Introduction

Horizontal strain-rate, vertical vorticity, and horizontal velocity observed at the surface of large ice sheets and ice shelves constitute field data essential to the study of large-scale ice-flow dynamics and mass balance. These data are traditionally acquired by implanting stake networks and surveying their deformation over a known time interval (Reference NyeNye, 1959; Reference Zumberge, Zumberge, Giovinetto, Kehle and ReidZumberge and others, 1960; Reference Drew and WhillansDrew and Whillans, 1984; Reference Thomas, Thomas, MacAyeal, Eilers, Gaylord, Hayes, Bentley and WashingtonThomas and others, in press). In practice, a one-year time interval is required to allow relative stake displacements in excess of measurement resolution. This one-year period necessitates the deployment of survey parties on two successive field seasons. This paper presents a technique, based on prior work by Reference NyeNye (1957, Reference Nye1959), that reduces the time Interval between initial and final survey and eliminates costly re-deployment of the survey party.

Typical stake networks used to measure the general flow of the Ross Ice Shelf and its grounded margins, for example, consist of four individual stakes arranged in either a rosette or a rectangular pattern (Reference Thomas, Thomas, MacAyeal, Eilers, Gaylord, Hayes, Bentley and WashingtonThomas and others, in press). These networks provide a degree of data redundancy because only three stakes are required to measure horizontal strain-rates and the vertical component of vorticity. In practice, no significance is assigned to apparent deformation gradients within the network, so this redundancy is generally treated by averaging the results of the four triangular sub-arrays (Reference Thomas, Thomas, MacAyeal, Eilers, Gaylord, Hayes, Bentley and WashingtonThomas and others, in press).

This study examines the consequence of expanded data redundancy within compact stake networks to determine whether the time interval between survey and re-survey can be shortened to less than one field season. Figure 1 shows an example of such a stake network. The results demonstrate that greater redundancy can shorten field operations and suppress aliasing errors that may occur when small-scale strain-rate fluctuations are superimposed on the average large-scale deformation. The stake-network design features that achieve these advantages may additionally be useful for certain remote-sensing programs that employ large numbers of natural surface features which can become obscured from one year to the next.

Fig. 1 The idealized n-leg rosette considered in this study for the purpose of error analysis consists of n outlying stakes surrounding a central stake (0) at equal angular intervals and with equal radial separation. Typical rosettes used on the Rose Ice Shelf had n = 3 (Reference Thomas, Thomas, MacAyeal, Eilers, Gaylord, Hayes, Bentley and WashingtonThomas and others, in press). Also shown above is a modified n-leg rosette that uses m + k < n stakes. This rosette design achieves n-legs by having k = n/m stakes, and may be more practical under actual field conditions.

Sources of Measurement Error

The horizontal strain-rate tensor components .(i = 1,2; j = 1,2) and vertical vorticity component are defined by the following expressions:

where u= (u₁, u_2, u₃) is the ice velocity and subscripts 1, 2, and 3 refer to a orthogonal coordinate system (x, y, z) having unit vectors ê_x, ê_y and ê_z such that ê_z is perpendicular to the geoid and ê_x and êy define the horizontal plane.

Measurement of stake displacements by conventional surveying equipment establishes the velocity gradients required to solve Equations (1) and (2) for

and Sufficient time must be allowed between initial and final survey so that the displacements will exceed measurement error and any unnatural stake disturbances caused by wind or sunlight. If, for example, ΔL ≈ 0.02 m is the stake position uncertainty, L ≈ 1.5 x 10³ m is the stake separation and 1 x 10⁻¹⁰ s⁻¹ is the strain-rate scale (typical for Ross Ice Shelf conditions), then a time interval of

is required between initial and final stake survey to achieve a displacement measurement accuracy of 1%. This time span is generally too long to be fit within a single field season; thus a second field season, incurring the duplicate cost of field-party re-deployment, is required.

An alternative to allowing one year between survey and re-survey is to plant more stakes in each network. From elementary statistics, the measurement error should reduce roughly as 1/√n where n is the number of stakes (or where n is the number of Independent measurements of a single stake) (Reference Mendenhall and ScheafferMendenhall and Scheaffer, 1974). It is thus conceivable that, by choosing n sufficiently high, Δt can be reduced to several days or weeks.

A second potential advantage gained by using stake arrays with large numbers of stakes is the reduction of aliasing errors caused by small-scale strain-rate and vorticity fluctuations superimposed on the large-scale deformation. Defining

as the strain-rate averaged over a suitably chosen large-scale distance and as the fluctuation around this average, the total strain-rate may be defined as

Uniaxial strain-rate data from the Ross Ice Shelf, shown in Figure 2, demonstrates that

is not necessarily smaller than and can vary over a length scale commensurate with the actual 1–2 km stake separations generally used on ice shelves (Reference Thomas, Thomas, MacAyeal, Eilers, Gaylord, Hayes, Bentley and WashingtonThomas and others, in press).

Many field programs are designed to measure

rather than however, it may not be possible to select a stake placement that will average out the effects of the unknown Aliasing errors will consequently arise in the measured value of The reduction of aliasing errors can be achieved through either increasing the stake separation or increasing the number of stakes to allow, in effect, averaging of individual aliasing errors. Planting stakes with large separations may be ruled out by equipment limitations, so increasing the number of stakes may again provide the best alternative.

Fig. 2 Uniaxial strain-rate measured on the Ross Ice Shelf at station C-16 (lat. 81°11′38″S., long. 189°30′09″W.) is plotted as a function of distance along a line bearing 321°T. The relative ice-shelf surface elevation along the line is also plotted. Open circles denote stake positions. The error level of the observed strain-rate is approximately 1.0 × 10⁻⁴/year.

Strain-Rate Estimation From An Over-Determined Data Set

An n-leg generalization of the 3-leg rosette stake network used by Reference Thomas, Thomas, MacAyeal, Eilers, Gaylord, Hayes, Bentley and WashingtonThomas and others (in press) on the Ross Ice Shelf will be examined to determine the advantages of redundancy and demonstrate the data-analysis technique. Each outlying stake of this rosette defines a position vector (x_i, y_i) i = 1,…,n that will change by (δx_i, δy_i) i = 1,…,n as a result of horizontal ice-shelf strain in the time Δt_i, i = 1,…,n between initial survey and re-survey. For purposes of this analysis, (i) the net strain is infinitesimal and (ii) the strain-rate variations over the length scale spanned by the stake network are considered as noise. Rosette design procedures other than those described here should be adopted if either of these conditions is not met.

The set of n observed stake displacements are converted into the four horizontal velocity component gradients ∂u_i/∂X_j, i,j = 1,2 by solving the following linear equation:

Here s is the column vector

representing the true local average of the horizontal velocity gradients, d is the column vector

that represents the measured stake displacements resulting from the true ice strain s, z is the column vector

that represents the disturbance of the true stake displacements resulting from inaccurate measurement and from actual strain-rate variation over the rosette, and

is the 2n x 4 stake-position matrix given by

The noise vector z is assumed to be a random variable characterized by the following 2n x 2n covarience matrix

In the above representation of <s> the error in measuring one stake is assumed, for simplicity, to be uncorrected with those of other stakes. The uncertainty of one stake’s displacement, however, may be greater than another’s. In more complex situations, errors in measuring different stakes may be correlated. In this circumstance, the data pre-weighting matrix

defined in the following section should be chosen to diagonalize

The matrix

contains all the information regarding the number and positioning of the rosette stakes. In general, will have a greater number of rows than columns because it maps a given element s of the “velocity-gradient vector space” having a dimensionality of 4 into an element d of the “data vector space” having a dimensionality equal to twice the number of stakes. Note, however, that not all possible elements of the data space are accessible through from an element of the velocity-gradient space. These inaccessible data elements constitute the measurement of noise and provide no information useful for calculating s. The objective of the data-analysis technique presented here is to select an inverse of that discards all inaccessible data elements.

Pre-Weighting and Scaling

Before deriving the inverse of

, Equation (5) is modified so that the covariance matrix has equal diagonal elements and the variables are scaled dimensionless quantities of order one (Reference WunschWunsch, 1978). The dimensionless weighting matrix defined by

where

is used to transform Equation (5) to

where

Observe that now To avoid unnecessary notational complication, however, the factors will be henceforth dropped from Equation (9).

The purpose of the weighting matrix

is to render each component of each observed stake displacement rate in a form having the same uncertainty. In practice, the displacement rate vectors (δ x_i/Δt, δ y_i/Δt_i) i=1,…,n are measured in polar coordinate (δ R_i/Δt_i, δ θ_i/Δt_i) i = 1,…,n where the uncertainty of δθ_i is considerably higher than δR_i/Rj. This disparity results from the practical limitations of conventional theodolite and electronic distance measuring devices. Typically, accuracies of ±0.5 x 10⁻² m and ±10″ are attainable for δR_i and δθ_i, respectively. Unless the outlying stakes are placed closer than c. 100 m from the central stake, these disparate measurement uncertainties will render the uncertainty of the tangential component of (δ x_i, δ y_i) greater than that of the radial component.

The advantage of using the weighting matrix

is that it allows the data elements of d representing radial displacements to weigh more heavily than those representing tangential displacements.

Non-dimensionalization of variables is accomplished by defining the length and strain-rate scales

and by transforming the variabies as follows:

where A_ij, ∂ u_i/∂ x_j, δ x_ℓ, δ y_ℓ, and z_ℓ are now of order on. Equation (9) may be expressed in terms of the non-dimensional uncertainty parameter

Scaling Equation (9) in this way will simplify the forthcoming discussion on how to best select Δt and L from prior estimates of

and σ.

Singular-Value Decomposition

The estimate

that minimizes the error defined by

is obtained by the singular-value decomposition method (Reference Lawson and HansonLawson and Hanson, 1974; Reference WunschWunch, 1978). According to this method,

is decomposed into the product of three associated matrices that are more readily invertable:

The 4 × 4 matrix

is diagonal and is composed of the eigenvalues λ_i, i=1,…,4 associated with the following linear equations

and

The normalized eigenvectors of

and , q_ℓ and r_ℓ respectively, comprise the columns of and the rows of respectively. The inverse matrices and are easily obtained by taking the transposes of and This simplicity results from the orthonormality of the eigenvectors q_ℓ and r_ℓ.

The inverse of

isFootnote *

where

is the 4 × 4 diagonal matrix composed of diagonal elements Г_ii ⁻¹ = 1/λ_i, i = 1,…,4. Equation (9) is solved by applying to the observed data d, so that

The expectation value of the error variance,

is given by

Equation (18) states that the error of each component of Se depends on the eigenvalues of

and the eigenvectors

Optimal Rosette Design

The task of designing the best rosette amounts to choosing the adjustable parameters Δt, L, and (x_i,y_i), i=1,…,n that satisfy a design criterion given by Equation (18) restated as

where E is the desired accuracy level (expressed in units scaled by

). Here, σ² is a parameter combining information about the surveying equipment and natural strain-rate variation, and is estimated or determined from prior data. In practice, it is best to choose L as long as possible (c. 2 km); therefore only n and Δt will be considered adjustable in the following discussion.

To derive the design criterion, the n-leg rosette will be idealized as a system of n outlying stakes placed regularly on a circle at angular intervals of Δθ = 2π/n. In practice, however, allowances must be made for deviations from this ideal configuration. Assuming that δR and Lδθ are the intrinsic radial and tangential uncertainties of observed stake displacements, the non-dimensional and pre-weighted version of the matrix

is given by:

expressed as a percentage of

is presented as a function of n and Δt in Figure 3 for parameter values L = 2 km and . Observe that this error value varies as (∆t√n)⁻¹. As shown in Figure 3, a choice of n = 13 would yield 3% accuracy within a time span of four weeks.

Fig. 3. The maximum error of the velocity gradients, expressed as percent of

is plotted as a function of n and Δt. Scale factors used in the calculation of these graphs are

= 10⁻¹⁰/s and L = 2.0 × 10³ m. Notice that 3% accuracy can be achieved by a 13-leg rosette in 28 d or by a 3-leg rosette in 84 d. Under some circumstances, it may be preferance to deploy rosettes having n >15 so that all measurements can be accomplished within one field season. This diagram, and other similar ones, can be used by field glaciologists to design their field program to maximize the data return for a given logistic effort.

Demonstration

A stake network planted up-stream of Crary Ice Rise on the Ross Ice Shelf demonstrates the data-analysis procedure associated with the n-leg rosette. Figure 4 shows the disposition of four three-leg rosettes (O,E,F,G) planted in 1983 as part of a field project by Robert Bindschadler and others to measure strainrates up-stream of an ice rise. The position of a much older three-leg rosette planted in 1973 (G₈) falls within the area enclosed by the four new rosettes. This older rosette was re-surveyed in 1974 as part of the Ross Ice Shelf Geophysical and Glaciological Survey (RIGGS) (Reference Thomas, Thomas, MacAyeal, Eilers, Gaylord, Hayes, Bentley and WashingtonThomas and others, in press); and yielded an accurate strain-rate determination (± 9%) with which data from the new rosettes may be compared.

Fig. 4. Four 3-leg rosette (0, E, F, G) planted upstream of Crary Ice Rise on the Ross Ice Shelf were re-surveyed after an average time period of 4.3 d. A fifth rosette (G₈) planted during RIGGS provides an accurate measurement of the actual strain-rate to be used as a reference. In this study, the four 3-leg rosettes are combined into one 12-leg rosette to see how the subsequent strain-rate measurement provided by the 4.2 d period compares with the actual strain-rate.

The four rosettes planted in 1983 were each surveyed and re-surveyed within a time interval spanning 4.2 d. This re-survey was conducted primarily to check survey equipment and to scan for possible zones of ultra-high deformation. Here, however, the re-survey data will be used to test the ability of a 12-leg rosette to yield an accurate measurement in a short time interval. The imaginary 12-leg rosette is synthesized from the four smaller rosettes by treating each of their central stakes as one. Actual strain-rate gradients expected within the field area will degrade the accuracy of the synthesized 12-leg rosettes because the smaller 3-leg rosettes are separated from each other by up to 15 km. Nevertheless, the synthesization provides an otherwise unavailable opportunity to demonstrate a 12-leg rosette measurement.

Figure 5 shows the stake disposition for the synthesized 12-leg rosette, and indicates the observed stake velocities relative to the combined central stakes. Except for two stakes corresponding to rosette 0, tangential displacements were not measured. Angular displacements required to observe the tangential components would have been below the accuracy level (±10″) of the theodolites used in this project, so any unknown tangential displacements were arbitrarily assumed to be zero.

Fig. 5. Stake positions of the 12-leg rosette are mapped along with the relative stake velocities. The principal axes of the strain-rate measured using the 12-leg rosette are compared with those measured during RIGGS (G₈) (Reference Thomas, Thomas, MacAyeal, Eilers, Gaylord, Hayes, Bentley and WashingtonThomas and others, in press). Also shown are the principal axes resulting from the 3-leg rosette E using a 3.9 d interval between survey and re-survey. Uncertainty of the observed velocities is variable. The shaded ellipse represents, for example, the velocity uncertainty for the three most outlying stakes. The ellipse should be oriented perpendicular to the position vector.

The results of the 12-leg rosette analysis are compared with the known strain-rates in Figure 5 and in Table I. The two principal strain-rate components (denoted

) given by the 12-leg rosette differ from the known values by 9% and 14%, respectively; and their orientations are displaced by 6^°, The unsatisfactory results produced by the 3-leg rosette (E) closest to G₈, using the data analysis procedure outlined by Reference Thomas, Thomas, MacAyeal, Eilers, Gaylord, Hayes, Bentley and WashingtonThomas and others (in press), are also shown in Figure 5 and Table I as a contrast to the results of the 12-leg rosette.

Practical Considerations

It would be a mistake to adopt rosettes with large n without first considering the limitations imposed by logistic support, survey technology, and the various benefits of a multi-year field program not emphasized in this study. It is, of course, always desirable to plant rosettes with large n. This desire conflicts, however, with the need to minimize the field-party work-load, especially when aircraft support limits the time a field party can spend at each field site. If field programs encompass a large number of field sites, the 3- or 4-leg rosette may be best because field operations would be likely to span several years regardless of whether individual sites were re-visited in the same or in a succeeding field season. For field programs covering a small area, such as that conducted up-stream of the Crary Ice Rise, rosettes with greater numbers of stakes may be preferable so that the entire field project can be accomplished in several weeks.

In view of the need to reduce the field-party workload, several design modifications to the n-leg rosette may prove useful. Figure 1 shows, for example, how planting k central stakes and m outlying stakes will achieve a high data redundancy while reducing the total number of stakes and the surveying workload. Surveying the m outlying stakes from k independent central locations yields k. m=n independent measurements of stake displacement. Only m + k< n stakes are required by this non-ideal n-leg rosette; and if appropriate measures are taken (e.g. assigning a separate corner-reflector prism to each outlying stake) the surveying can be accomplished within a reasonable time.

Conclusion

The stake-network designs that best measure surface strain-rates and vorticity encompass a large degree of redundancy. The more stakes included in each network, the less individual stake displacement errors will affect the overall measurement. The advantage gained at the expense of the redundancy is the reduced time interval between survey and re-survey.

Redundancy may also be a key element in the design of future field programs based on remote-sensing techniques such as airborne photogrammetry. Reference BrecherBrecher (1982), for example, has used photographs of natural features to identify the motion of the Byrd Glacier. These natural features are available in virtually unlimited number, and so provide, in principle, a natural equivalent of a large-n rosette. A shortened time interval between initial and final aerial photographic missions may reduce the camouflaging of natural features by drifting snow, and may, therefore, ultimately achieve greater accuracy by virtue of higher data redundancy.