Submersion Models

Submersion dose from an airborne radioactive plume is non-local. Photons at an observer can originate throughout the surrounding cloud, so the activity concentration at the observer is not, by itself, enough to describe the photon field. ensi_g14_calculator provides three ways to represent this effect:

Path	Quantity used for submersion	Intended role
ENSI G-14	Regulatory pseudo-dispersion \(\chi_{K,S}\)	Default regulatory calculation
Exact photon PKI	Line-by-line point-kernel dose coefficient	Reference and detailed point calculation
PKI surrogate	Energy-dependent equivalent dispersion \(\chi_{eq}(E)\)	Fast, vectorized approximation to the exact PKI

Scope and status

This package is an independent attempt to implement the complete ENSI-G14 directive. It is not an official ENSI product and has not been endorsed or certified by ENSI. The exact PKI and neural surrogate are optional package extensions; they are not methods prescribed by ENSI-G14.

ENSI G-14 Baseline

The physical Gaussian-plume dispersion factor \(\chi_K\) relates release rate to local activity concentration. For submersion, ENSI instead defines \(\chi_{K,S}\) as the concentration that would have to fill the half-space above an observer to produce the same immersion dose as the actual Gaussian plume. It is therefore a pseudo-dispersion or equivalent-dispersion quantity, not the local activity concentration.

For an elevated plume component, the implemented structure is

\[ \chi_{K,S} = \frac{KF_{sub,shape}\,KF_{sub,axis}} {\pi\sigma_y\sigma_z U_{H_a}}, \]

with an analogous ground-release component. KF_sub,shape represents the finite Gaussian cloud shape. KF_sub,axis represents the offset between the observer and the plume axis. The official G-14 Beilage states that the polynomial coefficients for both factors were derived from submersion dose calculations performed at CERN with the point-kernel method.

Energy Treatment

The polynomial geometry is evaluated with one prescribed attenuation coefficient,

\[ \mu = 0.01\ \mathrm{m}^{-1}, \]

corresponding to the 661 keV emission of Cs-137/Ba-137m. Photon-line energy is therefore not an input to the polynomial correction.

ENSI also specifies the nuclide-dependent factor \(k_{SPE}\) from the centroid energy \(SPE\) of the emitted gamma spectrum:

Centroid energy	\(k_{SPE}\)
\(SPE \geq 550\) keV	1.0
\(350 \leq SPE < 550\) keV	1.1
\(150 \leq SPE < 350\) keV	1.2
\(SPE < 150\) keV	1.3

In the package's regulatory dose path, \(k_{SPE}\) multiplies the tabulated immersion dose coefficient after \(\chi_{K,S}\) has been calculated. It can be understood as a coarse, zeroth-order compensation for the missing explicit energy dependence of the fixed-energy geometric correction. It does not make the plume geometry itself line-by-line energy dependent.

Why Add a Short-Distance Model?

The standard package workflow follows the G-14 evaluation convention and does not calculate submersion inside either a 200 m downwind limit or a 200 m radial limit. The fitted correction also imposes bounds on plume widths, plume-axis height, lateral distance, and far-downwind evaluation. Those rules are useful for a reproducible regulatory calculation, but they do not answer questions at arbitrary locations near a source.

The plots on this page deliberately lower the package's two submersion masks to 30 m to expose model behaviour around the source.

Exact Photon Point-Kernel Integral

The first new model evaluates finite-cloud photon transport directly. For a photon line with energy \(E\), yield \(y_E\), attenuation coefficient \(\mu_E\), buildup \(B(E,\mu_E r)\), and isotropic fluence-to-effective-dose conversion \(K(E)\), the coefficient at observer \(\mathbf{x}_O\) is

\[ C_E(\mathbf{x}_O) = \frac{y_E K(E)}{4\pi} \int_{\mathbb{R}^3} \frac{\chi(\mathbf{x})}{r^2} \exp(-\mu_E r) B(E,\mu_E r)\,\mathrm{d}^3x, \qquad r=|\mathbf{x}-\mathbf{x}_O|. \]

The implementation places spherical quadrature coordinates at the observer and substitutes \(s=\mu_E r\). The spherical Jacobian cancels the inverse-square term, leaving a smooth three-dimensional integral. Every photon line is evaluated using its own energy and yield, then summed for the nuclide.

This model:

evaluates observers at arbitrary 3-D coordinates, including below 200 m, at the source plane, and upwind of the source;
evaluates the physical reflected Gaussian plume at every quadrature point;
uses packaged air attenuation, buildup, photon spectra, and ISO isotropic fluence-to-effective-dose data;
masks cloud material below local terrain, but does not model attenuation by intervening terrain or buildings;
is photon-only and uses a separate dosimetric basis from the complete regulatory coefficient chain;
is much more computationally expensive than the ENSI polynomial model.

Neural PKI Surrogate

The second new model approximates the exact PKI while retaining fast NumPy array evaluation. Like the ENSI model, it predicts an equivalent dispersion rather than final dose. Unlike the ENSI polynomial, that equivalent dispersion is explicitly energy dependent.

For each photon line, define a semi-infinite half-space reference coefficient

\[ C_\infty(E) = \frac{K(E)}{2\mu_E} \int_0^{40}\exp(-s)B(E,s)\,\mathrm{d}s, \]

and define \(\chi_{eq}\) through

\[ C_E = y_E C_\infty(E)\chi_{eq}(E). \]

The network predicts \(\chi_{eq}(E)\). Photon yields, line coefficients, weather probabilities, wind speeds, and elevated/ground plume mixing remain explicit production arithmetic outside the network.

Training Design

The teacher is the exact spherical PKI applied to one reflected Gaussian plume component on a locally flat ground patch. Signed downwind coordinates are used so source-plane and upwind irradiation remain part of the learning problem.

The v8 input has 11 features describing:

observer offsets in photon mean-free-path units;
horizontal and vertical plume widths at a reference distance and twice that distance;
observer offsets relative to those plume widths;
photon energy and attenuation coefficient.

The transformed target is

\[ g=\chi_{eq}U\mu^2, \qquad t=\log\left(1+\frac{g}{g_0}\right). \]

The packaged multilayer perceptron has architecture 11 -> 128 -> 128 -> 128 -> 64 -> 1 with SiLU activations. Production inference is NumPy-only. The v8 configuration defines 4,096 plume groups with 64 observer/energy samples each, or 262,144 exact labels before a deterministic 80/10/10 group split. Grouping prevents observations from the same plume configuration leaking between training, validation, and test sets.

The current packaged model does not embed dataset hashes or a training-row manifest. Quantitative use should therefore identify the model artifact and retain the external training provenance rather than infer a sample count from the weights alone.

What Changes Between Models?

The atmospheric plume is the same. The difference is how each model converts that plume into the finite-cloud field relevant for submersion.

Physical, ENSI, and energy-dependent surrogate dispersion maps

The common-scale maps show local physical dispersion, ENSI's regulatory pseudo-dispersion, and surrogate-equivalent dispersion for Ar-41 and Tc-99m. The physical plume has no photon-energy dependence. The two surrogate maps do, because photon attenuation length and buildup change with energy.

Plume-axis comparison of physical, ENSI, and surrogate dispersion

The axis profiles make the same distinction in one dimension. The vertical line at 200 m marks the standard package cutoff; values inside it are shown only because this diagnostic uses a 30 m mask.

Exact Reference Versus Fast Models

Exact PKI, surrogate, ENSI, and physical-dispersion profiles

These profiles compare all four quantities for Ar-41 and Tc-99m on the plume axis and 500 m crosswind. Exact PKI can remain nonzero at a point where local dispersion is zero because photons can arrive from activity elsewhere in the downwind cloud. The timing annotation belongs to this plotted run and should not be treated as a portable benchmark, but it illustrates why exact quadrature is unsuitable for routine dense maps.

Surrogate Validation

PKI surrogate validation by observer region and phase space

The validation figure separates upwind, source-plane, and downwind points and shows residuals across dimensionless position, plume-relative offset, and energy. Very small exact coefficients produce large logarithmic residuals, and the observed error tail means the surrogate is not uniformly accurate. Use the exact PKI when a point is safety-significant or when surrogate behaviour is uncertain.

Choosing a Path

Use the default ENSI G-14 path for calculations intended to follow the directive. Use exact PKI for detailed photon comparisons and points where the regulatory geometry is unavailable or insufficient, especially below 200 m. Use the surrogate when many energy-dependent points are required, but retain exact-PKI spot checks and validation evidence.

The runnable photon PKI example shows the APIs. The plotting guide records the commands used to regenerate these figures.