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Four-dimensional modeling of recent vertical movements in the area of the southern california uplift
~ecto~opkysics, 52 (1979) 287-300 0 Elsevier Scientific Publishing Company,
287 Amsterdam
-Printed
in The Netherlands
FOUR-DIMENSIONAL MODELING OF RECENT VERTICAL MOVEMENTS IN THE AREA OF THE SOUTHERN CALIFORNIA UPLIFT
PETR VANleEK,
1 MICHAEL
R. ELLIOTT
2 and ROBERT
0. CASTLE
2
1 Department of Surveying Engineering, University of New Brunswick, Fredericton, (Canada) 2 U.S. Geological Survey, Menlo Park, Calif. (U.S.A.) (Accepted
for publication
N.B.
April 26,1978)
ABSTRACT VanrfZek, P., Elliott, M.R. and Castle, R.O., 1979. Four-dimensional modeling of Recent vertical movements in the area of the southern California uplift. In: C.A. Whitten, R. Green and B.K. Meade (Editors), Recent Crustal Movements, 1977. Tectonophysics, 52: 287-300. This paper describes an analytical technique that utilizes scattered geodetic relevelings and tide-gauge records to portray Recent vertical crustal movements that may have been characterized by spasmodic changes in velocity, The technique is based on the fitting of a time-varying algebraic surface of prescribed degree to the geodetic data treated as tilt elements and to tide-gauge readings treated as point movements. Desired variations in time can be selected as any combination of powers of vertical movement velocity and episodic events. The state of the modeled vertical displacement can be shown for any number of dates for visual display. Statistical confidence limits of the mode’ted displacements, derived from the density of measurements in both space and time, line length, and accuracy of input data, are also provided. The capabilities of the technique are demonstrated on selected data from the region of the southern California uplift.
INTRODUCTION
Vertical crustal movements defined by the results of repeated levelings can be calculated and represented in a variety of ways. The conceptually simplest and least equivocal technique consists of differencing observed elevations obtained through successive surveys over the same route. Because the extension of this technique beyond single-line comparisons generally requires systematic releveling over a closed network, it is usually impossible to relate areawide groups of such comparisons to a single control point (such as a primary tide station) whose stability can be independently assessed. Alternatively, because releveled segments are commonly scattered in both space and time, areally dist~buted vertical velocities or displacement can be calculated from
288
geodetically measured tilts through the fitting of these data to a mathematically defined surface or set of surfaces, such as an algebraic polynomial of specified degree. The most obvious advantages afforded by this technique are that it permits the use of otherwise unusable data and provides an objective basis for characterizing variations in the velocity configuration through time. The chief disadvantage of any surface-fitting technique is inherent in the smoothing process that discards or subdues short-wavelength features of potential si~ific~ce. Earlier adaptations of surface-fitting techniques to the reconstruction of vertical crustal movements have been developed around a simplifying assumption of constant vertical velocity at every point, i.e., zero acceleration. The resultant velocity surfaces are probably most valid in areas such as Chesapeake Bay (VaniEek and Christodulidis, 1974) or the maritime provinces of Canada (VaniEek, 1975; 1976) where the assumption of constant velocity can be tested through velocity misclosures or a knowledge of the Holocene deformation. Spasmodic or episodic, aseismic cmstal movements of the sort recognized in southern California (Castle et al., 1976) have provoked the development of the modeling technique described here, a technique that permits the fitting of a time-varying surface of a prescribed degree. By specifying the power in time as equal to one and by omitting episodic movements, the described model automatically reduces to one characterized by constant velocity. The data set for testing this newly conceived fob-dimensional model has been selected from repeated levelings carried out within the area of the southern California uplift during the period 1897-1934. We have chosen this data set because: (1) It is large enough that it can be used to assess the logic of the mathematical formulation, yet sufficiently small that it could be put in computerreadable form in something less than a lifetime. (2) The prescribed period includes independently identified episodes of dramatically changing vertical velocity. (3) The results can be compared with those obtained through a classic reconstruction involving a network of closed loops. MATHEMATICAL FORMULATION
The uplift surface, u, is sought in the follow~g
u(x,
Y,
t) =
kSl +2 5 5 = COkl;i(t)
i=O
i+
= qqclj
j-0
j'# 0
+ X(x, y, t)c
km1
CijkXiY'Tk(t)
form:
289
where: Tk(t) = tk
k = 1, n,,
)
t< bk
,/O’ Tk(t)
= \(t
bk)/(ek - bk) , bk < t <
-
k
ek
=
np + 1 , np + n,
t> ek
1,
and c is the n-vector of unknown coefficients for the previously chosen values n,, n, of maximum power in x and y. In these formulas, x and y are local horizontal Cartesian coordinates calculated from latitude and longitude through the following transformation equations: Cijk
y = (X - Xo)R cos $0
x=($-&P;
(2)
where (&,, X,) is the centroid of all bench marks and R is the mean radius of the earth. Time, t, is reckoned from a stipulated date, to, for which u(x, y, t,,) is everywhere equal to zero. In addition, bk, ek, for k = np + 1, nP + ne, are the dates of the beginning and end of n, movement episodes so that nt = np + n,. We note that the episodic movements are treated as linear within the duration of the episode. Observation equations for m releveled segments can now be written: Ah(x1,
= ids i+j#
~1,
~2,
x2,
jam
key
t2)
-
Cij,(3cIyi2
Y,,
Ah&l,
-xiY’l)[‘k(‘Z)
~2, ~2, tl)
-
= 4x,,
‘k(‘l)I
y1, x2> ~29 tl,
“(X1,
YI,
3t2,
t2)
Y2, tl,
t2)
0
= WI,
YI,
3c2,
Y29
t1,
t21c
+
(3)
r
where d denotes the m-vector of the differences Ah, and r is the residual vector. If m > n = (n, en, +n, +n,h we can find the solution, c, through the following normal equations: (BTC,‘B)c
= BTC,‘d
of leveled height differences, (4)
the method
of least squares which yields
(5) where Cd is a diagonal covariance matrix of the releveled segments. The diagonal elements equal the variances of individual height difference differences. In the computer program the normal equations are solved through orthogonalization. Clearly, the shift coefficients, co, cannot be determined from the releveled segments alone. Uplift u*(x, y,t) of at least one, but generally ng tide-gauges (x, y), must be determined from sea-level records at nd dates to allow us to evaluate the shift coefficients. The following ng - nd observation equations
290
can be then formulated: u*(Xi, _Yi,tj) =
T(tj)Co
+
X(Xi, Yi,
tj)C
+ r*
3
i = 1 , n,; j = 1 , nd
(6)
If n, . nd > nt, then the above equations may be solved for co, again using the method of least squares. The minimum condition imposed here is: min T*T(CU* + Cxc)-‘Y*
(7)
CO
where C,* and Cx, are covariance matrices of U* and Xc respectively. Otherwise, everything in eq. 6 is known. The covariance matrix, C,, of the coefficients c can be evaluated from the well-known formula:
cc =-;2,: (BTcdB)-’ Hence the covariance c,,
matrix,
Cx,, is given as:
= XC,XT
(9)
The covariance
ng
’
nd
matrix,
-
nt
Co, of the shift coefficients
(P(C,*
co is obtained
from:
+ C,,)-‘T)-’
(10)
We note that for each tide gauge the uplift U* must be determined satisfy the following condition:
so as to (11)
u*(x, Y, to) = 0 Finally, placement,
the standard deviation, u(x, y, t), of the computed vertical U(X, y, t), may be evaluated from the following relation:
4x, Y, t) = ddiag[7’(t)Co~(t)
+ X(x,
Y, t)CcXT@,
Y,
t)l
dis-
(12)
COMPUTER PROGRAM
The computer program was designed to solve the problem described in the preceding section. It has the following additional features: (1) Coefficients in orthogonal solution space that are below the prescribed noise level are removed through filtering. The noise level is prescribed as h times the coefficient’s own standard deviation, where h is input. (2) Episodes may be specified at the user’s option; if they are not, then nt = n,. (3) The vertical displacement (eq. 1) and associated standard deviation (eq. 12) surfaces are computed within a minimal rectangle covering all the releveled segments, on a regular grid with specified steps in 4 and h. The surfaces are produced for n, specified dates, t = Ti, where i = 1, n,. If there is no ri specified, a default value, 7, = to + 100 years, is chosen automatically.
291
(4) The surfaces may be plotted at the user’s discretion. The scale of the plot is an input parameter. (5) A plot of the location of the releveled segments may also be requested. (6) If the vertical displacement and standard deviation surfaces are plotted, then both the contour interval and maximum and minimum values may also be specified. The scale in both x and y directions is determined by the program in relation to the specified steps in 4 and h (A$ and Ax). (7) Vertical displacements of the tide stations, u*, are treated as optional information. If they are not supplied, both the shift coefficients (eq. 6) and
Dotted
Input
Data
Cards
Optional
Deck
Fig. 1. Diagrammatic puter program.
representation
of input
data
deck
for the
four-dimensional
com-
292
their covariance matrix (eq. 10) are all set to zero. The uplift surfaces are then held to zero at the centroid (&, X,) of the bench marks utilized. (8) Weights (02;) or weight factors (Wi) are also to be supplied for releveled segments. Weights are computed from the standard formulas (see, e.g., VaniEek, 1976). If weight factors are used, they should be introduced as: 2 for two first-order, 5 for one first-order and one second-order, 8 for two second-order, 17 for one first-order and one third-order, 20 for one secondorder and one third-order, and 32 for two third-order levelings. These weight factors are automatically converted into the proper weights for each releveled segment, where it is assumed in the conversion that first-order accuracy is equal to 1 mm X d(distance in kilometers). (9) Segments with residuals larger than a prescribed rejection criterion are automatically eliminated. A default value for the rejection criterion may be chosen (as has been done in this example), so that no segment is rejected. Assembly of the input data deck is shown in Fig. 1. INPUT DATA
The data set used to test the computer program is based on a broad distribution of repeated surveys that incorporates every combination of leveling precision stipulated in the preceding section. While we have not included all of the data collected within this area (Fig. 2) during the period 18971934, incorporation of these additional, relatively recently unearthed segments probably would contribute little additional strength to the solutions, for their distribution closely matches much of the data already included. Examination of the data distribution shows that the chief deficiencies are associated with the concentration of these data in two general time frames (around the turn of the century and the early to middle 1920s) and the poor spatial control west of Ventura. Input commands and features common to each series of fitted surfaces are : (1) Computation of vertical displacements at 5’ spacings in $J and 6’ spacings in h. (2) Use of a plotting projection in which the angular unit in 4 equals the angular unit in h. * (3) Adopting of to= 1897. (4) Plotting of cumulative vertical displacements from 1897 to 1902, 1908,1914,1924,1928, and 1934. (5) Inclusion of sea-level changes at the San Pedro Tide Station equal to zero through the period 1897-1934, to which a standard deviation of 3 cm has been assigned. * The horizontal distortion implicit in this projection produces a lengthening in X proportional to set $ and derives from a deficiency in the plotting subroutines; nevertheless, this distortion has virtuatly no effect on our chief goal, which is to test utility of the model,
-0 L I
50
,,fl’
r;/
-0
Approximate
EARSTOW
scale
,
00 Kilometers
-.--~-
- -
Fig. 2. Genera3ized map of southern California showing distribution of releveled segments and San Pedro tide station. Only those segmenta actually used in testing the four-dimensional computer program are shown here.
SAN1 A
BAKERSFIELD OwCr_
2 w
294
(6) Inclusion of a series of manufactured tilt segments obtained through differencing sea-level changes between San Pedro and San Diego, and weighted according to an estimated noise level of 3 cm (approximately equivalent to segments obtained through two successive second-order surveys). RESULTS
Because the solution depends on the degree of the fitted surface, which in turn affects the computer time needed, we have been constrained in the numbers of solutions attempted. On the other hand, preliminary classical reconstructions (as yet unpublished) provide a reasonable basis for selecting the degree of the fitted surface in both $J and X, and a somewhat poorer basis for selecting the power in t and the episodes. Experimentation with various powers in #, X, and t, and with the a-priori introduction of several episodes, suggests that the changing vertical displacement field within our area is reasonably approximated by a model with a maximum power of 2 in 4, 3 in h, 3 in t, and no episodes (Fig. 3). The four frames shown here (out of the six actually plotted) are clearly representative and provide a basis for comparison with our continuous profiling. Thus, the modeled solution indicates that the area contained within the lo-cm standard deviation contour sustained virtually no change in elevation between 1897 and 1902, other than modest uplift east of Los Angeles (Fig. 3A), a generalization that agrees with our conventional reconstruction. By 1914 uplift in excess of one standard deviation had propagated westward almost to Santa Barbara and had increased significantly east of Los Angeles (Fig. 3B). Comparison between the 1897-1924 and 1897-1914 plots (Figs. 3B and 3C) suggests a substantial increase in uplift between Los Angeles and the western edge of the map area between 1914 and 1924. On the other hand, continuous profiling between San Pedro and Bakersfield indicates that nearly all of this uplift occurred by 1914. The 1897-1934 displacement surface (Fig. 3D) suggests that most of the previously developed uplift collapsed and that the steep gradient along the north flank of the uplift shifted sharply southward during the period 1924-34. This interpretation, however, again conflicts with our conventional reconstruction, for we are reasonably certain that the .Los Angeles Basin experienced no more than a few centimeters of tectonic subsidence during this interval, The breakdown of the model implicit in the seeming continuation of uplift between 1914 and 1924 and the almost total collapse during the period 1924-34 almost certainly derives from the smooth interpolation of the movement in time as stipulated by a degree of 3 in time. Had we increased the power in t by one and added two episodes (1906-10 and 1924-28), uplift would have culminated by 1914 and the seemingly total coilapse between 1924 and 1934 would have been sharply diminished. Although the preceding evaluation of this clearly imperfect solution is
295
obviously subjective, the general validity of the solution is supported by the a-posteriori variance factor of 1.942 (compared with an assumed value of 1.000). Also only 12 coefficients out of 33 differ significantly from zero at the one standard deviation level. The a-posterior-i variance factor tWdr/ (m - n) assesses three effects that are inextricably entwined: (1) the selection of the model; (2) weighting of the data; (3) the data density and distribution in both space and time; hence it is impossible to say with certainty which of these three contributed most to the value of this quantity. Nevertheless, we strongly suspect that had the data been more evenly distributed both in space and time, the a-posteriori variance factor would have dropped substantially. In any event, to the extent that this factor is a measure of “goodness of fit”, the cited value supports the general validity of the 2, 3, 3 solution (VaniEek, 1975,1976). For comparison with the selected solution, we show a single frame (18971924) for the solution based on a m~imum power of 2 in @, 4 in h, 3 in t, and no episodes (Fig. 4). Differencing the two solutions through the subtraction of the vertical displacement field of Fig. 4 from that of Fig. 3C shows that, whereas the two displacement surfaces differ very little within the lo-cm standard deviation contour of either solution, they differ significantly away from the central part of the map area (Fig. 5). If it is accepted that the difference, within the region reasonably covered with data, between the two solutions is a variate with normal probability distribution, we might expect those areas in which the differences are greater than one standard deviation to cover about one-third of the map. This appears to be approximately the case here, Parenthetically, those areas in which the difference between the two solutions is greater than one standard deviation are generally coincident with the areas in which the 2, 3, 3 solution least closely accords with our conventional reconstruction. This seems to support our choice of the 2, 3,3 solution as relatively representative of the changing vertical displacement field within the area of our investigation. CONCLUSION
The results of the described tests of the four-dimensional modeling approach cannot be considered definitive. Given the restrictions imposed by our decision to limit the degree of the uplift surface, p~ticul~ly in time, and the use of only 352 segments, these results are, nonetheless, encouraging. Had we been able to obtain a few tens of additional segments randomly scattered through the western and northeastern parts of the area investigated, many and perhaps all of these solutions would have probably been improved significantly. Similarly, even a fragmentary knowledge of sea-level changes at Avila Beach (cf. Fig. 2) during the period 1897-1934 could have sharply constrained all of the attempted solutions. This tide gauge did not become a primary station until 1945.
2!3&
/
a
-_.-_..__.._d
X'zai ..--_ no'
--I..-
50
lOOKILOMETER
dPPROYIFvfATE SCRcF __._ -I,I_ _______.-l_lllll_l_ll
1 (~._"--.____11_--
-
3c
Fig. 3. Computed vertical displacement surfaces within the area of thd’southern Cailifornia uplift incorporating a power of 2 in latitude, a power of 3 in longitude, and a power of 3 in time. A. 1897-1902. B, 1897---1914,. C. 1897-1924. D. 1897--1934. Note: explanation applies to all four frames.
298
-
-;
-
.“--~-._.“”
300 ACKNOWLEDGEMENTS
The development of the was supported in part by G-408. We thank Robert C. comments and a meticulous
four-dimensional model described on this report U.S. Geological Survey Grant No. 14-08-0001Jachens and M. Darroll Wood for suggestions and review of the manuscript.
REFERENCES Castle, R.O., Church, J.P. and Elliott, M.R., 1976. Aseismic uplift in southern California. Science, 192: 251-253. VanZek, P., 1975. Vertical crustal movements in Nova Scotia as determined from scattered relevellings. Tectonophysics, 29: 183-189. VanrZek, P., 1976. Pattern of recent vertical crustal movements in Maritime Canada. Can. J. Earth Sci., 13: 661-667. Vanicek, P., and Christodulidis, D., 1974. A method for the evaluation of vertical crustal movement from scattered geodetic relevellings. Can. J. Earth Sci., 11: 605-610.
287 Amsterdam
-Printed
in The Netherlands
FOUR-DIMENSIONAL MODELING OF RECENT VERTICAL MOVEMENTS IN THE AREA OF THE SOUTHERN CALIFORNIA UPLIFT
PETR VANleEK,
1 MICHAEL
R. ELLIOTT
2 and ROBERT
0. CASTLE
2
1 Department of Surveying Engineering, University of New Brunswick, Fredericton, (Canada) 2 U.S. Geological Survey, Menlo Park, Calif. (U.S.A.) (Accepted
for publication
N.B.
April 26,1978)
ABSTRACT VanrfZek, P., Elliott, M.R. and Castle, R.O., 1979. Four-dimensional modeling of Recent vertical movements in the area of the southern California uplift. In: C.A. Whitten, R. Green and B.K. Meade (Editors), Recent Crustal Movements, 1977. Tectonophysics, 52: 287-300. This paper describes an analytical technique that utilizes scattered geodetic relevelings and tide-gauge records to portray Recent vertical crustal movements that may have been characterized by spasmodic changes in velocity, The technique is based on the fitting of a time-varying algebraic surface of prescribed degree to the geodetic data treated as tilt elements and to tide-gauge readings treated as point movements. Desired variations in time can be selected as any combination of powers of vertical movement velocity and episodic events. The state of the modeled vertical displacement can be shown for any number of dates for visual display. Statistical confidence limits of the mode’ted displacements, derived from the density of measurements in both space and time, line length, and accuracy of input data, are also provided. The capabilities of the technique are demonstrated on selected data from the region of the southern California uplift.
INTRODUCTION
Vertical crustal movements defined by the results of repeated levelings can be calculated and represented in a variety of ways. The conceptually simplest and least equivocal technique consists of differencing observed elevations obtained through successive surveys over the same route. Because the extension of this technique beyond single-line comparisons generally requires systematic releveling over a closed network, it is usually impossible to relate areawide groups of such comparisons to a single control point (such as a primary tide station) whose stability can be independently assessed. Alternatively, because releveled segments are commonly scattered in both space and time, areally dist~buted vertical velocities or displacement can be calculated from
288
geodetically measured tilts through the fitting of these data to a mathematically defined surface or set of surfaces, such as an algebraic polynomial of specified degree. The most obvious advantages afforded by this technique are that it permits the use of otherwise unusable data and provides an objective basis for characterizing variations in the velocity configuration through time. The chief disadvantage of any surface-fitting technique is inherent in the smoothing process that discards or subdues short-wavelength features of potential si~ific~ce. Earlier adaptations of surface-fitting techniques to the reconstruction of vertical crustal movements have been developed around a simplifying assumption of constant vertical velocity at every point, i.e., zero acceleration. The resultant velocity surfaces are probably most valid in areas such as Chesapeake Bay (VaniEek and Christodulidis, 1974) or the maritime provinces of Canada (VaniEek, 1975; 1976) where the assumption of constant velocity can be tested through velocity misclosures or a knowledge of the Holocene deformation. Spasmodic or episodic, aseismic cmstal movements of the sort recognized in southern California (Castle et al., 1976) have provoked the development of the modeling technique described here, a technique that permits the fitting of a time-varying surface of a prescribed degree. By specifying the power in time as equal to one and by omitting episodic movements, the described model automatically reduces to one characterized by constant velocity. The data set for testing this newly conceived fob-dimensional model has been selected from repeated levelings carried out within the area of the southern California uplift during the period 1897-1934. We have chosen this data set because: (1) It is large enough that it can be used to assess the logic of the mathematical formulation, yet sufficiently small that it could be put in computerreadable form in something less than a lifetime. (2) The prescribed period includes independently identified episodes of dramatically changing vertical velocity. (3) The results can be compared with those obtained through a classic reconstruction involving a network of closed loops. MATHEMATICAL FORMULATION
The uplift surface, u, is sought in the follow~g
u(x,
Y,
t) =
kSl +2 5 5 = COkl;i(t)
i=O
i+
= qqclj
j-0
j'# 0
+ X(x, y, t)c
km1
CijkXiY'Tk(t)
form:
289
where: Tk(t) = tk
k = 1, n,,
)
t< bk
,/O’ Tk(t)
= \(t
bk)/(ek - bk) , bk < t <
-
k
ek
=
np + 1 , np + n,
t> ek
1,
and c is the n-vector of unknown coefficients for the previously chosen values n,, n, of maximum power in x and y. In these formulas, x and y are local horizontal Cartesian coordinates calculated from latitude and longitude through the following transformation equations: Cijk
y = (X - Xo)R cos $0
x=($-&P;
(2)
where (&,, X,) is the centroid of all bench marks and R is the mean radius of the earth. Time, t, is reckoned from a stipulated date, to, for which u(x, y, t,,) is everywhere equal to zero. In addition, bk, ek, for k = np + 1, nP + ne, are the dates of the beginning and end of n, movement episodes so that nt = np + n,. We note that the episodic movements are treated as linear within the duration of the episode. Observation equations for m releveled segments can now be written: Ah(x1,
= ids i+j#
~1,
~2,
x2,
jam
key
t2)
-
Cij,(3cIyi2
Y,,
Ah&l,
-xiY’l)[‘k(‘Z)
~2, ~2, tl)
-
= 4x,,
‘k(‘l)I
y1, x2> ~29 tl,
“(X1,
YI,
3t2,
t2)
Y2, tl,
t2)
0
= WI,
YI,
3c2,
Y29
t1,
t21c
+
(3)
r
where d denotes the m-vector of the differences Ah, and r is the residual vector. If m > n = (n, en, +n, +n,h we can find the solution, c, through the following normal equations: (BTC,‘B)c
= BTC,‘d
of leveled height differences, (4)
the method
of least squares which yields
(5) where Cd is a diagonal covariance matrix of the releveled segments. The diagonal elements equal the variances of individual height difference differences. In the computer program the normal equations are solved through orthogonalization. Clearly, the shift coefficients, co, cannot be determined from the releveled segments alone. Uplift u*(x, y,t) of at least one, but generally ng tide-gauges (x, y), must be determined from sea-level records at nd dates to allow us to evaluate the shift coefficients. The following ng - nd observation equations
290
can be then formulated: u*(Xi, _Yi,tj) =
T(tj)Co
+
X(Xi, Yi,
tj)C
+ r*
3
i = 1 , n,; j = 1 , nd
(6)
If n, . nd > nt, then the above equations may be solved for co, again using the method of least squares. The minimum condition imposed here is: min T*T(CU* + Cxc)-‘Y*
(7)
CO
where C,* and Cx, are covariance matrices of U* and Xc respectively. Otherwise, everything in eq. 6 is known. The covariance matrix, C,, of the coefficients c can be evaluated from the well-known formula:
cc =-;2,: (BTcdB)-’ Hence the covariance c,,
matrix,
Cx,, is given as:
= XC,XT
(9)
The covariance
ng
’
nd
matrix,
-
nt
Co, of the shift coefficients
(P(C,*
co is obtained
from:
+ C,,)-‘T)-’
(10)
We note that for each tide gauge the uplift U* must be determined satisfy the following condition:
so as to (11)
u*(x, Y, to) = 0 Finally, placement,
the standard deviation, u(x, y, t), of the computed vertical U(X, y, t), may be evaluated from the following relation:
4x, Y, t) = ddiag[7’(t)Co~(t)
+ X(x,
Y, t)CcXT@,
Y,
t)l
dis-
(12)
COMPUTER PROGRAM
The computer program was designed to solve the problem described in the preceding section. It has the following additional features: (1) Coefficients in orthogonal solution space that are below the prescribed noise level are removed through filtering. The noise level is prescribed as h times the coefficient’s own standard deviation, where h is input. (2) Episodes may be specified at the user’s option; if they are not, then nt = n,. (3) The vertical displacement (eq. 1) and associated standard deviation (eq. 12) surfaces are computed within a minimal rectangle covering all the releveled segments, on a regular grid with specified steps in 4 and h. The surfaces are produced for n, specified dates, t = Ti, where i = 1, n,. If there is no ri specified, a default value, 7, = to + 100 years, is chosen automatically.
291
(4) The surfaces may be plotted at the user’s discretion. The scale of the plot is an input parameter. (5) A plot of the location of the releveled segments may also be requested. (6) If the vertical displacement and standard deviation surfaces are plotted, then both the contour interval and maximum and minimum values may also be specified. The scale in both x and y directions is determined by the program in relation to the specified steps in 4 and h (A$ and Ax). (7) Vertical displacements of the tide stations, u*, are treated as optional information. If they are not supplied, both the shift coefficients (eq. 6) and
Dotted
Input
Data
Cards
Optional
Deck
Fig. 1. Diagrammatic puter program.
representation
of input
data
deck
for the
four-dimensional
com-
292
their covariance matrix (eq. 10) are all set to zero. The uplift surfaces are then held to zero at the centroid (&, X,) of the bench marks utilized. (8) Weights (02;) or weight factors (Wi) are also to be supplied for releveled segments. Weights are computed from the standard formulas (see, e.g., VaniEek, 1976). If weight factors are used, they should be introduced as: 2 for two first-order, 5 for one first-order and one second-order, 8 for two second-order, 17 for one first-order and one third-order, 20 for one secondorder and one third-order, and 32 for two third-order levelings. These weight factors are automatically converted into the proper weights for each releveled segment, where it is assumed in the conversion that first-order accuracy is equal to 1 mm X d(distance in kilometers). (9) Segments with residuals larger than a prescribed rejection criterion are automatically eliminated. A default value for the rejection criterion may be chosen (as has been done in this example), so that no segment is rejected. Assembly of the input data deck is shown in Fig. 1. INPUT DATA
The data set used to test the computer program is based on a broad distribution of repeated surveys that incorporates every combination of leveling precision stipulated in the preceding section. While we have not included all of the data collected within this area (Fig. 2) during the period 18971934, incorporation of these additional, relatively recently unearthed segments probably would contribute little additional strength to the solutions, for their distribution closely matches much of the data already included. Examination of the data distribution shows that the chief deficiencies are associated with the concentration of these data in two general time frames (around the turn of the century and the early to middle 1920s) and the poor spatial control west of Ventura. Input commands and features common to each series of fitted surfaces are : (1) Computation of vertical displacements at 5’ spacings in $J and 6’ spacings in h. (2) Use of a plotting projection in which the angular unit in 4 equals the angular unit in h. * (3) Adopting of to= 1897. (4) Plotting of cumulative vertical displacements from 1897 to 1902, 1908,1914,1924,1928, and 1934. (5) Inclusion of sea-level changes at the San Pedro Tide Station equal to zero through the period 1897-1934, to which a standard deviation of 3 cm has been assigned. * The horizontal distortion implicit in this projection produces a lengthening in X proportional to set $ and derives from a deficiency in the plotting subroutines; nevertheless, this distortion has virtuatly no effect on our chief goal, which is to test utility of the model,
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Fig. 2. Genera3ized map of southern California showing distribution of releveled segments and San Pedro tide station. Only those segmenta actually used in testing the four-dimensional computer program are shown here.
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(6) Inclusion of a series of manufactured tilt segments obtained through differencing sea-level changes between San Pedro and San Diego, and weighted according to an estimated noise level of 3 cm (approximately equivalent to segments obtained through two successive second-order surveys). RESULTS
Because the solution depends on the degree of the fitted surface, which in turn affects the computer time needed, we have been constrained in the numbers of solutions attempted. On the other hand, preliminary classical reconstructions (as yet unpublished) provide a reasonable basis for selecting the degree of the fitted surface in both $J and X, and a somewhat poorer basis for selecting the power in t and the episodes. Experimentation with various powers in #, X, and t, and with the a-priori introduction of several episodes, suggests that the changing vertical displacement field within our area is reasonably approximated by a model with a maximum power of 2 in 4, 3 in h, 3 in t, and no episodes (Fig. 3). The four frames shown here (out of the six actually plotted) are clearly representative and provide a basis for comparison with our continuous profiling. Thus, the modeled solution indicates that the area contained within the lo-cm standard deviation contour sustained virtually no change in elevation between 1897 and 1902, other than modest uplift east of Los Angeles (Fig. 3A), a generalization that agrees with our conventional reconstruction. By 1914 uplift in excess of one standard deviation had propagated westward almost to Santa Barbara and had increased significantly east of Los Angeles (Fig. 3B). Comparison between the 1897-1924 and 1897-1914 plots (Figs. 3B and 3C) suggests a substantial increase in uplift between Los Angeles and the western edge of the map area between 1914 and 1924. On the other hand, continuous profiling between San Pedro and Bakersfield indicates that nearly all of this uplift occurred by 1914. The 1897-1934 displacement surface (Fig. 3D) suggests that most of the previously developed uplift collapsed and that the steep gradient along the north flank of the uplift shifted sharply southward during the period 1924-34. This interpretation, however, again conflicts with our conventional reconstruction, for we are reasonably certain that the .Los Angeles Basin experienced no more than a few centimeters of tectonic subsidence during this interval, The breakdown of the model implicit in the seeming continuation of uplift between 1914 and 1924 and the almost total collapse during the period 1924-34 almost certainly derives from the smooth interpolation of the movement in time as stipulated by a degree of 3 in time. Had we increased the power in t by one and added two episodes (1906-10 and 1924-28), uplift would have culminated by 1914 and the seemingly total coilapse between 1924 and 1934 would have been sharply diminished. Although the preceding evaluation of this clearly imperfect solution is
295
obviously subjective, the general validity of the solution is supported by the a-posteriori variance factor of 1.942 (compared with an assumed value of 1.000). Also only 12 coefficients out of 33 differ significantly from zero at the one standard deviation level. The a-posterior-i variance factor tWdr/ (m - n) assesses three effects that are inextricably entwined: (1) the selection of the model; (2) weighting of the data; (3) the data density and distribution in both space and time; hence it is impossible to say with certainty which of these three contributed most to the value of this quantity. Nevertheless, we strongly suspect that had the data been more evenly distributed both in space and time, the a-posteriori variance factor would have dropped substantially. In any event, to the extent that this factor is a measure of “goodness of fit”, the cited value supports the general validity of the 2, 3, 3 solution (VaniEek, 1975,1976). For comparison with the selected solution, we show a single frame (18971924) for the solution based on a m~imum power of 2 in @, 4 in h, 3 in t, and no episodes (Fig. 4). Differencing the two solutions through the subtraction of the vertical displacement field of Fig. 4 from that of Fig. 3C shows that, whereas the two displacement surfaces differ very little within the lo-cm standard deviation contour of either solution, they differ significantly away from the central part of the map area (Fig. 5). If it is accepted that the difference, within the region reasonably covered with data, between the two solutions is a variate with normal probability distribution, we might expect those areas in which the differences are greater than one standard deviation to cover about one-third of the map. This appears to be approximately the case here, Parenthetically, those areas in which the difference between the two solutions is greater than one standard deviation are generally coincident with the areas in which the 2, 3, 3 solution least closely accords with our conventional reconstruction. This seems to support our choice of the 2, 3,3 solution as relatively representative of the changing vertical displacement field within the area of our investigation. CONCLUSION
The results of the described tests of the four-dimensional modeling approach cannot be considered definitive. Given the restrictions imposed by our decision to limit the degree of the uplift surface, p~ticul~ly in time, and the use of only 352 segments, these results are, nonetheless, encouraging. Had we been able to obtain a few tens of additional segments randomly scattered through the western and northeastern parts of the area investigated, many and perhaps all of these solutions would have probably been improved significantly. Similarly, even a fragmentary knowledge of sea-level changes at Avila Beach (cf. Fig. 2) during the period 1897-1934 could have sharply constrained all of the attempted solutions. This tide gauge did not become a primary station until 1945.
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Fig. 3. Computed vertical displacement surfaces within the area of thd’southern Cailifornia uplift incorporating a power of 2 in latitude, a power of 3 in longitude, and a power of 3 in time. A. 1897-1902. B, 1897---1914,. C. 1897-1924. D. 1897--1934. Note: explanation applies to all four frames.
298
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300 ACKNOWLEDGEMENTS
The development of the was supported in part by G-408. We thank Robert C. comments and a meticulous
four-dimensional model described on this report U.S. Geological Survey Grant No. 14-08-0001Jachens and M. Darroll Wood for suggestions and review of the manuscript.
REFERENCES Castle, R.O., Church, J.P. and Elliott, M.R., 1976. Aseismic uplift in southern California. Science, 192: 251-253. VanZek, P., 1975. Vertical crustal movements in Nova Scotia as determined from scattered relevellings. Tectonophysics, 29: 183-189. VanrZek, P., 1976. Pattern of recent vertical crustal movements in Maritime Canada. Can. J. Earth Sci., 13: 661-667. Vanicek, P., and Christodulidis, D., 1974. A method for the evaluation of vertical crustal movement from scattered geodetic relevellings. Can. J. Earth Sci., 11: 605-610.