Uncertainty in modeled upper ocean heat content change

Abstract

This paper examines the uncertainty in the change in the heat content in the ocean component of a general circulation model. We describe the design and implementation of our statistical methodology. Using an ensemble of model runs and an emulator, we produce an estimate of the full probability distribution function (PDF) for the change in upper ocean heat in an Atmosphere/Ocean General Circulation Model, the Community Climate System Model v. 3, across a multi-dimensional input space. We show how the emulator of the GCM’s heat content change and hence, the PDF, can be validated and how implausible outcomes from the emulator can be identified when compared to observational estimates of the metric. In addition, the paper describes how the emulator outcomes and related uncertainty information might inform estimates of the same metric from a multi-model Coupled Model Intercomparison Project phase 3 ensemble. We illustrate how to (1) construct an ensemble based on experiment design methods, (2) construct and evaluate an emulator for a particular metric of a complex model, (3) validate the emulator using observational estimates and explore the input space with respect to implausible outcomes and (4) contribute to the understanding of uncertainties within a multi-model ensemble. Finally, we estimate the most likely value for heat content change and its uncertainty for the model, with respect to both observations and the uncertainty in the value for the input parameters.

Appendix

Symbol	Units	Definition
$\beta$		Regression parameters (vector)
$\Upgamma(\cdot)$		Gamma function
ρ0	kg/m3	Density
σ 2 obs		Variance related to observations
σ 2 cmip		Variance related to cmip models
σ 2 disc		Discrepancy
σ2		Variance prior
σ2*		Variance posterior
σ structural		Structural standard deviation
σ parameter		Parameter standard deviation
τ	Month	Model month
ν		Matern covariance shape parameter—set to 3/2
$\chi(\cdot,\cdot)$		Correlation function
A		n x n covariance matrix between D locations
B		Smoothing parameters (matrix)
b		Components of B
c p	J/(kgoK)	Specific heat
D		Matrix of all locations × (design points) (n × q)
D MD		Mahalanobis distance
dz	m	Model layer depth
dx	m	Zonal width of model grid cell
dy	m	Meridional width of model grid cell
$f(\cdot)$		Emulator function
$F(\cdot)$		Simulator function
$G(\cdot)$		Gaussian
H		Matrix of regression functions (h) at D
${\bf h}(\cdot)$		Regression function
I *2 mp		Implausibility score for cmip comparisons
I 2 mp		Implausibility score
$K_\nu(\cdot)$		Modified Bessel function
$m_o(\cdot)$		Prior mean process function
$m^*(\cdot)$		Posterior mean process function
N cmip		Number of CMIP simulations
n		Number of simulator runs at D locations
${\Updelta Q}$	J	Yearly heat content change
$\overline{\Updelta Q}$	J/yr	Mean of ${\Updelta Q}$
q		Length of input vector—the number of process parameters
T	oK	Temperature of model layer
t		Vector of covariances between a new point and $D' (\hbox{n} \times 1){\bf v}_{\bf o}(\cdot,\cdot)$ a
${\bf v}^{*}(\cdot,\cdot)$		Posterior covariance
x		Generic input parameter vector, length q
x val		Input parameter vector associated with validation location
Y		Generic simulator outcome
Y		Ensemble of simulator outcomes (vector)
Y val		Simulator outcome for validation
Y obs		Outcome related to observations
Y chip		Outcome related to cmip models
Y emul		Simulator outcome to use to create emulator

1. ^aPrior covariance
2. Scalar: roman; Vector, Matrix: bold roman
3. Bold indicates a vector of these values