba et al. 2010b, 2010c; Norman et al. 2013]. Any

result obtained with MC sampling utilized 20,000 trials.

The NCRP 153 [2006] report provides an overview of GCR model development and validation efforts.

No systematic errors are noted in any of the models (i.e. consistent over-prediction or under-prediction), and

overall model uncertainty is assessed to be within 15% below 5 GeV/n. This uncertainty estimate is consistent

with other assessments as well [O'Neill 2006, 2010; Mrigakshi et al. 2012].

In order to provide meaningful context and examples for the following discussion of uncertainty

propagation, relative uncertainty distributions will be chosen based on the NCRP 153 report. In particular, a

normal distribution with mean μ and standard deviation σ, denoted by N(μ, σ), will be utilized. The distribution

mean is assumed to be zero (i.e. μ = 0) and implicitly declares that the model shows no systematic over-

prediction or under-prediction of the measurements – consistent with NCRP 153 report. For most examples, the

standard deviation, σ, will be taken as 0.15 – corresponding to the 15% relative uncertainty estimate.

The final work in this uncertainty quantification effort will replace the assumed distribution and shape

parameters with rigorously developed uncertainty distributions drawn directly from comparisons to available

data.

MC Sampling Method

MC sampling is a straightforward, although computationally expensive, method for propagating

uncertainties in a GCR model to effective dose values behind shielding. First, it is important to note that the

GCR model relative uncertainty is defined here as

(1)


where U is the relative uncertainty, is the model result, and M is the measurement. Equation (1) does not

explicitly include measurement error, but could be accounted for using the method of Ferson and Hajagos

[2004] and would not significantly alter the present approach. The important feature of equation (1) in the

present context is that it takes the form of a relative difference quotient.

If a sufficient number of measurement data are available, the set of relative uncertainties calculated

with equation (1) form a distribution, or probability density function (PDF). Let ϕZ(E) be a nominal GCR

energy spectrum (boundary condition) for ion Z, and suppose Ui is a randomly sampled uncertainty value from

the PDF, then rearrangement of equation (1) yields the expression for the perturbed boundary condition

(2)

where M has been replaced with .

According to equation (1), a relative uncertainty value of U = 0.15 implies that the nominal model

result is 15% larger than the true (measured) value. In this case, equation (2) would decrease the model result.

The perturbed boundary condition may therefore be viewed as a corrected spectrum, with the correction factor

represented as a multiplicative term applied to the nominal result. If this sampling process is repeated many

times, and each perturbed boundary condition is propagated through to effective dose, the result will be a

distribution of effective dose values.

As a first example of utilizing equation (2), the GCR model relative uncertainty distribution is assumed

to be energy-independent (i.e. the uncertainty in the nominal GCR boundary condition is the same at all

energies). The relative uncertainty distribution is assumed to be N(0, 0.15) (i.e. a normal distribution with a

mean of zero and a standard deviation of 0.15). In this case, a relative uncertainty value is sampled from the

distribution and applied to the nominal proton boundary condition as in equation (2). The perturbed boundary

condition is transported through shielding and tissue, and effective dose is computed. This process is repeated

for each sampled uncertainty value.

Results are shown in Figure 1. In all subsequent figures showing effective dose PDFs, the printed

values with a bracketed interval correspond to the distribution median and 95% confidence level (CL). The skew

towards larger values in the perturbed proton spectra and resulting effective dose values is consistent with


equations (1) and (2). The relative uncertainty definition in equation (1) is bounded below by -1, and the

corresponding perturbed boundary conditions are bounded below by zero.

Recall that setting the mean of the relative uncertainty distribution to zero implicitly declares that the

nominal GCR model result has no systematic error (over-prediction or under-prediction). Consequently, the

median of the distribution of effective dose values should be similar to the effective dose value resulting from

the nominal GCR boundary condition, to within sampling error. Behind 20 g/cm2 of aluminum, the nominal

GCR proton spectra produced an effective dose of 194.3 mSv/year, and the median of the effective dose

distribution shown in the right pane of Figure 1 was 195.4 mSv/year (0.5% relative difference within sampling

error). If there were a systematic error in the GCR model this would appear in the resulting effective

dose distribution as a noticeable shift away from the nominal result.

A more realistic example is to consider (piece-wise) energy-dependent relative uncertainty

distributions. Such an example is relevant because, for instance, model uncertainties are unlikely to be the same

at energies below 500 MeV/n as they are above 5 GeV/n. The piece-wise description requires energy groups to

be defined and uncertainty distributions to be determined in each energy group. Hereafter, the energy groups

defined by Slaba and Blattnig [2014] will be used. For clarity, these groups are given in Table 1.

Even if separate relative uncertainty distributions are known for defined energy groups, sampling from

these distributions requires correlation coefficients to be considered. If the uncertainty distributions in each

energy bin are assumed to be completely uncorrelated, it implies that model errors in one energy region are

independent of errors in another energy region. Uncorrelated sampling from multiple distributions increases the

amount of uncertainty cancellation that occurs during sampling, resulting in a narrower distribution of exposure

quantities. Conversely, if the uncertainty distributions in each energy group are assumed to be completely

correlated, it implies that model errors in one energy region are related to model errors in another region.

Correlated sampling from multiple distributions reduces cancellation, resulting in a broader distribution of

exposure quantities.

Some level of correlation between GCR model relative uncertainties in each energy region is expected

due to the parametric nature of the models. It should also be noted that, for a specific time period or solar

activity level, two different models may exhibit different systematic errors across the energy spectrum.

Therefore, the correlation coefficients between different energy regions depend on the GCR model and level of

solar activity over which data are being considered. Correlation will be discussed in more detail in the next

work. Presently, results will be given for completely correlated and uncorrelated sampling to highlight and


bound the impact on the distribution of exposure quantities.

To introduce the concept of correlation, the energy domain is separated into the groups defined in

Table 1. In each energy bin, the uncertainty distribution is assumed to be N(0, 0.15). Correlated and uncorrelated

sampling is implemented. In this specific example, correlated sampling means that when an uncertainty value is

sampled for the first energy group, the same uncertainty value is used in all other groups. Uncorrelated

sampling, for this example, means that the uncertainty value is chosen independently in each energy group.

Results are given in Figure 2 for 20 g/cm2 of aluminum shielding. The results labeled "Energy

independent" (red dashed line) were taken directly from the right pane of Figure 1. The results obtained with

correlated sampling are almost identical to the energy independent results, and those generated with

uncorrelated sampling produce a narrower distribution, as previously stated. It is expected that, in this case, the

"Energy independent" results are identical to those generated with correlated sampling since the uncertainty

distribution in each energy bin is identical. Both sets of results have been provided in Figure 2 to show that

computational sampling procedures have been properly implemented and results behave as expected.

It is also instructive to show that the current uncertainty propagation analysis yields results that are

consistent with previous sensitivity analyses. As already discussed, it has been shown that GCR ions with Z > 2

and energy below 500 MeV/n induce less than 5% of the total effective dose behind shielding. Consequently,

large uncertainties in the low energy region should have minimal impact on the distribution of effective dose

values, even with very little shielding.

Results are shown in Figure 3 for no shielding. This shielding level was chosen to further highlight

that fluctuations in the low energy portion of the GCR spectrum have little impact on the resulting distribution

of effective dose values. The uncertainty distribution in the lowest two energy bins is assumed to be N(0, 0.30),

and the uncertainty in the upper three energy bins is kept at N(0, 0.15). Correlated sampling is used. Increasing

the standard deviation in the lowest two energy bins by a factor of two allows much larger perturbations to occur

in the low energy portion of the spectrum. Yet, it is clear in Figure 3 that these fluctuations have little impact on

the distribution of effective doses. In the following section, it is shown that the computationally expensive MC

method of uncertainty propagation is equivalent to a more efficient approach utilizing results from previous

sensitivity analyses.


Fast Uncertainty Propagation

In this section, an efficient method for propagating relative uncertainties in the GCR boundary

condition into effective dose behind shielding is presented. The mathematical formulation and computational

examples establish equivalence between the MC approach detailed in the previous section and the efficient

method presented in this section.

Let T represent the operator that translates a boundary condition spectrum into effective dose behind

some shielding geometry. In mathematical terms,

(3)

where is the boundary condition, and DZ is the effective dose. This operator includes the Boltzmann

transport equation [Wilson et al. 1991], dose equivalent integral [ICRP 1990], and summation equation for

effective dose [ICRP 2007]. It can easily be shown that all of these operations are linear, and hence, the operator

T is linear.

Suppose the distribution of model uncertainties in a given energy group (j) is known, and let ( )( )jZ E

and be the nominal and perturbed boundary conditions for ion Z and energy group j, respectively. As an

example, the nominal boundary condition over a specific energy group may be written as

(4)

According to equation (2), these boundary condition spectra are related through the equation

(5)

where U(j)

is an uncertainty value in energy group j. Applying the operator T to both sides of equation (5) yields


(6)

The total effective dose is obtained by summing equation (6) over all energy groups. Applying this summation

produces the equation

(7)

where is the total effective dose delivered by the perturbed boundary condition. Equation (7) may be further

modified so that

(8)

where D is the total effective dose delivered by the model boundary condition, and wj is the weight quantifying

the relative contribution of energy group j to the total effective dose. Since the measured (true) boundary

condition and corresponding effective dose is not known in general, equation (8) may be viewed as a corrected

effective dose obtained from the nominal GCR boundary condition. A procedure for deriving the weights, wj,

has been described by Slaba and Blattnig [2014], and weights for specific shielding materials, shielding

thicknesses, and solar activity have been provided in Tables 1 - 3 of that paper.

The relative error propagated into effective dose as a result of uncertainties in the nominal GCR

boundary condition is given by the equation

(9)

Replacing in equation (9) with the expression given in equation (8) produces


(10)

Equation (9) quantifies the relative error propagated into effective dose as a result of uncertainties in the

nominal GCR boundary condition. Equation (10) allows the relative uncertainty on effective dose to be

expressed as a weighted average of the GCR model relative uncertainties in each energy group.

The weights for effective dose have been computed [Slaba and Blattnig 2014] for specific shielding

scenarios and solar activity levels, and represent the relative contribution of each boundary energy group to

effective dose behind shielding. These weights depend on the shielding configuration and solar activity. If one

were interested in propagating boundary condition uncertainties into some other exposure quantity, the

corresponding weights would need to be developed. In principle, this could be accomplished using the

procedure outlined in the previous work [Slaba and Blattnig 2014].

Equation (10) may be used to efficiently propagate entire uncertainty distributions in each energy

group. In this case, U(j)

values would be sampled repeatedly from the uncertainty distributions in each energy

group (accounting for correlations as discussed in the previous section). The result would be a distribution of UD

values representing the distribution of uncertainties on effective dose caused by energy-dependent uncertainties

in the GCR boundary condition. Of course, sampling uncertainty values from a distribution and evaluating

equation (10) is much faster than a detailed radiation transport and effective dose calculation, as outlined in the

previous section.

Perhaps more importantly, equation (8) establishes an efficient method for propagating GCR model

relative uncertainties into effective dose values behind shielding. Given an effective dose value corresponding to

a nominal GCR model environment, equation (8) allows a distribution, or error bars, to be easily placed around

the nominal result. All that is needed are the weights that have already been provided by Slaba and Blattnig

[2014] and the energy-dependent uncertainty distribution for the GCR model.

In order to numerically verify that equation (8) provides the same results as the MC approach outlined

in the previous section, the energy groups and weights from Slaba and Blattnig [2014] are used. The relative

uncertainty distributions are assumed to be N(0, 0.3) in the lowest two energy groups and N(0, 0.15) in the


highest three energy groups. Correlated sampling is used, and equation (8) is used to compute the resulting

distribution in effective dose values. This distribution may be directly compared with the distribution in Figure 3

(green curve). Results are shown in Figure 4. Clearly, the computationally expensive MC method outlined in the

previous section and the efficient method described in this section give almost identical results.

3. Summary and Conclusions

In this paper, two computational approaches for propagating GCR model uncertainties into effective

dose values behind shielding were presented. The first method utilized a direct MC approach wherein a nominal

GCR model boundary condition was perturbed with a sampled uncertainty value. The perturbed, or corrected,

boundary condition was then transported using HZETRN-π/EM, and effective dose was computed. After 20,000

trials, the end-result is a distribution of effective dose values. Energy dependent uncertainty distributions

methods were also discussed along with correlated and uncorrelated sampling methods. In general, correlated

sampling should be used, and correlation coefficients may be determined after comparing GCR model results to

available measurement databases.

The second method for propagating GCR model uncertainties into effective dose values behind

shielding is simpler and more efficient. This approach was mathematically formulated and utilizes the weights

developed in the previous sensitivity analysis [Slaba and Blattnig 2014]. Examples were shown to establish

equivalence between the computationally intensive MC approach and the more efficient method. The main

result of this work is establishing a simple and efficient computational method for propagating GCR model

uncertainties into effective dose values behind shielding. Future analyses focused on GCR environments,

including risk analysis, may therefore easily include error bars, or complete distribution information, around any

point estimate of effective dose. The computational procedure could be easily extended and applied to other

exposure quantities and geometries allowing error bars to be placed around quantities suitable for transport code

comparison [Zeitlin et al. 2013a, 2013b; Hassler et al. 2014a].The only information that is needed are the

weights that may be computed using the method described by [Slaba and Blattnig 2014], along with GCR model

uncertainty distributions in the pre-defined energy bins.

The following work will present GCR model validation metrics and apply the metrics using an

extensive database of GCR measurements. This will allow the assumed normal distribution and shape

parameters chosen in this work to be replaced with real uncertainty distributions.


4. Acknowledgments

This work was supported by the Human Research Program under the Human Exploration and

Operations Mission Directorate of NASA. The data presented in this paper may be obtained by contacting the

authors.

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Figure 1. The left pane shows a nominal proton GCR spectrum (solid black line) for October 1976 along with

perturbed boundary condition spectra corresponding to different sampled uncertainties (dashed lines). The right

pane shows the resulting distribution in effective dose values behind 20 g/cm2 of aluminum.


Figure 2. PDF of effective dose values during October 1976 behind 20 g/cm2 of aluminum shielding. The results

labeled "Energy dependent (correlated)" (solid green) and those labeled "Energy independent" (dashed red) are

almost identical.


Figure 3. PDF of effective dose values during October 1976 with no shielding. The results labeled "Energy

independent" (dashed red) were generated by assuming an uncertainty distribution of N(0, 0.15) for all energies.

The results labeled "Energy dependent (correlated)" (solid green) were generated by assuming an N(0, 0.30)

uncertainty distributions in the lowest two energy groups and an N(0, 0.15) uncertainty distribution the highest

three energy groups.


Figure 4. PDF of effective dose values during October 1976 with no shielding. The results labeled "Energy

dependent" (solid green) were generated with a full uncertainty propagation; results labeled "Fast method"

(dashed red) were obtained using equation (8).


Table 1. Energy groups used in the current uncertainty propagation analysis.

Energy group label Lower bound (MeV/n) Upper bound (MeV/n)

E(1)

0 250

E(2)

250 500

E(3)

500 1500

E(4)

1500 4000

E(5)

4000 ∞

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GCR environmental models II: Uncertainty propagation methods for GCR environments

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