Extrapolation and smoothing of strain data

Data taken in the field consisted of distances and angles of targets relative to a fixed reference stake. As a first step in the data reduction, these angles were converted to angles relative to true north as measured on Julian day 81 (21 March). Rotations of the array were not taken into account for strain calculations, so that the coordinate system used for this study is slightly different from the true north coordinate system (the maximum difference is, however, less than 5 deg). Rotations of the array were, of course, included in the vorticity calculations. The time scale was converted to G.M.T. by using four G.M.T. calibration times obtained in the field to find a least-squares rate for the clock used in the measurements. The data point times were recomputed with this rate and then all data (both angles and distances) were linearly interpolated and resampled every hour. Using this new data set at one-hour intervals, the time series was smoothed with a low-pass filter having a transition band width with periods from 8.0 to 6.15 h. The smoothed time series was then resampled every third hour. If there was reason to expect that the data contained a large number of time intervals greater than 3 h, a low-pass filter having a transition band from 20 to 11.4 h was used before resampling. Both filters had less than 0.16% side-lobe errors and consisted of 81 symmetric weights designed according to the procedure discussed by Reference HiblerHibler (1972).

This process of interpolation followed by smoothing may be viewed as a consistent way of constructing a smooth curve (with no high-frequency components past a reasonable cut-off dictated by the average sampling rate) through the randomly spaced data points. Alternatively the curve may be considered an accurate representation of the low-frequency portion of the linearly interpolated curve.

Examples of the smooth curves generated by this process are shown in Figure 2, Curve a, which results from the filter with the higher frequency cut-off, follows the data quite closely. This indicates that there is little variance associated with periods shorter than 6 h.

Fig. 2.

Typical restais of the interpolation and smoothing process used to generate equi-spaced values for Ike strain analysis. Curve a was obtained with a smoothing filter transition band from 8.0 to 6.15 h and curve b with a transition band from 20.0 to 11.4 h.

Experimental error estimation

The primary source of error in the measurements is the uncertainty of target angles. Since angular measurements were generally accurate to ±0.000 3 rad (± 1 min) and distance errors were small, we estimated the x and y position errors of each measurement of a target at distance r and angle θ to be

(1)

Since measurements three hours apart were subtracted to obtain velocities we estimated x and y velocity errors to be

(2)

with Δt = 3 h. This is a slight over-estimation of errors for the difference between two numbers with uncorrelated errors. However, since there may be some errors due to interpolation between inequally spaced points that are not removed by smoothing, we have made a conservative error estimation. The experimental errors given by Equation (2) were then used as input to obtain the experimental error on the strain-rate tensor as discussed in the next section.

Least-squares computational technique

To understand conceptually the least-squares strain and vorticity calculations, it is useful to visualize a contour plot of the x (ory) velocity component of the ice. In essence the computer program used to calculate the deformation rate fits a planar surface through the contour plot with the slope angles of the planes yielding the strain-rates and voriicities. Since the actual velocity components will deviate from a perfect plane, there will be some uncertainty in the slope angles of the plane. We refer to the average deviation οf the velocity components from the plane as the residual fluctuation error and the uncertainty in the slope angles of the plane is referred to as the inhomogeneity variation. In addition, once the least-squares equation for the plane as a function of say N velocity measurements is determined, the estimated experimental errors may be inserted to obtain estimates of slope uncertainties due only to experimental errors. The remainder of the slope uncertainty may then be identified with non-linearities in the ice velocity field.

To formulate this conceptual model mathematically we proceed as follows. Using tensor notation for the strain-rate, the strain-rate tensor ė and vorticity ϖ are defined by

where v_i is the ith velocity component of the ice pack (considered as a continuum) and i, j = 1, 2 since we are only concerned with the horizontal motion of the sea ice. Considering N targets whose positions are being measured, we denote by the measured jth velocity component of the ith target, and by θ_i and r_i the polar co-ordinates of the ith target relative to an arbitrary origin.

As a model to explain the velocities of the N targets, we consider the ice velocity field (on the scale of about 20 km) to consist of a continuum velocity component, which varies in a uniform linear manner, (specified by and ϖ) plus a random fluctuation component. Mathematically our model is expressed by the equation

(3)

where

(4)

and A ₁, A ₂ are constants representing the continuum velocity components at the origin. In Equation (3), z_i is the fluctuation component plus any measurement error and E(u_i) = x_ij ε_j is the “expected” value of u_i since E(z_i) = 0. The least-squares estimates of є_i, denoted by ê _i, are obtained by minimizing ∑ z_i ².

To do this we differentiate

with respect to ε_k which yields the matrix equation for the least-squares estimates of ε_i (Reference Jenkins and WattsJenkins and Watts, 1968, p. 132):

(5)

where M = X ^TX and T denotes transpose.

When using these least-squares equations, it should be noted that adding a constant rotation to all angles only changes the vorticity ϖ. This can be demonstrated by noting that in Equation (3) changing ϖ to changes u_i to u_i’ where

This, however, is equivalent to adding a constant rotation per unit time to all angles. In a similar manner, adding a constant velocity to all points changes only A ₁, and A ₂

Since we have only a finite number of random velocity measurements (each with some random fluctuation “error”) with which to calculate there will be some uncertainty or error in . TO estimate this variation or uncertainty it is necessary to calculate the covariance matrix of which is easily shown (Jenkins and Watts, 1968, p. 134) to be given by

(6)

where and V_ij = cov (z_i, z_j). In our case z_i consists of two parts

where z_i ^M is the measurement error (which we estimate to be given by Equation (2)) and z_i ^I is the fluctuation component of the ice motion due to the fact that the actual ice velocity field is non-linear. If we assume that z_i ^M and z_i ^I are uncorrelated, then the variation in ê _i due only to measurement errors, call it C_ij ^M, may be estimated using Equation (6) with V_ij ^M = cov (z_i ^M, z_i ^M). Using Equation (2) we estimate V_ij ^M to be

(7)

To find the variation of some linear combination of the due to the total “error” z_i, we make the usual assumption that the z_i are uncorrelated with the same mean and variance so that cov (z_i, z_j) = δ_ij σ ² where the point estimator of σ ² (denoted by s ²) is given by the residual error

(8)

where .

In this case Equation (6) reduces to

(9)

Given a linear combination of then the deviation of from is such that

(10)

has the t distribution with 2N—6 degrees of freedom, assuming that the errors are normally distributed (Reference Bennett and FranklinBennett and Franklin, 1954, p. 250). Consequently, confidence limits for the estimated strain may be obtained using a t-distribution table.

In addition to the estimated strain, we are also concerned with the velocity fluctuation component z_i ^I which is strictly speaking not an error, but represents the variation from linearity of the velocity field over the region sampled. In the cases we have studied, the estimated residual error z_i obtained from Equation (8) was generally found to be larger than the average estimated value of experimental error z_i ^M from Equation (2). Consequently as a matter of convenience, we will often refer to the residual error obtained from Equation (8) as the residual fluctuation error.

Also, as a matter of notation we will refer to the uncertainty in , as the inhomogeneity error. It should be remembered that the inhomogeneity error is an estimate of the uncertainty in the least-squares estimated strain and consequently will depend some-what on the number of samples used. The residual fluctuation error, on the other hand, should be relatively independent of the number of sample points used.

Time series of the strain tensor

Since the outer array targets were measured more often than other targets, the outer array provides a more detailed time series for analysis. Results of least-squares calculations using the outer array are shown in Figures 3 and 4 which present the two invariants (Reference NyeNye, 1957) (denoted by and ) of the strain-rate tensor both separately and in the form of the divergence rate (sum of the principal components) and the maximum shear rate (half the difference of the principal components). For the strain-rates, the angles of the strain lines at the beginning of each 3 h interval were used in the least-squares calculations. The continuous errors in Figures 3 and 4, which are the ∆ values shown below each curve, represent the inhomogeneity error (calculated using Equations (8) and (9)) which is due primarily to velocity fluctuations. The small error bar on the divergence-rate curve represents the maximum experimental error, which was calculated by first calculating the experimental

Fig. 3.

(a)Least-squares divergence rate and accompanying inhomogeneity error and (b)maximum shear rate and inhomogeneity error. The small error bar in (a)represents the maximum uncertainty due to measurement error.

Fig. 4.

(a)Principal axis components of the least squares strain-rate tensor and (b)inhomogeneity errors.

error for each point in the time series according to Equations (6) and (7) and then finding the maximum error over the time series. For a more compact summary of the relative magnitude of the strain-rates and strain-rate variation errors we list the r.m.s. values for the various time series in Table II.

Table II

Table II. R.M.S. Strain-Rates, Strain-Rate Variation Errors and Experimental Errors for Outer Array.

It is clear from the figure and from Table II that in general strain-rates are of greater magnitude than their respective strain-rate inhomogeneity errors. The figures also indicate that the inhomogeneity error generally increases with increasing strain-rate. It is important to remember that the inhomogeneity errors shown are for six targets and would be smaller for a larger number of targets. This is analogous to the error on the slope of a simple least-square line, which becomes less as more points are added even though the standard error of the estimate may remain the same. In particular, for the same residual error for each velocity, an equilateral strain triangle would have a divergence-rate error c. 1.7 times as large as that shown in Figures 3 and 4.

One striking aspect of the deformation, best illustrated in Figure 4 is that in the principal-axis coordinate system most of the expansion or contraction is taking place along one axis. Moreover, there is usually contraction along one axis and extension along another. Another salient characteristic of the ice deformation is that the ice motion appears to consist of deformational events which occur every several days and usually consist of dilation followed by convergence.

The time series of the strain-rate generally shows rather rapid variations in strain which are probably due to the random bumping and yielding of floes as well as the random opening of leads. Under our idealized model consisting of ice fluctuations superimposed upon a continuum, these high-frequency motions should primarily represent fluctuations. The maximum observed divergence rate is seen to be (0.16% ±0.09%) per hour and the maximum convergence rate (0.15% ± 0.09%) per hour. The largest maximum shear rate is (o.16% ± 0.09%) per hour.

Non-linear velocity fluctuations

The least-squares calculation, besides yielding the average strain-rates, also give a measure of the non-linearity in the velocity field through the residual fluctuation error. This residual error can be viewed as a fluctuation in the velocity field from the ideal continuum value. The magnitude of these fluctuations is important for determining the size of a measurement array necessary for accurately measuring the average strain-rate. In terms of our continuum model, the fluctuation contribution together with the average strain-rate yields a characteristic length above which the pack ice may be considered a continuum and below which the ice motion of individual floes and leads becomes dominant. Such a characteristic length is estimated by determining on the average the length over which the fluctuation is almost the same as the continuum velocity variations.

For a best estimation of the fluctuation error we utilized the combined array consisting of 11 targets plus the. origin. Since some targets in the combined array were not measured as frequently as those in the outer array, the linearly extrapolated data were smoothed with a filter having a transition band from 20.0 to 11.4 h as discussed in the previous section on data analysis.

The resulting residual errors from the combined array and strain tensor components in a (north, west) co-ordinate system are shown in Figure 5. In order to put the residual error and deformation rates in perspective to the overall motion of the pack ice, we have also shown, in Figure 5, plots of the velocity components of the central point of the array. These velocity plots were obtained using drift data obtained by Thorndike and others (1972) from satellite navigation fixes. We have smoothed the velocity data with the same filter used in the strain calculation. In the processing of the data carried out by Thorndike, some smoothing was also carried out, but this smoothing only affected the higher frequencies past the transition

Fig. 5.

Comparison of (a)mesoscale divergence rate, (b) north–south strain-rate, (

), (c)east–west strain-rate (

), (d)shear rate (

), (e)least-squares residual fluctuation error, (f)east–west velocity of the camp, (g)north-south velocity of the camp, and (h)speed of camp. All curves were smoothed with a low-pass filter having a transition band from 20.0 to 11.4 h

band of our filter. For a more concise comparison of the data, r.m.s. values of the various curves are given in Table III.

Table III

Strain-Hates, Residual Errors, and Central-Point Velocities (from Fig. 5).

As can be seen from Table III, the characteristic length (residual error/strain-rate) over which the velocity change due to strain-rate is of the same magnitude as the residual fluctuation error is of the order of 10 km. A second characteristic length of some interest is the length over which the velocity difference according to the strain-rate equals the average drift velocity. This length is seen to be about 1 000 km, roughly similar to the size of the Pacific Gyre. Consequently we see that the residual error is rather insignificant in terms of the absolute velocity for a point but becomes more critical in terms of the velocity gradient or deformation rate. In terms of a continuum model the results suggest that on a scale larger than about 10 km the pack ice begins to behave like a continuum, whereas on a smaller scale the individual particle behavior begins to dominate the observed motion.

The magnitude of the residual error also allows us to estimate the effect of fluctuations on strain-rates from arrays of different sizes. For example, we can view the strain measured by a simple area, say a triangle, to consist of two components: (1) the continuum strain-rate and (2) the fluctuation component. Assuming that the residual error is approximately constant, independent of the size of the triangle, then the fluctuation contribution to the strain-rate will be of greater relative magnitude for small triangles. In addition, since the fluctuation component would be expected to consist of rather rapid bumping motions, strain-rates would behave in a more erratic manner as the measuring triangle becomes smaller. This effect is apparent in plots of individual triangles. In Figure 6, for example, we illustrate the net divergences (essentially the areas) of three nested triangles as a function of time. The more rapid, large-magnitude motion of the small triangles is apparent. The curves do, however, illustrate a general correlation of strain events consisting of dilation followed by convergence. Generally these results suggest that although the smaller triangles are averaging over very few leads and/or floes, over a period of several days the ice on a small scale would be expected to diverge if the pack is generally diverging and converge if the pack is converging.

Time-series comparison

In order that both time series be smoothed in the same manner, all deformation rates were smoothed with a low-pass filter with a transition band from 21 to 84 h. This smoothing also allowed a time series for the mesoscale vorticity to be constructed by adding in the camp rotation, a step that is difficult without extensive smoothing because azimuthal measurements were typically made only once a day. To obtain the camp rotation for the mesoscale calculation, a time series R = (sin ϕ) (longitude —azimuth) (with longitude increasing in an easterly direction, see Reference NyeNye (1974)) where ϕ is the latitude, was constructed by linearly extrapolating the satellite position and azimuth measurements reported by Thorndike and others (1972). This time series was smoothed by the same filter that was applied to the deformation data. This rotation rate was added to the vorticity calculation in the camp coordinate system to obtain the true vorticity as discussed in a previous section. For mesoscale data in the camp coordinate system, the least-squares results for the outer array were used. In addition, several days of earlier data (taken before the array was complete) consisting of only four targets (1, 2, 3, 8) plus the center point were used to extend the time series.

Fig. 7.

Comparison of (a)mesoscale and macroscale divergence rates, (b)vorticities, and (c)maximum shear rates. The dashed lines represent macroscale data and the solid lines mesoscale data.

In Figure 7 we show a meso–macro comparison of the divergence rates, vorticities and maximum shear rates. The dashed lines represent the macroscale data. Due to a malfunction of one of the satellite navigation units, there is a gap of several days in the macroscale data which was bridged by Thorndike using a Kalman filter. This gap is indicated in Figure 7. For a quantitative comparison of the curves we give in Table IV correlation coefficients and r.m.s. values for the various curves up to the gap in the macroscale data. The standard error for the correlation coefficients is based upon a number of degrees of freedom equal to the number of points correlated times the fraction of the spectrum passed by the filter.

Table IV

Mesoscale and Macroscale R.M.S. Deformation Rates and Correlation Coefficients (from Fig. 7).

Figure 7 and Table IV indicate that there are significant correlations between the time series for deformation measured at different scales. Visual examination of the curves suggests that the correlation is due to the presence of similar strain events over periods of several days, Since these events are often of different amplitude and occur at slightly different times, the correlation is not complete, especially at higher temporal frequencies. The results also show the deformation rates to have comparable amplitudes, with the mesoscale amplitude generally being slightly larger. The comparison generally indicates that both mesoscale and macroscale arrays are measuring similar continuum motions of the ice pack.

Spectral densities

Some of the important differences between the nature of the mesoscale and macroscale deformations are illustrated by the spectral densities which we calculated using the lag product method as discussed, for example, by Reference RaynerRayner (1971, p. 94). In Figure 8a we show the spectra of the mesoscale divergence rate and shear (not maximum shear) and in Figure 8b the spectra of the macroscale divergence rate, shear and vorticity. Because of the inadequacy of the lime series for the camp rotation at higher frequencies, it is not possible to construct a mesoscale vorticity spectrum. Also, because of the Kalman filter smoothing of the macroscale data, the macroscale spectra are valid only up to frequencies with about 15 h periods. Since there were differing data gaps, the spectra do not come from the same time periods, but were calculated from day 88 to 113 for the mesoscale data and from day 81 to 101 for the macroscale data.

From Figure 8 we see that the macroscale spectra generally contain less variance at higher frequencies than the mesoscale spectra. Such a result is commensurate with viewing the deformation as a continuum signal plus a fluctuation component with the fluctuation magnitude dropping off inversely with the size of the array. This follows because the fluctuation signal, being of a random nature, would be expected to have a greater high-frequency variance.

An interesting aspect of the mesoscale spectra (especially the divergence rate) is the presence of a significant spectral peak at about 12 h periods. Whether such a peak is contained in the macroscale data cannot be ascertained because the smoothing employed by Thorndike effectively filtered out any such oscillation. A possible explanation for this peak is a variation in water currents due to inertial oscillations Reference Hunkins(Hunkins, 1967). Measurements of ocean currents by Newton and Coachman Reference Newton and Coachman(1973) during previous AIDJEX pilot studies have indicated 12 h cycles in the currents with the oscillation displaying coherence over distances up to 20 km. Different drag coefficients for different ice floes could couple with

Fig. 8.

Power spectra of (a) mesoscale divergence rate and shear rate, (b) macroscale divergence rate, shear rate and vorticity.

these currents to create a differential ice motion. There is also evidence of an approximately 24 h cycle in the macroscale divergence rate spectrum (which may possibly be present in the mesoscale data). The source of this peak is not at present understood.

The general fall-off of the spectra in Figure 8 is also relevant for sampling considerations. The shape of the spectra generally suggests that sampling intervals up to 10 h (with accurate measurements) would yield low-frequency information without intolerable aliasing. A more direct test of this can be made by sampling the data at larger intervals before smoothing and comparing this to data smoothed before resampling. Such comparisons have been made for the mesoscale data Reference Hibler, Hibler, Weeks, Kovaes and Ackley(Hibler and others, 1973[a]) and support the conclusion that accurate samples every eight hours are adequate for resolving the low-frequency components of the time series required for comparison with synoptic meteorological variations which generally occur over a time scale of several days Reference Monin(Monin, 1972).