p yjz A ; z V
ð
Þ/p z A j y
ð
Þp y jz V
ð
Þ:
ð1Þ
The first conditional distribution p(z A | y) is a data model representing the airborne data as a function of the dose-rate distribution y. We assume a spatial
weighted averaging function of the dose rate map:
z A;j ¼
X
i2C j
w i;j y i þ e j :
ð2Þ
where C j represents the pixels within the spatial averaging range, w i,j is the weight
determined by the distance between i-th pixel and j-th airborne data point, and e j is
an error associated with each data point. We assume an inverse square distance
function for the weights, w i,j . The weight can be computed by radiation transport
equations [11].
The range is typically considered to be equal to the flight height, following Torii
et al. [6]. We assume that the error e j includes not only measurement errors associated with hardware (such as instrument noises) but also the uncertainty associated
with other factors (such as the variable height of buildings, small-scale spatial
variability). In this example, we assume that e j follows an independent normal
distribution with zero-mean and the error variance r A can be determined from the
correlation analysis between two datasets.
As a process model, we assume that y is a multivariate Gaussian field described
by geostatistical parameters. For simplicity, we assume that the ground-based
measurements have insignificant errors compared to the airborne measurements,
and hence we use the ground measurements as conditional points to constrain the
distribution of y as p(y|z V ). Since two conditional distributions are both multivariate
Gaussian, we can derive an analytical form of this posterior distribution as a
multivariate normal distribution with mean Q
−1
g and covariance Q
−1 , where
Q = R c
−1 +A
T
D
−1 A and l = l c + A
T D
−1 z A [7]. In Q and g, l c and R c are the
conditional mean and covariance given the ground-based data (z V ) and geostatistical parameters. D is the data-error covariance matrix; each of the diagonal components is r A . The matrix A is n-by-m A sparse matrices, where A ij = w ij if i-th pixel
is within the range C j ; otherwise A ij is 0. We may directly sample y or estimate the
mean (or expected) dose map by Q
−1
g. Although the example shown here is quite
simple, we may add more complexity, such as, for example, physics-based radiation
transport models (instead of weighted averaging) to represent the airborne data or
data-derived correlations between dose rates and land use or topography [2].
3 Demonstration
In this example, we applied the developed method to the air-dose-rate data collected
in Fukushima, Japan. We used the ground-based car-borne data from the second
car-borne survey (December 5–28, 2011; http://radioactivity.nsr.go.jp/en/contents/
60
H.M. Wainwright et al.
ð
Þ/p z A j y
ð
Þp y jz V
ð
Þ:
ð1Þ
The first conditional distribution p(z A | y) is a data model representing the airborne data as a function of the dose-rate distribution y. We assume a spatial
weighted averaging function of the dose rate map:
z A;j ¼
X
i2C j
w i;j y i þ e j :
ð2Þ
where C j represents the pixels within the spatial averaging range, w i,j is the weight
determined by the distance between i-th pixel and j-th airborne data point, and e j is
an error associated with each data point. We assume an inverse square distance
function for the weights, w i,j . The weight can be computed by radiation transport
equations [11].
The range is typically considered to be equal to the flight height, following Torii
et al. [6]. We assume that the error e j includes not only measurement errors associated with hardware (such as instrument noises) but also the uncertainty associated
with other factors (such as the variable height of buildings, small-scale spatial
variability). In this example, we assume that e j follows an independent normal
distribution with zero-mean and the error variance r A can be determined from the
correlation analysis between two datasets.
As a process model, we assume that y is a multivariate Gaussian field described
by geostatistical parameters. For simplicity, we assume that the ground-based
measurements have insignificant errors compared to the airborne measurements,
and hence we use the ground measurements as conditional points to constrain the
distribution of y as p(y|z V ). Since two conditional distributions are both multivariate
Gaussian, we can derive an analytical form of this posterior distribution as a
multivariate normal distribution with mean Q
−1
g and covariance Q
−1 , where
Q = R c
−1 +A
T
D
−1 A and l = l c + A
T D
−1 z A [7]. In Q and g, l c and R c are the
conditional mean and covariance given the ground-based data (z V ) and geostatistical parameters. D is the data-error covariance matrix; each of the diagonal components is r A . The matrix A is n-by-m A sparse matrices, where A ij = w ij if i-th pixel
is within the range C j ; otherwise A ij is 0. We may directly sample y or estimate the
mean (or expected) dose map by Q
−1
g. Although the example shown here is quite
simple, we may add more complexity, such as, for example, physics-based radiation
transport models (instead of weighted averaging) to represent the airborne data or
data-derived correlations between dose rates and land use or topography [2].
3 Demonstration
In this example, we applied the developed method to the air-dose-rate data collected
in Fukushima, Japan. We used the ground-based car-borne data from the second
car-borne survey (December 5–28, 2011; http://radioactivity.nsr.go.jp/en/contents/
60
H.M. Wainwright et al.