A sensitivity method for ice floe trajectory calculations

A sensitivity method for ice floe trajectory calculations

N. R. THOMSON AND J. F. SYKES Department of Civil Engineering, University of Waterloo, Waterloo, Ont., Canada N2L 3Gl

Received July 25, 1991

Revised manuscript accepted November 29, 1991

The simulation of an ice floe trajectory normally involves the solving of a physically based governing equation in order to determine the temporal variation of ice floe velocities. As part of this calculation process, a sensitivity analysis is usually undertaken to determine the response of the solution to variations in the system or input parameters; thus providing a quantitative procedure to help interpret the results of the simulation. In this paper, a versatile sensitivity method is described, which can be employed in this sensitivity analysis. This sensitivity method is based on the adjoint operator technique which involves the calculation of an adjoint state variable. The required sensitivities are determined from the temporal variation of this adjoint state variable and the ice floe velocity. The results from a comparison between a numerical and an analytical solution for a one-dimensional free drift ice floe example are presented. In addition, an example dealing with the sensitivity of the distance between a drilling location and an ice floe trajectory to changes in the input parameters is discussed.

Key words: free drift, sensitivity, adjoint, ice floe, transient, iceberg.

La simulation de la trajectoire d'un floe de glace comporte normalement la resolution d'une equation dominante afin de determiner la variation temporelle des vitesses des floes de glace. Dans le cadre de ce processus de calcul, une analyse de sensibilite est habituellement entreprise afin de verifier comment la solution se comporte lorsque le systkme ou les paramktres d'entree sont soumis des variations; il s'agit donc d'une methode quantitative qui facilite l'interpretation des rksultats de la simulation. Dans cet article, une methode polyvalente pouvant &tre utiliske dans l'analyse de sensibilite est decrite. Cette methode est fondee sur la technique de l'operateur adjoint qui necessite le calcul d'une variable d'etat adjoint. Les sensibilites necessaires sont dkterminees a partir de la variation de cette variable d'etat adjoint dans le temps et de la vitesse des floes de glace. Les resultats d'une comparaison entre une solution numerique et une solution analytique dans le cas d'un exemple unidimensionnel de floes de glace libre sont presentes. De plus, un exemple traitant de la sensibilite de la distance entre un lieu de forage et la trajectoire d'un floe de glace aux modifications des paramktres d'entree est discuti.

Mots elks : derive libre, sensibilite, adjoint, floe de glace, transitoire, iceberg. [Traduit par la redaction]

Can. J. Civ. Eng. 19, 573-585 (1992)

Introduction In support of various offshore activities, the need often

arises for the prediction of an ice floe or iceberg trajectory. This trajectory is usually determined from a temporal inte- gration of an estimate of the transient ice floe or iceberg velocity under free drift conditions. The prediction of the transient velocity of an ice floe or an iceberg has received considerable treatment in the literature. In the case of ice floes, a distinction has been clearly made between situations where ice floes move freely (e.g., McPhee 1980) and situations where the interaction between ice floes becomes important (e.g., Thomson et al. 1988; Coon 1980; Hibler 1979). For iceberg motion, both deterministic approaches (Smith and Banke 1981; El-Tahan el al. 1983) and statistical approaches (Garrett 1985) have been proposed. Although the governing equations that describe the transient free drift velocity of an ice floe and an iceberg are not identical, the parallel exists, since they both can be treated as individual particles. In this paper, only free drift ice floe motion is con- sidered; however, the techniques discussed herein can be directly applied to free drift iceberg motion.

T o determine the transient ice floe velocity, a number of available simulation models may be utilized; however, the same questions regarding changes in the model results due

NOTE: Written discussion of this paper is welcomed and will be received by the Editor until December 31, 1992 (address inside front cover). Prinlcd in Canada / lmprimc au Canada

to variations in the input parameters can be raised. Nor- mally, in order to deal with this concern, a sensitivity analysis is conducted, which typically involves the changing of model inputs or parameters in an ad hoc fashion and then conducting additional model simulations. Although this approach may be adequate in some situations, it generally requires a considerable amount of computational effort to produce satisfactory results. In this paper, an alternative methodology is developed for a transient free drift ice floe model. This approach is based on the adjoint operator technique, which requires the equivalent of one additional model simulation in order to determine the sensitivity of transient ice floe velocity to changes in all the model input parameters. The adjoint method involves solving a set of adjoint state equations derived from either the continuous or the discrete form of the governing equations. The solution of this set of equations provides an adjoint function which is uniquely related to each of the dependent variables.

The results of a sensitivity analysis of this form provide an enormous amount of information about which model input parameters have the greatest effect on the ice floe velocity calculations and hence the ice floe trajectory. This information can aid greatly in an operational setting where data collection efforts are very expensive and where it is very important to get the best return from the information collected.

In the following sections of this paper, the governing equations and the solution methodology for a transient free

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574 CAN. J. CIV. ENG. VOL. 19, 1992

drift ice floe model are presented, the devlopment of the adjoint state equation is discussed, and the results from a comparison of the discrete adjoint solution with an analytical solution of the continuous adjoint sensitivity equation are presented for a one-dimensional free drift scenario. Finally, the sensitivity of the distance between a drilling location and an ice floe trajectory to changes in the input parameters is presented.

Governing equations and solution methodology The transient free drift model utilized in this paper is based

on a momentum balance equation which accounts for the transfer of momentum to an ice floe from wind and ocean currents. The general form of this model may be written as

with

and

where m is the mass per unit area of the ice floe; u represents the ice velocity in the Cartesian coordinate directions; t is time, which is restricted to the interval [to, tf], where to is the initial simulation time and tf is the final simulation time; pa is the density of air at the ice surface; Ca is the air drag coefficient; ua represents either surface or geostrophic winds; pa is the wind turning angle; p, is the density of water; C, is the water drag coefficient; u, represents the geostrophic ocean current velocity; 6, is the ocean turning angle; fc is the Coriolis parameter defined as 2w sin 4, where w is the rotation rate of the Earth and 4 is the latitude; k represents the unit of normal vector perpendicular to the Cartesian coordinate system or the plane of motion; and 11 represents the 2-norm. In the governing equation El], it is assumed that the air or wind velocity is much greater than the ice velocity. If the known wind field is neostro~hic.

A .

then pa is defined as the angle between the surface air stress direction and the geostrophic wind direction; however, if the surface winds are known, then pa is zero since the surface winds are colinear with the applied stress (McPhee 1980). The ocean turning angle, P,, may depend on a number of factors such as water-column stability, ice floe velocity, and ice surface roughness (McPhee 1980).

Accompanying [I] is the initial condition expressed by

[2] u(t) = uo for t = to

where to is the known initial time and uo represents the initial ice velocity state at time to. The initial ice velocity state, uo, may be equated to some known velocity state, or determined from an equilibrium or steady-state analysis. Depending on the ice characterization and external forcing,

the specification of the initial ice velocity may play an extremely important role in the simulation of the ice floe velocity.

To solve [I] in conjunction with [2], a fully implicit, backward-in-time finite difference scheme may be employed to yield the following discrete representation of [I]:

with A representing a 2 x 2 coefficient matrix comprised of the entries

a,, = -a2, = -mfc - p,C,((u - ~ $ 1 1 2 sin Pw

and f representing a 2 x 1 load vector comprised of the entries

k fl = ~acallu,kll2(uk cos Pa - uax sin Pa)

where uk is the discrete ice floe velocity, Atk = tk - tk- ' is the time step increment, the superscripts k and k - 1 represent the current time level and past time level respectively, the subscripts ax and ay represent the Cartesian coordinate components of the surface or geostrophic winds, and the subscripts wx and wy represent the Cartesian coordinate components of the ocean current velocity. Equation [3], which is nonlinear in nature, is used recursively to determine the components of the ice floe velocity at each time level by starting with the initial conditions given by [2]. At each time level, [3] must be solved by an appropriate iterative procedure such as the Picard method or the Newton- Raphson method (Zienkiewicz 1977) until some defined convergence tolerance is satisfied. In this study, the following convergence criteria, which represents the sum of the relative changes between iteration levels of the components of the ice floe velocity, are employed:

with

where n represents the iteration level and E is a prescribed convergence tolerance. For nonlinear problems, the Newton-Raphson method usually provides a much better convergence rate than the first-order convergence rate of the Picard method. However, the computational effort required to solve [3] by either iterative scheme is minimal. In this analysis the Newton-Raphson procedure is utilized. Although

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THOMSON AND SYKES 575

the details of this procedure are fairly straightforward, it is important to introduce some of the details here, since they are directly applicable to the presentation of the adjoint state equation discussed later in this paper. Consider [3] written as a set of nonlinear algebraic equations such as

The typical Newton-Raphson expression may be derived as a result of a truncated Taylor series expansion of hk(uk) around uk as expressed by

where J represents the Jacobian matrix of [5], which is comprised of the entries presented in Appendix A, and (6"lT = (6;, 6,"). Because the Jacobian is known at the current iteration level, [6] represents a set of linear equations which can be solved easily (e.g., using Cramer's rule) to determine An, and hence un.

Sensitivity methods A sensitivity analysis performed on the transient free drift

calculations described in the preceding section would attempt to address the question of how sensitive these calculations are to changes in the model input parameters or system parameters. In such a sensitivity analysis, attention is usually focused on the calculations that are particular to a specific area or time of interest. For example, the speed of the ice floe at one particular point in time may be of interest, but usually it is the ice floe trajectory that is of more concern, since this will dictate if emergency procedures should be initiated. The ice floe trajectory is estimated by integrating the ice floe velocity over the simulation period. Mathematically, this takes the form

where xT(t) = (xl, x2) represents the position of the ice floe at time t, xo is the initial position of the ice floe, and s is a dummy variable.

A convenient way to represent a measure such as an ice floe trajectory is to define a model response or performance function. The general form of the continuous performance function, which is related to [I], is given by

where f [u(t), a , t] is a specified function of the ice floe velocity, of time t, and of the vector of model parameters a. This vector a contains the following model parameters: the mass of the ice floe, m; the air drag coefficient, C,; the components of the surface or geostrophic wind, u,; the wind turning angle, Pa; the water drag coefficient, C,; the components of the ocean current velocity, u,; the ocean current turning angle, P,; the Coriolis parameter, f,; and the initial ice floe velocity, uo. An equivalent representation of [8], which is related to the discrete system [3], is given by

where N i s the total number of time levels used to represent the simulation period; and d(uk, a , k) is a scalar function of the discrete ice floe velocity, uk, the vector of system parameters, a , and the time level, k. The choice of acceptable performance functions is unlimited; however, if the sensitivity of the speed of an ice floe at a particular point in time, say, t^ E (to, tf), was desired, then the continuous performance function [8] would take on the form

and the discrete performance function [9] would take the form

where k is the time level that corresponds to ( 6(t - t^) is the Dirac delta function, and hk,k is the Kronecker delta.

The results of a sensitivity analysis are sensitivity coefficients which represent the change in a performance function, P , to a change in the parameters a and may be written as dp/daT. These sensitivity coefficients are only valid in a neighbourhood around the set of parameters used in the simulation for which the response in P to changes in a remains linear. To determine these sensitivity coefficients, one of the following three possible methods may be employed: the direct parameter sampling method, the sensitivity equation method, or the adjoint operator method.

The direct parameter sampling method involves perturb- ing the parameters in a one-by-one and resolving the problem in order to recalculate the change in the performance function. Since the governing equation or its discrete equivalent must be resolved for each parameter, an excessive amount of computational effort may be expended if a large number of parameters are going to be investigated or if the simulation time is long. Moreover, the choice of the magnitude of the perturbation may create problems. For example, if the perturbation is too small in magnitude, then the effect of the performance function may be lost due to computer precision. If the magnitude of the perturbation is too large, then the linear assumption may be violated.

The sensitivity equation method requires that the partial derivative of the performance function with respect to the parameter under investigation be taken; however, the resulting equation must be resolved for each parameter under investigation similar to the direct parameter sampling method.

The final method that may be employed to determine the sensitivity coefficients involves the use of adjoint states or variables. This method requires the development of an associated, or auxiliary, equation which governs the adjoint variable. With this method, only two solutions are required: one for the ice floe velocity (the forward problem) and one for the adjoint state variable (the backward problem). Sen- sitivity coefficients for the performance function under investigation can then be calculated from the solution of these two problems for all parameters of interest. In this

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576 CAN. J. CIV. ENG. VOL. 19. 1992

study, the adjoint sensitivity method is used to determine the required sensitivity coefficients.

The required adjoint state equation can be derived from either the continunous or the discrete representation of the governing equation. Since the discrete representation is utilized to solve for the ice floe velocities, it is prudent to concentrate only on the discrete representation of the adjoint state equations. Following the procedure outlined in Appen- dix B, the discrete adjoint state equation, which is related to [3], may be written as

[12] ( J ~ ) ~ $ k - l = rk for k = N, N - 1, ..., 1

with the entries in rk defined as

where i has replaced k to represent the time level in the summation appearing in the discrete performance function, and $x and $, represent the adjoint state variable corresponding to the components of the ice floe velocity ux and u,, respectively. Equation [12] represents the discrete governing equation for the adjoint state variables. These variables are determined recursively by starting with the known final adjoint state variables ($N = O), and working backward through the time levels. The coefficient matrix in [12] is the transpose of the Jacobian matrix in the Newton-Raphson iterative scheme used to determine the ice floe velocities as part of the forward problem (see equation [6]). This similarity between the forward problem and the backward problem allows the convenient implementation of this sensitivity method, since much of the numerical framework has been constructed as part of the solution for the forward problem. The actual physical interpretation of the adjoint state variable, $, depends on the choice of the performance function employed in the sensitivity analysis. For example, if the performance function was the x-component of ice floe velocity, then $x and $, represent the x-component of the ice floe acceleration due to a unit applied stress in the x-component and y-component directions respectively.

The last term on the right-hand side of [13] and [14] represents the forcing or load on the adjoint state system. The temporal location and value of this load is determined from the performance function under investigation. For example, if the performance function was the speed of the ice floe at the end of the simulation period, as represented by [ l l ] with k = N, then the last term on the right-hand side of [13] and [14] would be equal to u;/IIuNII2/Atk and u~/1IuNII2/Atk respectively for the time level k = N, and null for all other time levels. As long as the performance function remains unchanged, no additional adjoint state variables need to be determined.

Once the adjoint state variables have been determined at all time levels for a given performance function, the sensitivity coefficients for the required parameters may be calculated from

TABLE 1. Physical data for example 1

Parameter Description Value

P \V Water density 1000.0 kg/m3 Cbv Water drag coefficient 0.005 Pa Air density 1.2 kg/m3 ca Air drag coefficient 0.001 % Wind speed 5.0 m/s rn Ice mass per unit area 2000 kg/m2 uo Initial ice floe velocity 0.1549 m/s

where dP/da, represents the change in the performance function to a change in the lth parameter in the vector of parameters a. Equation [15] is repeatedly employed for each parameter under investigation. For example, if the ice floe mass per unit area is one of the parameters under investigation and the scalar function d(uk, a , k ) is not a function of the ice floe mass, then [15] would take the form

where dP/dm represents the change in the performance function (e.g., the speed of the ice floe) to a change in the ice floe mass per unit area. The entries in dhk/da, for the remaining parameters in a are presented in Appendix C.

The generated sensitivity coefficients can be normalized by the following expression so that a comparison between all of the parameters may be undertaken:

This normalized sensitivity coefficient, S,, represents the percentage change in the performance function due to a 1 Yo change in the parameter a,.

Example 1. One-dimensional transient free drift The purpose of this example is to illustrate some of the

properties of adjoint state solutions, and the utility of the resulting sensitivity coefficients. The governing one- dimensional transient free drift equation is defined as

with an initial condition defined by

[19] u( t ) = uo for t = to

Notice in [18] that the ocean current component which normally appears in the water stress term has been set to 0.

The selected continuous performance function for this example is the value of ice velocity at some time t^ as expressed by

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Analytical

CTS-12

CTS-120 ...........

VTS-23

I I I I I

Time ( h ) FIG. 1. Ice floe velocity for example 1.

J '0 J '0

The sensitivity coefficients produced by this choice of per-

with

formance function yield information on the change in ice velocity at time f t o a change in the system parameters for pWcw " all time less than and equal to Using this continuous per- c 2 =

formance function in conjunction with [18] and [19] results in a continuous adjoint state governing equation of the form

d ll, c1 + C2uo

[21] m - 2pwCwu(t)$(t) = 6(t - f ) A = [C, - C2#,I

with the final condition

[221 $(tf) = 0

where $(t) is the adjoint variable associated with the ice velocity. To solve the adjoint state variable $, [21] must be solved backward in time from the final condition [22]. Once the adjoint state variable has been determined for all time, the appropriate sensitivity coefficients may be calculated from

The first four terms on the right-hand side of [23] represent the transient contributions to the sensitivity coefficient from the ice mass, the air drag coefficient, the wind speed, and the water drag coefficient, respectively. The final term reflects the contribution from the initial condition, uo.

The analytical solution to the governing one-dimensional free drift equation [18] is given by

For large values of time, [24] approaches the steady state free drift solution of

Using [24], the analytical solution to the continuous adjoint state equation [21] and final condition [22] is given by

( 0 otherwise

For time equal to f in [26], the adjoint state variable reduces to l/m. Using [26], in conjunction with [24], the required sensitivity coefficients may be determined by performing the appropriate integral evaluations of [23].

A list of the physical and numerical parameters used in this example are contained in Table 1. As an initial condition, the steady state free drift ice velocity of 0.1549 m/s, which corresponds to a wind speed of 10.0 m/s, was used. Since the driving force for the forward problem is based on a wind speed of 5 m/s (see Table l), a period of approximately 6 h is required (due to inertial effects) for the ice velocity to reach steady state. This temporal period of 6 h

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Analytical

- CTS-12 . - - - - - .

CTS- 1 20 ............

- VTS-23

-

I !

I I

Time (h)

FIG. 2. Adjoint variable for example 1.

constitutes the temporal domain over which the results of the analytical and numerical solutions will be compared. The performance function was selected as the ice velocity at the end of 3 h. The three numerical solutions utilized in this comparison differ only in the choice of temporal discretization. Two of the numerical solutions involve constant time steps, the first is based on 12 time steps of 30 min each and is denoted by CTS-12, while the second is based on 120 time steps of 3 min each and is denoted by CTS-120. The third numerical solution employs variable time steps of decreasing length prior to temporal location of the performance function. This numerical solution, denoted by VTS-23, is comprised of 23 time steps broken down into the following discretization: 3 time steps of 30 min each, 2 time steps of 15 min each, 8 time steps of 6 min each, 4 time steps of 3 min each, and, finally, 6 time steps of 30 min each.

The results of the forward problem for the analytical and numerical solutions are shown in Fig. 1. The CTS-120 solution and the analytical solution compare extremely well, while the CTS-12 and VTS-23 solutions produce slightly larger ice velocities at the end of 3 h. However, all solutions approach the steady-state value of 0.07747 m/s and are generally of acceptable quality (i.e., little or no numerical problems).

The adjoint state variable or importance function corresponding to each of the forward solutions is presented in Fig. 2. Notice how the CTS-120 and VTS-23 solutions compare with the analytical solution. Since the CTS-12 solution is based on larger time steps, the resulting importance function suffers in comparison to the analytical solution. Spe- cifically, the sharp adjoint state peak at a time of 3 h is impossible for the CTS-12 solution to attain, and thus the solution is subject to dampening and numerical dispersion. The overall shape of the analytical importance function is characterized by the choice of physical parameters. For instance, the peak of the importance function may be decreased by increasing the ice mass.

TABLE 2. Normalized sensitivity coefficients and performance function values for example 1

Analytical CTS-12 CTS-120 VTS-23 Parameter solution solution solution solution

Civ -0.5145 -0.5127 -0.5151 -0.5153 c a 0.4720 0.4359 0.4685 0.4485 u a 0.9439 0.8717 0.9369 0.8970 m 0.04254 0.07683 0.04660 0.06679 uo 0.01356 0.05144 0.01650 0.03620 P 0.07825 0.08002 0.07839 0.07928

The normalized sensitivity coefficients for the analytical and numerical solutions as well as their respective performance function values are listed in Table 2. The sensitivity coefficients corresponding to the water drag coefficient, the air drag coefficient, the wind speed, and the ice mass have been integrated over all time for the analytical solution and summed over all time levels for the numerical solution. Gen- erally, as the number of time steps increases, the normalized sensitivity coefficients determined by the numerical solutions converge to those determined by the analytical solution. The normalized sensitivity coefficients provide information on how a change in a given parameter, a,, will effect the performance function (i.e., the ice velocity at the end of 3 h of simulation). For example, with the VTS-23 solution, a 1% increase in the water drag coefficient would cause the performance function to decrease by - 0.5153%, while a 1 % increase in the wind speed would cause the performance function to increase by 0.8970%. Notice that, for the performance function and parameters selected in this example, both the ice mass and initial ice velocity are of little importance relative to the wind speed, the air drag coefficient, and the water drag coefficient. This ability to discern the relative importance of each parameter (based on the magnitude and sign of the normalized sensitivity coefficient) gives an indica-

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TABLE 3. Validation of the sensitivity coefficients for the analytical solution

1 % perturbation 10% perturbation Normalized sensitivity Calculated Expected Calculated Expected

Parameter coefficient (m/s) (m/s) (m/s) (m/s)

TABLE 4. Validation of the sensitivity coefficients for the CTS-12 numerical solution



TABLE 5. Validation of the sensitivity coefficients for the CTS-120 numerical solution



TABLE 6. Validation of the sensitivity coefficients for the VTS-23 numerical solution


Parameter coefficient (m/s) (m/s) (m/s> (m/s)

tion of where further data collection or re-analysis of the present data should be conducted in order to provide a more reliable model result.

Using the normalized sensitivity coefficients from Table 2, the expected change in a performance function using the first-order approximation was compared with the actual or calculated change in the performance function as a means of sensitivity coefficient verification. The results of a 1% and a 10% perturbation for each of the parameters are tabulated in Tables 3,4, 5, and 6 for the analytical, CTS-12, CTS-120, and VTS-23 solutions, respectively. Comparisons (to the fourth significant digit) between the calculated and

expected performance function values were within a relative error of 0.4% or an absolute error of 0.00029 m/s for all solutions. All four solutions produced approximately the same total absolute error (summed over all parameters and Yo perturbations) of 0.0005 m/s.

Notice that the variable time stepping solution, VTS-23, produced results comparable to the constant time stepping solution, CTS-120. Although the computational effort required for this example is trivial, computational savings, improved accuracy (relative to the analytical solution), and reduction in numerical difficulties may be realized if a variable time stepping scheme is employed.

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Calculated Trajectory - . .

-

-

-

I I I I

FIG. 3. Ice floe trajectory for example 2.

Based on these results, it is clear that a good comparison between the numerical and analytical solutions is not a requirement for sensitivity coefficient validation. Moreover, the sensitivity coefficients determined by the numerical solutions are dependent on physical parameters as well as on the numerical discretization scheme. It is this latter dependence that is important to note, since this infers that the sensitivity coefficients are relevant in a numerical sense only.

Example 2. Ice floe trajectory Ice drift and meteorological data were acquired by Gulf

Canada Limited at the Kulluk drilling caisson (70" 14' 52"N, 134O09'30"W) in the Beaufort Sea during the summer of 1985. Ice floe targets were visible from marine radar to a range of approximately 3 nautical miles. Position measure- ments were made 15 min to 1 h apart, although regular 20 min observations were common. Generally, three targets were observed simultaneously and tracked for periods that ranged from an hour to nearly a day; however, because of the short radar range, many of the recorded tracks were only a few hours in length as the floes passed by the rig. No information was provided on the size, shape, and thickness of the observed floes. All ice floe positions were referenced to a Cartesian coordinate system, centred at the drilling caisson. Ten metre winds were available hourly from the Kulluk, recorded in nautical miles per hour and to the nearest 10". These wind data were converted to Cartesian components.

From the data described above, an ice floe trajectory that was observed on August 2 was selected to exemplify how the sensitivity method discussed in this paper may be applied t o an ice floe trajectory calculation. Figure 3 presents the observed ice floe trajectory along with the calculated ice floe trajectory over a 5-h interval on August 2, 1985. The parameters used to calculate the ice floe trajectory shown in this figure are tabulated in Table 7. Since no information on the magnitude and direction of the tidal currents were available, a stationary ocean and a turning angle of 23" (e.g., McPhee)

TABLE 7. Data for example 2

Parameter Description

Ice mass per unit area Air drag coefficient Air density Wind speed Wind turning angle Water drag coefficient Water density Ocean current Ocean turning angle Coriolis parameter x-component of initial velocity y-component of initial velocity Time step increment

Value

1620 kg/m2 0.001

1.2 kg/m3 Variable

0.0 0.007

1000.0 kg/m3 0.0 m/s

23" 0.00014 s-I -0.60 m/s 0.001 m/s

3 min

was assumed. The air drag coefficient, the water drag coefficient, and the ice thickness were obtained through a parameter sampling calibration procedure. The initial ice floe velocity, uo, was calculated from ice floe position measure- ments made just prior to the start of the simulation period. Standard values were assumed for the remaining parameters.

During this 5-h period, the winds were from the east with a mean x-component of -4.6 m/s with a variance of 0.33 m2/s2, and a mean y-component of 0.5 m/s with a variance of 0.89 m2/s2.

The concern of the operator of the drilling caisson in this example is focused on how close the ice floe will come to the drilling caisson. To represent this concern, a performance function which relates the distance between the ice floe and the drilling caisson may be employed to determine which model parameters are most important or most affect the calculation of this distance. This performance function may be written as

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FIG. 4. Adjoint variable for example 2.

a > -0.5 + c .- 0 .- 2 -1

-1.5

-6 0 1 2 3 4 5

Time (h)

/ - / /

- - - - - - - - - / - - / - . , , ,

L- - - - - --____--,

I I I I

..-..-..-.. -._ /water drag coefficient ,..I -..

- ./.' -..-.. 0

.0.'

./'

/ : , . , : - y-component of wind , . - - _ - - - - - _ _ _ - - - _ _ - - - - - - - - - _ - \-1---------- alr drag coefficient

.............. .._, '.. '._, ........... -

.:\ ..' x-component of wind ..........

I I I I

0 1 2 3 4 5

Time (h)

FIG. 5. Temporal variation of the normalized sensitivity coefficients in example 2.

where the point (x,,, x2,) is the location of the drilling caisson, which is the origin in this example, and the point (xy, XY) is the location of the ice floe at the end of the simulation period. For the ice floe trajectory in this example, the corresponding value of this performance function ([27]) is 2.642 km.

The adjoint variable corresponding to this choice of performance function over the 5-h simulation period is presented in Fig. 4. Note that throughout the entire simulation period the x-component of the adjoint variable is positive, while the y-component of the adjoint variable is negative. This implies that, over the course of the simulation period, an increase in the stress applied to the ice floe in the

x-direction will increase the distance between the drilling caisson and the ice floe, while an increase in the stress applied to the ice floe in the y-direction will decrease this distance. This result is intuitively obvious, since an increase in the x-direction of stress on the ice floe will cause the ice floe trajectory to shift more to the right of the origin (location of the drilling caisson), thus increasing the distance between the ice floe and the drilling caisson.

Using the distribution of the adjoint state variable, the normalized sensitivity coefficients for eight parameters were determined at each time level. Figure 5 presents the temporal variation of these calculated normalized sensitivity coefficients for six of the eight parameters. Since the sensitivity

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TABLE 8. Normalized sensitivity coefficients for example 2

Parameter Description Value

UU x-component of wind C a Air drag coefficient Clv Water drag coefficient UOX x-component of initial velocity

y-component of wind Pw Ocean turning angle m Ice mass per unit area

U o ~ y-component of initial velocity

coefficients for the remaining two parameters (i.e., the components of the initial ice floe velocity) only exist at the first time level, they are not included in Fig. 5. Notice that the distance between the drilling caisson and the ice floe at the end of the simulation period is most sensitive to changes in the magnitude of the parameters during the first portion of the simulation period, and this sensitivity to changes in the magnitude of the parameters decreases as the simulation period increases. This is an important result, since in an operational framework this is where accurate definition of these parameters is most important.

The normalized sensitivity coefficients determined at each time level have been summed over all time levels and are lited in Table 8 in decreasing order of importance to the performance function. The parameter that is the most important is the x-component of wind, which has a normalized sensitivity of - 0.335. This value represents the percentage change in the performance function due to a 1% increase in the magnitude of the x-component of wind at each time level. Since the x-component of wind throughout the 5-h simulation is negative, an increase in the magnitude of this parameter over all the time levels will cause the ice floe at the end of the simulation to be closer to the drilling caisson. Note that the distance to the drilling caisson from the ice floe position at the end of the simulation is least sensitive to changes in the ocean turning angle, the ice mass per unit area, and the y-component of the initial ice velocity. Although these parameters are important for the simulation of the ice floe trajectory, changes in their magnitude around the values used in this example do not affect the performance function as much as the other parameters.

For this normalized sensitivity coefficient summary, it is clear that the parameters that correspond to the air stress (in particular, the x-component) have the greatest impact on the distance between the drilling caisson and the ice floe position at the end of the simulation period. Therefore for this example, data collection efforts should be concentrated on defining the wind field, since variations in the wind field have the greatest influence on the performance function.

Conclusions The deterministic simulation of an ice floe is facilitated

with the use of the adjoint operator technique for sensitivity analysis. Using this method, the sensitivity of a performance measure to all of the system or model input parameters can be efficiently determined as a function of the solution of the forward problem and the adjoint or backward problem. The forward problem is based on the momentum balance equation representing the forces applied to the ice floe. The

state variable in the resulting governing equation is the ice floe velocity. The adjoint state variable is determined from the solution of the adjoint state equation which is solved backward in time; that is, the solution is solved from the final time to the initial time. While the forward problem is nonlinear with respect to the ice floe velocity, the backward problem is linear. The sensitivity coefficients that are calculated using the adjoint state variable and the ice floe velocity distribution are local derivatives. These derivatives are a function of the parameter set used to calculate the ice floe velocities.

The performance measures discussed in this paper include the ice velocity at a selected time, the ice floe trajectory, and the deviation of the calculated ice floe position from a selected point-of-interest. Other performance measures that may be of interest include the sum-of-the-squares of the difference between the observed and calculated ice floe positions along an ice floe trajectory. Sensitivity coefficients that are generated using this measure may be used in parameter optimization schemes, as they provide information on how the input parameters can be adjusted in order to reduce the difference between the observed and calculated ice floe trajectories.

The calculated sensitivity coefficients provide useful information on the relative importance of each of the input parameters. When these sensitivity coefficients are normalized, they can be ranked in order to indicate which of the system parameters require the most careful resolution. For the analyses presented in this paper, the largest normalized sensitivity coefficients were calculated for the wind velocity and the drag coefficients. The ice floe motion was relatively insensitive to the ice mass and the initial ice velocity. However, the relative importance of a given parameter is problem specific.

Coon, M.D. 1980. A review of AIDJEX modelling. Proceedings of ICSI/AIDJEX Symposium on Sea Ice Processes and Models, University of Washington, Seattle, Wash., pp. 12-27.

El-Tahan, H.W., El-Tahan, M., and Venkatesh, S. 1983. Factors controlling iceberg drift and design of an iceberg drift prediction system. Proceedings of the 7th International Conference on Port and Ocean Engineering under Arctic Conditions, Helsinki, Finland, pp. 263-275.

Garrett, C . 1985. Statistical prediction of iceberg trajectories. Cold Regions Science and Technology, 11: 255-266.

Hibler, W.D., 111. 1979. A dynamic thermodynamic sea ice model. Journal of Physical Oceanography, 9(4): 815-846.

McPhee, M.G., 1980. An analysis of pack ice drift in summer. Proceedings of ICSI/AIDJEX Symposium on Sea Ice Processes and Models, University of Washington, Seattle, Wash., pp. 62-75.

Smith, S.D., and Banke, E.G. 1981. A numerical model of iceberg drift. Proceedings of the 6th International Conference on Port and Ocean Engineering under Arctic Conditions, UniversitC Laval, QuCbec, Que., pp. 1001-1011.

Thomson, N.R., and Sykes, J.F. 1990. Sensitivity and uncertainty analysis of a short-term sea ice motion model. Journal of Geo- physical Research, 95(C2): 1713-1739.

Thomson, N.R., Sykes, J.F., and McKenna, R.F. 1988. Short-term ice motion modeling with application to the Beaufort Sea. Journal of Geophysical Research, 93(C9): 6819-6836.

Zienkiewicz, O.C. 1977. The finite element method. 3rd ed. McGraw-Hill Book Company, New York, N.Y.

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THOMSON AND SYKES 583

Appendix A. Jacobian entries The entries in the Jacobian as expressed in [6] are given by

.k ah1 m k k J i 1 = z = - + a l l + ~,C,yX[(ux - ~ W X )

x At

k cos a , - ( u , - u$,) sin q V j

.k - ah1 k J12 - a;- = a12 + P W C W Y Y [ ( ~ ~ - ~ Z X ) COS

Y

- (u,k - u i y ) sin

.k - a h 2 J21 - = 0 2 1 + pwcWy~[(u$ - U $ X ) sin a ,

+ (us$ - u i y ) cos a,]

.k - a h 2 m k J22 - a ~ ; - = 7 + a22 + pwCW>[(ux - u i x )

Y At

sin a, + (u(: - u i y ) cos a,]

with

where J. is a 2N x 1 arbitrary and differentiable vector representing the adjoint state variables corresponding to the ice floe velocities at each time step, and h is a 2N x 1 vector representing the solution to the ice floe problem at each time step. Each 2 x 1 subvector in h consists of the nonlinear algebraic equation given by [5] multiplied by the respective time step interval.

To determine the adjoint state variable, and thus the required sensitivity coefficients, consider the partial derivative of the augmented performance function [Bl] with respect to an arbitrary member of the parameter vector a, say a / , which yields

where

U =

~ ~ ~ e n d i x B. ~evelopment of the adjoint state equation is the 2N x 1 vector representing the ice floe velocities at Following a methodology similar to Thomson and Sykes each time step. Since the vector J. is arbitrary, the expres-

(1990), consider the following augmented version of the sion multiplying au/aal in [B2] vanishes if the following general form of the discrete performance function as expression is satisfied:

with This expression represents the discrete adjoint state equation. For example, in an analysis that involves only three time steps, this expression would take the form

Expanding [B4] in terms of a recurrence relationship produces

a h k + l T T lB51 A ~ * [ ~ ] ~ S ~ - ' = (auklT - ~ t ~ + ' [ ~ ] for k = N, N - 1 , ..., J

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584 CAN. J. CIV. ENG. VOL. 19, 1992

with 4. Wind turning angle (Pa):

[B6] $ N = 0

where i has replaced k to represent the time level in the sum-

3 a P a

= ~ a ~ a l l u t 1 1 2 ( ~ L sin P a - uty cos Pa)

mation appearing in the discrete performance function. Simplifying [B5] by replacing the matrix that appears on the a h 2 k

a P a = P ~ C ~ I ~ U ~ I I ~ ( - U U COS P a + ~ : y sin Pa)

left-hand side by the transpose of the Jacobian matrix (see [6]), expanding the matrix that appears in the first term on 5. Water drag coefficent (C,): the right-hand side by using [5], and dividing through by Atk yields q ah 1 = p W l b k - uklI2 (uxk - ukx) COs bw [B7] ( J ~ ) ~ $ ~ - ' = r k for k = N , N- 1, ..., 1

with the entries in r k defined as - (u,k - ukY) sin P,

ah2 - - a c w

(4 - 4 3 sin P w +

I + (u,k - u i y ) COS Pw

N

r: = $$; + L 4 [ c d(u i , a , i )At i a t k auy i = l 1 6. x-component of ocean current (u,,):

I Equation [B7] is used to solve for the adjoint variable at - - ah1

auwx - - P ~ c ~ ~ ~ u ~ - ukw112 COS P w

each of the time step by recursively working backward through the time levels.

The remaining portion of [B2], which is presented below, (uxk - u i x ) COS PW is used to calculate the required sensitivity coefficients once the adjoint state variables have been determined. - (u,k - u i y ) sin Pw 1

Appendix C. Entries in ahk/aal + P ~ C ~ ~ Y ~ ~ ( ~ x k - ukx) sin PW The entries in ahk/aal required for the summation in [15]

[ + (u," - uky) COS P, - mfc

for the parameters in a are 1. Air drag coefficient (C,):

7. y-component of ocean current (u,):

I - - - -pallutl l2(ub cos a - u k sin P,) a c a - - ah1

auwy - P ~ c ~ ~ ~ u ~ - 4 1 1 2 sin Pw

3 = - pallu:l12(u: sin 8, + u k cos Pa) a c a

2. x-component of wind (u,): _I

- - ah' - - P a ~ a l l ~ : I 1 2 cOS P a - ~ a c a ~ a r ( ~ L COS P a au, 3 = -p,~\vl luk - ~ $ 1 1 ~ cos PW auwy

- uty sin Pa) ( ~ x k - ~ i x ) sin P,

3 = -pacallut112 sin Pa - P ~ C ~ T = ( ~ L sin P a au, + (u,k - uky) COS Pw 1

+ uty cos Pa)

3. y-component of wind (uav): 8. Ocean turning angle (P,):

3 ah 1

= pacallu:112 sin P a - PacaYay(~L COS P a - = - p , ~ ~ j l u ~ - ukl12[(u: - ukX) sin 8,

aua.v aPw

1

- uty sin Pa) + (u,k - ukY) cos Pw 1 + utY cos Pa> + (u,k - ukY) sin P,

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9. Coriolis parameter (f ,): 3 ah2 m = o ; -

ah1 ah2 k auoy auoy at'

aft - -m(u,k - ~ 6 , ) ; -

aft = m(ux - u:,) with

k 10. x-component of initial ice floe velocity (uOx): uax . uk Yax = - 9 Yay = A

ah1 - m . ah2 1 1 ~ : 1 1 ~ nuil12 - - - - - - 0 auox at1' auox

- (u; - u;,>. - (.Y" - 4,) 11. y-component of initial ice floe velocity (u,,): Ywx = 3 Y w , =

IIuk - -611~ IIuk - - $ ] I 2

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Documents

A sensitivity method for ice floe trajectory calculations