ground-based measurements. Geostatistical approaches are used to identify spatial
correlations and represent small-scale heterogeneity that is not resolved in the
coarse-resolution airborne data. We employ a Bayesian hierarchical model for
integrating the low-resolution airborne data and sparse ground-based data. We
demonstrate our approach using the airborne and car-borne datasets collected in
Fukushima City, Japan, in 2012. This integration aims to provide a more resolved
and integrated dose-rate map, and also quantify the uncertainty associated with the
map for modeling and for policy planning. This approach is now being used to map
the radiation dose rate in the Fukushima region, particularly in the evacuation zone
[11, 12].
2 Methodology
Our approach is based on a Bayesian hierarchical model [7, 8], which typically
consists of three types of statistical submodels: (1) data models, (2) process models,
and (3) prior models. The process models describe the spatial pattern (or map) of
dose rates within the domain, given the parameters defined by the prior models.
Geostatistical models are often used as process models based on the spatial
heterogeneity structure identified by available datasets. The data models connect
this pattern and the actual data, given measurement errors. These data models can
represent, for example, a direct ground-based measurement or a function of the
pattern—for example, spatial averaging over a certain area for a low-resolution
airborne dataset. The prior models determine the distributions or ranges of the
parameters based on pre-existing information. The overall model—a series of statistical submodels—is flexible and expandable, able to include complex correlations
(such as correlations with land use, soil texture, or topography) or various observations. Once all the submodels are developed, we can estimate the parameters, as
well as the radiation map and its confidence interval, using sampling methods or
optimization methods. To fully quantify the uncertainty, here we will use the
Markov-chain Monte-Carlo method. When the domain size and the number of
pixels are large, optimization-based methods will be used to obtain the mean
estimate and its asymptotic confidence intervals.
In this chapter, we show one simple example; the integration of airborne and
car-borne radiation measurements. We assume that each point of the airborne
measurements is the weighted average of radiations from radionuclides distributed
over the ground. To develop an integrated map, we denote the radiation dose rate at
i-th pixel by y i , where i = 1,…,n. We also denote two vectors, representing the
airborne data z A (each data point is represented by z A,j , where j = 1,…,m A ) and
car-borne data z V (each data point is represented by z V,j , where j = 1,…,m V ). The
goal is to estimate the posterior distribution of the radiation dose-rate map y (i.e.,
the vector representing the radiation dose rate at all the pixels) conditioned on two
datasets (z A and z V ), written as p(y|z A , z V ). By applying Bayes’ rule, we can rewrite
this posterior distribution as:
A Multiscale Bayesian Data Integration …
59
correlations and represent small-scale heterogeneity that is not resolved in the
coarse-resolution airborne data. We employ a Bayesian hierarchical model for
integrating the low-resolution airborne data and sparse ground-based data. We
demonstrate our approach using the airborne and car-borne datasets collected in
Fukushima City, Japan, in 2012. This integration aims to provide a more resolved
and integrated dose-rate map, and also quantify the uncertainty associated with the
map for modeling and for policy planning. This approach is now being used to map
the radiation dose rate in the Fukushima region, particularly in the evacuation zone
[11, 12].
2 Methodology
Our approach is based on a Bayesian hierarchical model [7, 8], which typically
consists of three types of statistical submodels: (1) data models, (2) process models,
and (3) prior models. The process models describe the spatial pattern (or map) of
dose rates within the domain, given the parameters defined by the prior models.
Geostatistical models are often used as process models based on the spatial
heterogeneity structure identified by available datasets. The data models connect
this pattern and the actual data, given measurement errors. These data models can
represent, for example, a direct ground-based measurement or a function of the
pattern—for example, spatial averaging over a certain area for a low-resolution
airborne dataset. The prior models determine the distributions or ranges of the
parameters based on pre-existing information. The overall model—a series of statistical submodels—is flexible and expandable, able to include complex correlations
(such as correlations with land use, soil texture, or topography) or various observations. Once all the submodels are developed, we can estimate the parameters, as
well as the radiation map and its confidence interval, using sampling methods or
optimization methods. To fully quantify the uncertainty, here we will use the
Markov-chain Monte-Carlo method. When the domain size and the number of
pixels are large, optimization-based methods will be used to obtain the mean
estimate and its asymptotic confidence intervals.
In this chapter, we show one simple example; the integration of airborne and
car-borne radiation measurements. We assume that each point of the airborne
measurements is the weighted average of radiations from radionuclides distributed
over the ground. To develop an integrated map, we denote the radiation dose rate at
i-th pixel by y i , where i = 1,…,n. We also denote two vectors, representing the
airborne data z A (each data point is represented by z A,j , where j = 1,…,m A ) and
car-borne data z V (each data point is represented by z V,j , where j = 1,…,m V ). The
goal is to estimate the posterior distribution of the radiation dose-rate map y (i.e.,
the vector representing the radiation dose rate at all the pixels) conditioned on two
datasets (z A and z V ), written as p(y|z A , z V ). By applying Bayes’ rule, we can rewrite
this posterior distribution as:
A Multiscale Bayesian Data Integration …
59