## Analytical Sensitivity Analysis

I have started working on a toy model and plan to include analytical sensitivity analysis as part of the methods. These notes, and an associated extended discussion that I have up-loaded, serve as a short introduction to the subject. The file is here.

**Update October 11, 2011.** These reports by J. R. Bates explore sensitivity and feedback of several aspects of simple climate models.

J. R. Bates, Some considerations of the concept of climate feedback, Quarterly Journal of the Royal Meteorological Society, 133: 545–560 (2007)

Published online in Wiley InterScience (www.interscience.wiley.com) DOI: 10.1002/qj.62

ABSTRACT: A conceptual study of climate feedbacks is carried out using two simple linear two-zone models and the commonly-used zero-dimensional model to which they reduce under simplifying assumptions. The term ‘feedback’ is used in many different senses in the climate literature. Two prototype usages, stability-altering feedback (defined in terms of a system’s asymptotic response to an impulsive forcing, negative when stability-enhancing) and sensitivity-altering feedback (defined in terms of a system’s steady-state response to a step-function forcing, negative when sensitivity-diminishing) have been isolated for study. These two climate feedback concepts are viewed against the background of control theory, which provides a generalized feedback perspective embracing all forms of forcing and which is often seen as providing the paradigm for the concept of feedback as used in climate studies.

The relationship between the prototype climate feedbacks is simple in the context of the zero-dimensional model. Here, the stability-altering and sensitivity-altering feedbacks provided by a given interaction are of the same sign, and the sign of the stability-altering feedback as measured by initial tendencies always coincides with its sign as measured by the defining asymptotic tendencies. Even in this simple model, however, the sign of the prototype climate feedbacks can be opposite to the sign of the system’s feedback as defined in control theory.

In the two-zone models, the relationship between the prototype climate feedbacks is not so simple. It is shown that, contrary to the common assumption, these feedbacks can be of opposite signs. Moreover, the sign of the stability-altering feedback as measured by initial tendencies can be opposite to its sign as measured by asymptotic tendencies. It is further shown that there is no simple relationship between the sign of either of the prototype climate feedbacks in the two-zone models and the sign of these models’ feedback as defined in control theory.

These results point to the need for greater precision and explicitness in the definition and use of the term ‘climate feedback’, both to facilitate interdisciplinary dialogue in relation to feedback and to guard against erroneous inferences within the climate field. Explicit definitions of the two prototype categories of climate feedback studied here are proposed.

Copyright 2007 Royal Meteorological Society

J. Ray Bates, Climate stability and sensitivity in some simple conceptual models, Climate Dynamics, 2010. Published Online December 2010. DOI 10.1007/s00382-010-0966-0

Abstract

A theoretical investigation of climate stability and sensitivity is carried out using three simple linearized models based on the top-of-the-atmosphere energy budget. The simplest is the zero-dimensional model (ZDM) commonly used as a conceptual basis for climate sensitivity and feedback studies. The others are two-zone models with tropics and extratropics of equal area; in the first of these (Model A), the dynamical heat transport (DHT) between the zones is implicit, in the second (Model B) it is explicitly parameterized. It is found that the stability and sensitivity properties of the ZDM and Model A are very similar, both depending only on the global-mean radiative response coefficient and the global-mean forcing. The corresponding properties of Model B are more complex, depending asymmetrically on the separate tropical and extratropical values of these quantities, as well as on the DHT coefficient. Adopting Model B as a benchmark, conditions are found under which the validity of the ZDM and Model A as climate sensitivity models holds. It is shown that parameter ranges of physical interest exist for which such validity may not hold. The 2 × CO2 sensitivities of the simple models are studied and compared. Possible implications of the results for sensitivities derived from GCMs and palaeoclimate data are suggested. Sensitivities for more general scenarios that include negative forcing in the tropics (due to aerosols, inadvertent or geoengineered) are also studied. Some unexpected outcomes are found in this case. These include the possibility of a negative global-mean temperature response to a positive global-mean forcing, and vice versa. [pay-walled]

J. R. Bates, On climate stability, climate sensitivity and the dynamics of the enhanced greenhouse effect, Danish Center for Earth System Science, DCESS REPORT Number 3, 2003.

Abstract

The dynamics of the enhanced greenhouse effect resulting from a CO2 increase are studied using a simple two-zone hemispheric atmosphere-ocean model on an aquaplanet that is simple enough to allow analytical solution. The model’s sensitivity to forcing is viewed against the background of its stability to free perturbations. Free perturbations in SST, regarded as representative of temperature perturbations in the mixed layers beneath, are subject to a destabilizing influence from the effects of the water vapor infrared radiative (WVIR) feedback and are stabilized by evaporation, which results in moist convection and precipitation that deposit the latent heat removed from the surface above the level of the main water vapor absorbers, whence it is radiated to space. The rate of evaporation depends on the surface wind strength and the air-sea humidity deficit. In the model, the former is parameterized in terms of the atmospheric angular momentum (AM) transport, which depends on the SSTs in both zones, and the latter in terms of the local SST through the Clausius-Clapeyron relationship. Using estimates of the parameters derived from observation and detailed radiative model calculations, the model gives an equilibrium temperature increase for a CO2 doubling that lies within the range of that given by GCMs. As in the GCMs, it is found that the warming is greatest in the extratropics. Unlike the case of the GCMs, the mechanism of the warming in the simple model can be fully understood. The model’s equilibrium sensitivity is found to be inversely proportional to the value of the stability determinant (which measures the product of the decay rates of the fast and slow normal modes) and to be strongly influenced by the strength of a ventilation feedback. Both of these factors are sensitively dependent on the strength of the extratropical WVIR feedback, and the ventilation feedback in addition depends critically on the latitudinal distribution of the surface forcing

J. R. BATES, A dynamical stabilizer in the climate system: a mechanism suggested by a simple model, Tellus (1999), 51A, 349–372.

Abstract

A simple zonally averaged hemispheric model of the climate system is constructed, based on energy equations for two ocean basins separated at 30° latitude with the surface fluxes calculated explicitly. A combination of empirical input and theoretical calculation is used to determine an annual mean equilibrium climate for the model and to study its stability with respect to small perturbations. The insolation, the mean albedos and the equilibrium temperatures for the two model zones are prescribed from observation. The principal agent of interaction between the zones is the vertically integrated poleward transport of atmospheric angular momentum across their common boundary. This is parameterized using an empirical formula derived from a multiyear atmospheric data set. The surface winds are derived from the angular momentum transport assuming the atmosphere to be in a state of dynamic balance on the climatic timescales of interest. A further assumption that the air–sea temperature difference and low level relative humidity remain fixed at their mean observed values then allows the surface fluxes of latent and sensible heat to be calculated. Results from a radiative model, which show a positive lower tropospheric water vapour/infrared radiative feedback on SST perturbations in both zones, are used to calculate the net upward infrared radiative fluxes at the surface. In the model’s equilibrium climate, the principal processes balancing the solar radiation absorbed at the surface are evaporation in the tropical zone and net infrared radiation in the extratropical zone. The stability of small perturbations about the equilibrium is studied using a linearized form of the ocean energy equations. Ice-albedo and cloud feedbacks are omitted and attention is focussed on the competing effects of the water vapour/infrared radiative feedback and the turbulent surface flux and oceanic heat transport feedbacks associated with the angular momentum cycle. The perturbation equations involve inter-zone coupling and have coefficients dependent on the values of the equilibrium fluxes and the sensitivity of the angular momentum transport. Analytical solutions for the perturbations are obtained. These provide criteria for the stability of the equilibrium climate. If the evaporative feedback on SST perturbations is omitted, the equilibrium climate is unstable due to the influence of the water vapour/infrared radiative feedback, which dominates over the effects of the sensible heat and ocean heat transport feedbacks. The inclusion of evaporation gives a negative feedback which is of sufficient strength to stabilize the system. The stabilizing mechanism involves wind and humidity factors in the evaporative fluxes that are of comparable magnitude. Both factors involve the angular momentum transport. In including angular momentum and calculating the surface fluxes explicitly, the model presented here differs from the many simple climate models based on the Budyko–Sellers formulation. In that formulation, an atmospheric energy balance equation is used to eliminate surface fluxes in favour of top-of-the-atmosphere radiative fluxes and meridional atmospheric energy transports. In the resulting models, infrared radiation appears as a stabilizing influence on SST perturbations and the dynamical stabilizing mechanism found here cannot be identified. [pay-walled]

**Update November 9, 2010.** here is additional information about applications of ASA to climate sciences.

Dan G. Cacuci, Mihaela Ionescu-Bujor, and Michael Navon, “Sensitivity and Uncertainty Analysis, Volume II: Applications to Large-Scale Systems”, CRC Press, 2005.

Chapter V. Using the ASAP to Gain New Insights into Paradigm Atmospheric Sciences Problems

Chapter VI. Adjoint Sensitivity Analysis Procedure for Operational Meteorological Applications

**Update February 8, 2010.** I have found that there are papers on this subject directly related to models of the Earth’s climate systems. I’m not surprised and especially that Dan Cacuci was on the case back in the early 1980s.

Here are a few good references:

Matthew C. G. Hall, Dan G. Cacuci and M. E. Schlesinger, “Sensitivity Analysis of a Radiative-Convective Model by the Adjoint Method”, Journal of the Atmospheric Sciences, Vol. 39, pp. 2038-2050, 1982.

Matthew C. G. Hall and Dan G. Cacuci, “Physical interpretation of the Adjoint Functions for Sensitivity Analysis of Atmospheric Models”, Journal of the Atmospheric Sciences, Vol. 40, pp. 2537-2546, 1983.

Dan G. Cacuci and Matthew C. G. Hall, “Efficient Estimation of Feedback Effects with Application to Climate Models”, Journal of the Atmospheric Sciences, Vol. 41, pp. 2063-2068, 1984.

I. M. Held and M. J. Suarez, “A Two-Level Primitive Equation Atmospheric Model Designed for Climatic Sensitivity Experiments”, Journal of the Atmospheric Sciences, Vol. 39, pp. 206-229, 1978.

S. Wanabe and R. T. Wetherald, “Thermal Equilibrium of the Atmosphere with a given Distribution of Relative Humidity”, Journal of the Atmospheric Sciences, Vol. 24, pp. 241-259, 1967.

S. Wanabe and R. T. Wetherald, “On the Distribution of Climate Change Resulting from an Increase in CO2 Content of the Atmosphere”, Journal of the Atmospheric Sciences, Vol. 37, pp. 99-118, 1980.

T. L. Bell, “Climate Sensitivity from Fluctuation Dissipation: Some Simple Model Tests”, Journal of the Atmospheric Sciences, Vol. 37, pp. 1700-1707, 1980.

Isaac M. Held and Brian J. Soden, “Water Vapor Feedback and Global Warming”, Annual Review of Energy and Environment, Vol. 25, pp. 441-475, 2000.

Brian J. Soden and Isaac M. Held, “An Assessment of Climate Feedbacks in Coupled Ocean–Atmosphere Models”, Journal of Climate, Vol. 19, pp. 3354-3360, 2006.

Brian J. Soden, Anthony J. Broccoli, and Richard S. Hemler, On the Use of Cloud Forcing to Estimate Cloud Feedback”, Journal of Climate, Vol. 17, No. 19, pp. 3661-3665, 2004.

REFERENCES

Cess, R. D., and G. L. Potter, 1988: A methodology for understanding

and intercomparing atmospheric climate feedback processes in

general circulation models. J. Geophys. Res., 93, 8305–8314.

Cess, R. D., and Coauthors, 1990: Intercomparison and interpretation of

climate feedback processes in 19 atmospheric GCMs. J. Geophys. Res., 95, 16 601–16 615.

Cess, R. D., and Coauthors, 1996: Cloud feedback in atmospheric general

circulation models: An update. J. Geophys. Res., 101, 12,791–794.

Colman, R., 2003: A comparison of climate feedbacks in GCMs.

Climate Dyn., 20, 865–873.

Colman, R., and B. J., McAvaney, 1997: A study of general circulation

model climate feedbacks determined from perturbed SST experiments.

J. Geophys. Res., 102, 19,383–19,402.

Colman, R., S. B. Power, and B. J. McAvaney, 1997: Non-linear climate

feedbacks from perturbed SST experiments. Climate Dynamics 13, 10, 717–731.

Cubasch, U., and R. D. Cess, 1990: Processes and modeling. Climate

Change: The IPCC Scientific Assessment, J. T. Houghton, G. J.

Jenkins, and J. J. Ephraums, Eds., Cambridge University Press,

365 pp.

Gates, W. L., J. F. B. Mitchell, G. J. Boer, U. Cubasch, and V. P.

Meleshko, 1992: Climate modeling climate prediction, and model

validation. Climate Change 1992: The Supplementary Report

to the IPCC Scientific Assessment, J. T. Houghton, B. A. Callander,

and S. K. Varney, Eds., Cambridge University Press, 200 pp.

GFDL Global Atmospheric Model Development Team, 2004: The

new GFDL global atmospheric and land model (AM2–LM2):

Evaluation with prescribed SST simulations. J. Climate, in press.

Held, I. M., and B. J. Soden, 2000: Water vapor feedback and global warming. Annu. Rev. Energy Environ., 25, 441–475.

Le Treut, H., Z. X. Li, and M. Forichon, 1994: Sensitivity of the

LMD general circulation model to greenhouse gas forcing associated with two different cloud water parameterizations. J. Climate, 7, 1827–1841.

Mitchell, J. F. B., and W. J. Ingram, 1992: Carbon dioxide and climate:

Mechanisms of changes in cloud. J. Climate, 5, 5–21.

Ramanathan, V., R. D. Cess, E. F. Harrison, P. Minnis, B. R. Barkstrom,

E. Ahmad, and D. Hartmann, 1989: Cloud radiative-forcing

and climate: Results from the Earth Radiation Budget Experiment.

Science, 243, 57–63.

Tsushima, Y., and S. Manabe, 2001: Influence of cloud feedback on annual variation of global-mean surface temperature. J. Geophys. Res., 106, 22 635–22 646.

Wetherald, R. T., and S. Manabe, 1988: Cloud feedback processes in a general circulation model. J. Atmos. Sci., 45, 1397–1415.

Zhang, M. H., R. D. Cess, J. J. Hack, and J. T. Kiehl, 1994: Diagnosticstudy of climate feedback processes in atmospheric GCMs. J. Geophys. Res., 99, 5525–5537.

**Summary**

These notes introduce a few of the ideas and concepts associated with sensitivity analysis for algebraic and ordinary differential equations. By sensitivity I mean what are the effects of changes in the numerical values of the parameters in a system of equations relative to a response function of interest. The response function can take any mathematical form, but I will focus on the values of the dependent variables of the equation system.

Mathematical methods for identification of, and quantifying the importance of, parameters in complex mathematical models of physical phenomena and processes are discussed. These methods are useful for calculating the sensitivity of the models and numerical solution methods embedded into computer software to the parameters associated with the application areas for the computer codes. Sensitivity investigations, a form of ‘what if’ analysis, are an integral part of the application of computer software. Generally we are almost always asking a question like, what is the effect on the calculated result (a system response) of changes in the parameters of the model equations and application. The mathematical methods discussed here are designed to provide an answer and additionally are also useful in uncertainly, parameter estimation, and optimization analyses.

Of the several methods available for calculating the sensitivity, the discrete adjoint sensitivity method (DASM) is the one that is most applicable, from both theoretical and practical aspects, to models and solution methods already coded into software. From the theoretical view, the DASM methodology is based on exactly the model equations and solutions methods used in the software, which are generally finite difference equation (FDE) approximations to the continuous equations. The adjoint approach also allows efficient investigations into alternative response functions and in particular very efficient investigations of many parameters for a given response. From the practical viewpoint, especially when implicit numerical solution methods are used, the amount of additional coding needed for the sensitivity methodology is relatively small. Finally, the solutions for the sensitivities are very low-cost calculations.

Some of the benefits expected from applications of analytical sensitivity analysis methods to models and software are as follows. Analytical sensitivity analysis, especially the discrete adjoint method applied to finite-difference equations, can be used to achieve a significant reduction in the number of computer runs needed to complete application analyses. Analytical sensitivity analysis is also useful for the case of determination and optimization of model and correlation parameters from experimental data when models embedded into computer codes are used as the method of finding the parameters. Application of sensitivity analyses will significantly improve the objectivity and efficiency of model and correlation development.

**Introduction and Background**

Identification of, and quantifying the importance of, parameters in complex mathematical models of physical phenomena and processes is an integral part of the application of computer software. Two kinds of parameters are introduced into almost all mathematical models of physical phenomena and engineering equipment: those that are well-based theory, and those of a more empirical, or heuristic nature. The former kind of parameter includes equation-of-state and thermophysical and transport properties of materials, for examples. And although these are well-founded in theory, the exact value might be uncertain for a variety of reasons. The second kind of parameter is associated with engineering models and empirical correlations of physical processes. The numerical value of parameters in the correlations, or even the form of the correlating functions, might be uncertain due to the nature of engineering models and correlations.

Additional parameters of interest are introduced by the use of software in analyses. The geometry of the equipment and systems that are the object of applications of the software is generally well-established. Sometimes, however, the effects of changes in the geometry on the results of an analysis might be of interest. Finally, the continuous equations are usually not solved by the software. Approximate solutions to discrete finite-difference approximations to the continuous equations are usually solved in the software. The effects of changes in parameters associated with finite-difference methods on the solution are usually of interest. The effects of changes in the numerical values of the discrete temporal and spatial increments, and stopping criteria for iterative methods, for examples, are generally investigated.

Questions almost always arise concerning which of the many parameters in the models and methods are the most important and what are the effects of the uncertainty of the numerical values of the important parameters relative to some calculated response of interest. The responses of interest range for the local-instantaneous values of the principal dependent variables, to auxiliary calculations of quantities of interest, to integral functionals of the dependent variables, to global functionals of all the models and methods in the code and an entire calculations. A general notion of a response function will be given below in these notes.

Of course the interesting case for me was Lorenz in chaotic regime .

I must reread it one more time . Why does it oscillate ?

[WORDPRESS HASHCASH] The poster sent us ‘0 which is not a hashcash value.

Comment by Tom Vonk | March 23, 2009 |

Data at this link

https://computation.llnl.gov/casc/sundials/documentation/cvs_examples/node3.html

provides a numerical benchmark for verification of calculation of sensitivities of a simple DE system (Robertson kinetics example) but a validation of numerical values with analytical would also be useful.

The sensitivities of this simple DAE system

y’ = 1/5 (x – y ) (1)

0 = -p1 y^2 – p2 + x

are straight forward

d y’/dp1 = y^2/5

d y’/dp2 = 1/5

d x/dp1 = y^2 (2)

d x/dp2 = 1

and can be approximated by the ‘forward’ method in a numerical solution of (1).

The analytical solution of (1) is

y(t) = b0 + b4 (b1 + b2 exp(-b3 t)) / (b1 – b2 exp(-b3 t)) (3)

x(t) = p1 y(t)^2 + p2

a = -4 ( p1 p2 + 1 ) + 5

b5 = sqrt(a)

b0 = 1/(2 p1)

b1 = 5 + b5

b2 = 5 – b5

b3 = b5/5

b4 = -b5/(2 p1)

p1 and p2 real such that -4 ( p1 p2 +1 ) + 5 > 0, say p1 = -5.837, p2 = 2.458.

(3) satisfies (1) but the analytical sensitivities calculated from (3) have values different from (2).

Comment by John Wasson | October 12, 2009 |

Hello John,

Thanks for the info and very useful link. I have almost no experience with DAE systems, so I’m not familiar with this problem at all.

Maybe I’ll take a look into it as time from my day job permits.

Comment by Dan Hughes | October 23, 2009 |

You might also consider complex step methods to calculate sensitivities. It’s become popular in the CFD community because it avoids subtractive cancellation problems of normal finite difference approaches, and gives you a sensitivity that is very accurate and consistent with your discrete approximation.

Comment by jstults | January 13, 2010 |

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