Mathematical models are utilized to approximate various highly complex engineering, physical, environmental, social, and economic phenomena. Model parameters exerting the most influence on model results are identified through a ‘sensitivity analysis’. A comprehensive review is presented of more than a dozen sensitivity analysis methods. This review is intended for those not intimately familiar with statistics or the techniques utilized for sensitivity analysis of computer models. The most fundamental of sensitivity techniques utilizes partial differentiation whereas the simplest approach requires varying parameter values one-at-a-time. Correlation analysis is used to determine relationships between independent and dependent variables. Regression analysis provides the most comprehensive sensitivity measure and is commonly utilized to build response surfaces that approximate complex models.

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A review of techniques for parameter sensitivity analysis of environmental models

Two general algorithms for solving constrained minimization problems are presented and discussed in the context of analysis and assimilation of meteorological observations. In both algorithms, the original constrained problem is transformed by appropriate modifications into one unconstrained problem, or into a sequence of unconstrained problems. The main advantage of proceeding in this way is that the new unconstrained problems can be solved by classical descent algorithms, thus avoiding the need of directly solving the Euler-Lagrange equations of the original constrained problem. The first algorithm presented in the augmented lagrangian algorithm. It generalizes the more classical penalty and duality algorithms. The second algorithm, inspired from optimal control techniques, is based on an appropriate use of an adjoint dynamical equation, and seems to be particularly well adapted to the assimilation of observations distributed in time. Simple numerical examples show the ability of these algorithms to solve non-linear minimization problems of the type encountered in meteorology. Their possible use in more complex situations is discussed, in particular in terms of their computational cost. DOI: 10.1111/j.1600-0870.1986.tb00459.x

https://onlinelibrary.wiley.com/doi/pdf/10.1111/j.1600-0870.1986.tb00459.x

Variational algorithms for analysis and assimilation of meteorological observations: theoretical aspects

The following variational approach is taken to the problem of assimilation of meteorological observations: find the solution of the assimilating model which minimizes a given scalar function measuring the ‘distance’ between a model solution and the available observations. It is shown how the ‘adjoint equations’ of the model can be used to compute explicitly the ‘gradient’ of the distance function with respect to the model's initial conditions. the computation of one gradient requires one forward integration of the full model equations over the time interval on which the observations are available, followed by one backward integration of the adjoint equations. Successive gradients thus computed are introduced into a descent algorithm in order to determine the initial conditions which define the minimizing model solution.

The theory is applied to the vorticity equation. Successful numerical experiments performed on a Haurwitz wave are described.

Variational Assimilation of Meteorological Observations With the Adjoint Vorticity Equation. I: Theory

Anticipating the opportunity to make supplementary observations at locations that can depend upon the current weather situation, the question is posed as to what strategy should be adopted to select the locations, if the greatest improvement in analyses and forecasts is to be realized. To seek a preliminary answer, the authors introduce a model consisting of 40 ordinary differential equations, with the dependent variables representing values of some atmospheric quantity at 40 sites spaced equally about a latitude circle. The equations contain quadratic, linear, and constant terms representing advection, dissipation, and external forcing. Numerical integration indicates that small errors (differences between solutions) tend to double in about 2 days. Localized errors tend to spread eastward as they grow, encircling the globe after about 14 days. In the experiments presented, 20 consecutive sites lie over the ocean and 20 over land. A particular solution is chosen as the true weather. Every 6 h obs...

https://journals.ametsoc.org/doi/pdf/10.1175/1520-0469%281998%29055%3C0399%3AOSFSWO%3E2.0.CO%3B2

Optimal Sites for Supplementary Weather Observations: Simulation with a Small Model

Hydraulic transients in closed conduits have been a subject of both theoretical stud intense practical interest for more than one hundred years. While straightforward in te of the one-dimensional nature of pipe networks, the full description of transient fluid fl pose interesting problems in fluid dynamics. For example, the response of the turbu structure and strength to transient waves in pipes and the loss of flow axisymme pipes due to hydrodynamic instabilities are currently not understood. Yet, such u standing is important for modeling energy dissipation and water quality in transient flows. This paper presents an overview of both historic developments and presen research and practice in the field of hydraulic transients. In particular, the paper cusses mass and momentum equations for one-dimensional Flows, wavespeed, nu solutions for one-dimensional problems, wall shear stress models; two-dimensional and momentum equations, turbulence models, numerical solutions for two-dimen problems, boundary conditions, transient analysis software, and future practical an search needs in water hammer. The presentation emphasizes the assumptions and tions involved in various governing equations so as to illuminate the range of applic ity as well as the limitations of these equations. Understanding the limitations of cu models is essential for (i) interpreting their results, (ii) judging the reliability of the da obtained from them, (iii) minimizing misuse of water-hammer models in both research practice, and (iv) delineating the contribution of physical processes from the contribu of numerical artifacts to the results of waterhammer models. There are 134 refrences in this review article.@DOI: 10.1115/1.1828050 #

A Review of Water Hammer Theory and Practice

The adjoint method of sensitivity analysis is demonstrated on a radiative-convective climate model. A single adjoint calculation, which requires about the same computation time as the original model suffices to calculate sensitivities of surface air temperature to all 312 model parameters. The uses of these sensitivities are discussed and illustrated. The sensitivities accurately predict the effect on surface air temperature of small variations in the model parameters. Relative sensitivities are used to rank the importance of all the parameters. Several of the sensitivities to parameters customarily considered in previous works (e.g., solar constant, surface albedo, relative humidity, CO2 concentration) are reproduced, but the largest sensitivities are to constants used to compute the saturation vapor pressure of water. The uncertainties in the model results are expressed formally in terms of all the sensitivities and parameter covariances. For results that cannot readily be compared with observa...

Sensitivity Analysis of a Radiative-Convective Model by the Adjoint Method

The adjoint functions for an atmospheric model are the solution to a system of equations derived from a differential form of the model's equations. The adjoint functions can be used to calculate efficiently the sensitivity of one of the model's results to variations in any of the model's parameters. This paper shows that the adjoint functions themselves can be interpreted, as the sensitivity of a result to instantaneous perturbations of the model's dependent variables. This interpretation is illustrated for a radiative convective model, although the interpretation holds equally well for general circulation models. The adjoint functions are used to reveal the three time scales associated with 1) convective adjustment, 2) heat transfer between the atmosphere and space and 3) heat transfer between the ground and atmosphere. Calculating the eigenvalues and eigenvectors of the matrix of derivatives occurring in the set of adjoint equations reveals similar physical information without actually solving ...

Physical Interpretation of the Adjoint Functions for Sensitivity Analysis of Atmospheric Models

This work presents an efficient method for calculating the sensitivity of a mathematical model's result to feedback. Feedback is defined in terms of an operator acting on the model's dependent variables. The sensitivity to feedback is defined as a functional derivative, and a method is presented to evaluate this derivative using adjoint functions. Typically, this method allows the individual effect of many different feedbacks to be estimated with a total additional computing time comparable to only one recalculation. The effects on a C02-doubling experiment of actually incorporating surface albedo and water vapor feedbacks in a radiative-convective model are compared with sensitivities calculated using adjoint functions. These sensitivities predict the actual effects of feedback with at least the correct sign and order of magnitude. It is anticipated that this method of estimating the effect of feedback will be useful for more complex models where extensive recalculations for each of a variety of...

Efficient estimation of feedback effects with application to climate models

A demonstration is presented of the adjoint method of sensitivity analysis on a radiative-convective climate model with a diurnal cycle. A single adjoint calculation, which requires about the same computation time as the original model, suffices to calculate sensitivities of average surface air temperature to all 312 model parameters. The sensitivities accurately predict the effect on average surface air temperature of small variations in the model parameters. Relative sensitivities rank the importance of all the parameters, the most important parameters being those that characterize solar radiation and determine transmission functions. The uncertainties in the model results are expressed formally in terms of all the sensitivities and parameter covariances. For this model, a 10% standard deviation in surface albedo causes the increase in average surface air temperature after doubling CO/sub 2/ to have a standard deviation of 0.24/sup 0/K about the expected value of 1.66/sup 0/K. For results that cannot readily be compared with observation (for example, the results of a CO/sub 2/ doubling experiment), this method of uncertainty analysis is the only systematic way to estimate the reliability of model results. This estimate of the reliability, though, will only be realistic if all important physical processes and feedback mechanisms havemore » been included, however, crudely, in the model. The radiative-convective model contains complex on-off nonlinear processes of the type found in general circulation models. Therefore, the fact that the adjoint method works successfully and efficiently for the radiative-convective model provides valuable information about subsequent application of the method to general circulation models.« less

Dan G. Cacuci

Papers

Sensitivity Analysis of a Radiative-Convective Model by the Adjoint Method

Physical Interpretation of the Adjoint Functions for Sensitivity Analysis of Atmospheric Models

Efficient estimation of feedback effects with application to climate models

Systematic Analysis of Climatic Model Sensitivity to Parameters and Processes