A modified form of the Lepping - Argentiero single spacecraft, shock normal determination procedure is presented. The modified method incorporates a simple predictor-corrector algorithm which allows a faster convergence rate and the use of average values of the parameters for the starting vector

Acuna, M. H.

Lepping, R. P.

English

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NASA
Technical Memorandum 86097
MODIFICATION TO SHOCK
FITTING PROGRAM
M. H. Acuna and R. P. Lepping
APRIL 1984
National Aeronautics and
Space Administration
Goddard Space Flight Center
Greenbelt, Maryland 20771
https://ntrs.nasa.gov/search.jsp?R=19840018285 2020-03-20T22:05:11+00:00Z
MODIFICATION TO SHOCK FITTING PROGRAM
M. H. Acuna and R. P. Lapping
Planetary Magnetospheres Branch
Laboratory for Extraterrestrial Physics
Greenbelt, Maryland 20771
April 1984
ABSTRACT
A modified form of the Lepping - Argentiero single spacecraft, shock
normal determination procedure is presented. The modified method incorporates
a simple predictor-corrector algorithm which allows a faster convergence rate
and the use of average values of the parameters for the starting vector.
INTRODUCTION
A technique and associated computer program developed by Lepping and
Argentiero (1971) to least-squares fit a sub-set of the Rankine-Hugoniot
equations to shock-related plasma and magnetic field data, is modified in this
note for the purpose of increasing the program's speed and efficiency and
making unnecessary the need for multiple starting conditions in the iterative
scheme. Originally the fitting scheme employed a standard Newton-Raphson
numerical iterative procedure to solve simultaneously a set of eight nonlinear
equations, denoted equations (36) in the original paper and equations (1)
below. The iterative technique is described by Deutsch (19.65). When
convergence did not result, one was required to repeat the procedure using, a
different "starting vector" (Z ), the eight components of which constituted
the independent plasma and field variables to be adjusted. The simple
modification suggested here, as well as making the use of more than one
starting vector unnecessary, in general speeds up convergence by eliminating
occasional large "overshoots" in the iterative procedure.
Formulation
**>
We denote 1 as the exact solution of
= 0 (i = 1, ...... 8) (1)
.
where
11 N(i) , .. '
L(2) = I l I. - (2)(
i=1 J=1 \ a,
with 2 = (X., X_, ... XQ), and where Y.(lj) (i = 1, .. 11; j = 1 .. N(i)) are
I C 0 1
2
the input data for the eleven physical quantities of interest, N ( i ) data
points each, usually arranged in the following order:
B 1x« V 31z' B2x' V B22' V2y ' V V2z ' V1z» Nr N2' and V2x ' V1x
(i.e., components of the magnetic field, plasma velocity differences and
densities, where the subscripts 1 and 2 represent pre-shock and post-shock
states, respectively), and.
ai (i = 1, .. 11)
are the "sigma noise parameters" used as weights in the least squares
procedure and usually composed principally of the rms deviations of the
measured physical quantities of interest (Y.'s), and finally
Xt (i = 1, ... 8)
are the independent variables (components of Z) to be solved for, physically
representing the first eight quantities of Y^^. Note that Xg (2^ ), XIQ (=N2)
and Xn (= 2^x - V1x) are related to Xi (i = l, ... 8) by the three lowest
order Rankine-Hugoniot equations, which play the key role of providing a
constraint on the nonlinear least-squares process [see equations (18), (19),
and (20) in Lepping and Argentiero (1971)].
We denote Z (n = 1, n ) as an estimate of Z at iteration step n.fj m 9 A
The components of Z , the starting vector, are commonly the averages of the
first eight components, Y.. Then at step n
When a "best estimate" of Z is obtained, the magnetic field components, i.e.,
the first six components of that estimate of Z, are used in the magnetic
coplanarity formula (Colburn and Sonett, 1966) to obtain a "best estimate" of
the shock normal.
The original computer program is described in the Appendix of Lepping and
Argentiero (1970).
Modification of Scheme
At each step n the quantity L, given by equation (2) and referred to as
the least squares "loss function", can be evaluated. Our basic purpose is to
minimize L(Z), as in equation (1). In order to avoid divergence of \&Zn\ and
L(2), we make a slight modification of the iteration procedure by allowing a
choice of two branches in the fitting program, which are defined by the
following:
First branch - If at step n
-
L < L , , then
n n— i
as in the original scheme.
Second branch - If at step n
-
L > L , , then
n n— 1
=
 (Ln-1/Ln>
This continues step by step until n equals a pre-chosen integer, M say, or
until
( U 2 n l / | 2 n _ . , | ) < e (6)
for some sufficiently small e > 0. Notice that the second branch differs from
the unmodified first branch only in that the length of each step along
AZ (unmodified) is shortened according to the ratio of the new to the old loss
function.
We have found in numerous cases that this simple modification has
decreased, and sometimes significantly, the time needed to run the f i t t ing
program and/or the operator 's efforts in finding a proper starting vector. In
extreme cases, for a given t , the original program would occasionally wander
around the neighborhood of L , with there being no apparent hope of
satisfying (6) for a reasonable t, especially when the gradient of L(Z) was
shallow near L . , i.e., for poorly conditioned cases. This modification
should eliminate that problem. Although not a unique solution to the
convergence problem, this type of modification is attractively simple.
Example
Here we show the benefit of using the modified program by comparing its
results with those of the unmodified program for a somewhat poorly behaved
case; we have encountered much worse cases but they are not typical and
therefore not good examples.
We examine a shock observed on Pioneer 6 at 2058 U.T . on March 22, 1966,
which was first studied by Chao (1970); he graphically displays and discusses
the data, which are from the Goddard magnetometer and the MIT plasma probe
onboard the spacecraft. The right side ("DATA") of Table 1 shows the
tabulated data points constituting 12.5 minutes of upstream field data, 14
minutes of downstream field data (partially decimated), and ±22.5 minutes of
all plasma quantities. The analysis intervals were chosen on the basis of
apparent steadiness of the data. All data are given in an R-T-N coordinate
system, centered at the spacecraft, where ft is positive radially away from the
sun, T is perpendicular to ft , parallel to the ecliptic plane and positive in
the direction of the earth's motion about the sun, and ft = ft x T. On the left
side of the table are best estimate ( B . E . ) values from both the modified and
unmodified programs, straight averages ( A V G ) of the quantities, and the
associated sigma noise parameters (SIG), which in this case were simply rms
deviations of the data values. The modified program converged to the 3.E.
value in 3 iterations where the A V G ' s were used for the components of the
starting vector. By contrast the unmodified program required 11 steps when
again the AVG starting vector was used. In the unmodified program the loss
function L had the undersirable feature of oscillating in value, as shown in
Table 2; the table also shows the monotonically decreasing values of L that
the modified program produced, as expected. The table also gives a related
quality factor defined below the table. We stress the case where A V G ' s were
used for the starting vector, Zo, since that is obviously the easiest and most
commonly used choice, as stated above. However, for this shock several other
apparently reasonable 2 fs were attempted using the unmodified version of the
program, and some did not give convergence after 15 interation steps; the
program was set to stop at the 15th step.
It must also be emphasized that the selection of an optimum fitting
coordinate reference frame is important in assuring proper and speedy
convergence. The implementation of the modified and unmodified procedures
have included interactive facilities to conveniently rotate the data into one
of three orthogonal alignments such as to assure the optimum selection of
dependent and independent variables in the fitting procedure. In general, the
quantities exhibiting the greatest variability or uncertainty should be
selected as the dependent variables to be least-squares estimated.
The best estimate shock normal, based on the B.E. values in Table 1,
is
fig
 £> = <n B , nr nM) = (0.94, -0.15, 0.31 ),
and the associated 95* certainty error cone half-angle was 7.7° (Lapping and
Argentiero, 1971). [If AVG value fields had been used, the error cone
half-angle would have been 3.3 times larger!]. By comparison Chao's estimate
of the shock normal was
nCHAO = (0'84' •°'10' °'54)*
differing by 13.6° from our estimated direction. The shock was a rather
typical oblique one (at 1 AU), whose B.E. normal was 59° from the upstream
field direction § and whose magnitude ratio across the shock was
We stress how different the velocity difference vector W = ( W R , WT, WN)
is between the B.E. and AVG values, especially in the WN component, as Table 1
shows. The angle between Wg£ and W is 17.1°. If one were to use "velocity
coplanarity" (Abrahara-Shrauner, 1972) and AVG values, the resulting shock
normal would be
= (0
'
69
' -°-
20
- °-
70)
'
which differs by 26.4° from our best estimate normal; the author warns that
this method is an approximation. Our added warning is that using average
values is often inadequate. For example, we see that by using the B.E. W and
velocity coplanarity we obtain
nvel S'^BE^BE1 = (°*86' ~°'23' ° 'U 6 ) '
which is only 9.8° from n,,.,. Obviously in all of the above we have made the
DC.
tacit assumption, argued in Lepping and Argentiero (1971), that fL_ is indeed
generally the best estimated shock normal from data from a single spacecraft.
Many such examples can be found to show the desirability of using the
modified version of the 'best-fit1 technique.
Acknowledgement
We thank K. Behannon for reading and commenting on this note.
References
Abraham-Shrauner, B., Determination of magnetohydrodynamic shock normals, J.
Geophys. Res . , 77, 1972.
it.
Chao, J.-K., Interplanetary collisionless shock waves, Ph.D. Thesis, M.I.T.
Cambridge, MA, CSR-TR-70-3, February 1970.
Colburn, D. C. and C. P. Sonett, Discontinuities in the solar wind, Space Sci.
Rev.,, i, 439, 1966.
Deutsch, R., Estimation theory, p.70, Prentice-Hall, Englewood Cliffs, NJ,
1965.
Lepping, R. P. and P. D. Argentiero, Improved shock normals obtained from
combined magnetic field and plasma data from a single spacecraft, NASA
X-692-70-276, NASA, 1970.
Lepping, R. P. and P. D. Argentiero, Single spacecraft method of estimating
. shock normals, J. Geophys. Res., 76, 4349, 1971.
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TABLE 2
Loss Function Values
Iteration Step
0 {Starting vector
1
2
3
4
5
6
7
8
9
10
11
L-unmodified (Q*)
1,095
19,360
2,985
2,749
7,336
805
5,013
1,156
1,639
127
139
108
(.295)
(.070)
(.178)
(.186)
(.114)
( .344)
(.138)
( . 287 )
(.241)
( . 865 )
( . 826 )
(.938)
L-modified (Q*)
455 (.457)
174 (.739)
111 ( . 925 )
108 (.938)
*Q = Quality = (NT /L) , where N., is the total number of data points, 95 in
this case. Either two sucessive Q's of 0.85 or larger, or Q of 0.90 or
larger, is usually considered a successful convergence, provided reasonable
"sigraa parameters" were used.
BIBLIOGRAPHIC DATA SHEET
1. Report No.
TM 86097
2. Government Accession No. 3. Recipient's Catalog No.
4. Title and Subtitle
Modification to Shock Fitting Program
5. Report Date
Q. Performing Organization Code
695
7. Author(s)
M. H. Acuna, R. P. Lapping
8. Performing Organization Report No.
9. Performing Organization Name and Address
Planetary Magnetospheres Branch
Laboratory for Extraterrestrial Physics
Goddard Space Flight Center
Greenbelt, Maryland 20771
10. Work Unit No.
11. Contract or Grant No.
13. Type of Report and Period Covered
12. Sponsoring Agency Name and Address
14. Sponsoring Agency Code
15. Supplementary Notes
16. Abstract
A modified form of the Lepping - Argentiero single spacecraft, shock
normal determination procedure is presented. The modified method
incorporates a simple predictor-corrector algorithm which allows a faster
convergence rate and the use of average values of the parameters for the
starting vector.
17. Key Words (Selected by Author(s)) 18. Distribution Statement
19. Security Classif. (of this report)
Unclassfied
20. Security Classif. (of this page)
Un r* 1.
21. No. of Pages
. 11
22. Price'
"For sale bv "ft? National Technical Ir.-crrnajion Service, Sanngiieid. Virginia GSFC 25-ui ( i o / y / i


Modification to shock fitting program

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