A method of processing data provided by a gradient ... - Springer Link

Phys. Oceanogr., Vol. 4, No. 4, pp.283--290 (1993)

9 vsP ~993.

A method of processing data provided by a gradient-distributed temperature sensor* V. V. FOMIN Abstract - - This paper describes an algorithm for processing data provided by a gradient-distributed temperature sensor, based on the regularization method, to estimate internal wave parameters in the thermocline. Measurements of the average temperature in the layer, temperature data at its boundaries, as well as the vertical temperature profile at the initial moment of time serve as input data. Data on the error-level of parameter measurements and an a priori assumption as to the monotonic declining of temperature with depth is applied. Examples of computations are given illustrating the validity of the algorithm.

To measure internal wave parameters in the thermocline, measuring systems featuring the gradient-distributed temperature sensor are being extensively used by oceanographers [1-3]. Treatment of the compiled data requires calculation of the weight-average vertical velocity values, as a result of which the thermocline's vertical displacement amplitudes, averaged over depth, are determined. To extract more information from the /n s/tu experiments on evaluation of the internal wave parameters, a numerical algorithm based on the regularization technique is described for processing temperature data compiled by the gradient-distributed temperature sensor. With some restrictions being imposed, the algorithm makes possible calculations of isotherm position within the observed layer. Consider the problem of determining the temperature T(z, t), and vertical velocity, w(z, t), fields within the layer z C [h(t) - l, h(t)] at the time interval t E [0, to] using the equation

OT

OT

0--/ + w-ff~z = O,

(1)

T(z, O) = T.(z) ,

(2)

by the initial condition by the temperature data at the upper and lower boundaries of the layer (measurements at one point) T(h(t) - l, t) = TB(t), T(h(t), t) = Tg(t), (3) and by the layer's mean temperature (indications of the distributed temperature sensors)

To(t) = l

1

h(O f T(z,t)dz.

(4)

h(t)-l

In equations (1)-(4), the z-axis is directed vertically downward; l is a constant and is the sensor's length, and h(t) is the known lower depth of the sensor determined by a pressure gauge. *Translated by V. Puchkin. UDK 551.466.

284

V. V. Fomin

Substitute the independent variables z' = (z - h(t) + l)/l ,

t' = t / t o ,

z', t' 9 [0, 1].

(5)

With (5) considered, we derive the following relations (for simplicity, primes will be discarded) OT

OT

O---t-+ w, 0---~ = O, T(1,t) = TH(t),

T(O,t) = TB(t),

(6) T(0, z) = T.(z),

(7)

1

To(t) = fT(z,t) dz,

(8)

o

w.

= to(w

- ~dh) / t .

(9)

The functions of (6)-(8) will be presented as a sum of two terms T ( z , t) = To(z, t) + Tl(Z, t),

w . ( z , t) = wo(z, t) + wl(z, t),

10To

+ TB(t),

To(z,t) = - A ( t ) z

wo(z,t) - A(t) Oz '

A(t) = TB(t) - T n ( t ) .

(10) (11) (12)

Here, To is the linear approximation of the vertical temperature profile within the layer housing the sensor, and wo is the vertical velocity derived by introducing To into (6). Introduce a new unknown quantity ~(z, t) ~(z,t) w l ( z , t ) = A ( A - OT,/Oz) "

(13)

Upon introducing (10)-(13) into (6)-(8), we have a boun~tary value problem for Tl(z, t) OT1

OT1

O---t- + wo Oz

k~

A

(14)

Tl(z, 0) = T,(z) - To(z, 0) = fo(z),

(15)

7'1(0, t) = 711(1,t) = 0.

(16)

Relation (8) then assumes the form 1

(17) o

Multiply (14) by A(t) and integrate by parts over the segment z E [0, 1]. With (16) and (17) taken into account, we derive an integrated relation 1

f a y ( z , t) dz - d AdtB o

- f(t).

(18)

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285

As follows from (14) and (16), at the layer boundaries ~(z, t) and Tl(z, t) are related by the conditions k~(0, t) = q(0, t), ~(1, t) = q(1, t), (19) where q(z, t) = awo~

.

(20)

With respect to the unknown quantities Ta and ~, relations (14)-(20) represent a linear system of equations. The algorithm for temperature data processing involves numerical resolving of this system of equations. Assume that 9 and T1 and the input data (TB, Tc, TH, and T.) are the restricted, sufficiently smooth functions within the domain, where they are being determined, and at any z, t E [0, 1], the inequalities below are realized for them TH < To + TI < TB,

(21)

0T1 .... < A

(22)

Oz

= TB- T~.

Conditions (21) and (22) agree with the assumption of the monotonic decrease of temperature with depth, required for w l to be restricted. Upon simple transformation considering (11), we obtain an inequality from (21): -A(1 - z) < 711 < Az. Its integration over the segment z E [0, 1] and applying of (17) leads to a quantitative restricting of the measured parameters 1 [B[ < ~ a , (23) this being equivalent to realization of the condition TH < Tc < TB.

Now let us construct a numerical algorithm to resolve the problem (14)-(20). A regular grid will be applied to a segment of the t-axis: tk = k A t , k = 0 , . . . , M , and A t M = 1. Approximate (14)-(16) and (18)-(20) by the system 2x M of one-dimensional equations cure sole the function ~ok(z) = Tl(z, tk), ~ok(z) = ~(z, tk): ~ok -- ~ok-1 + wk

At

1 k d z - - ~ q~ '

(24)

qo~ = f0(z),

qok(0) = qO~(1) = 0,

(25)

1

Ek~ k

= f

k~l'(z) dz = 1 It=t~ - ~ A B ~=tk-1 = f k ,

(26)

0

@~(0) = qk(0),

@~(1) = qk(1).

(27)

With k being fixed, equation (26) represents an integral equation of the first kind, whose solution is sensitive to minor modifications of the right-hand part of the equation. For such equations to be resolved, regularizing algorithms are applied. In keeping with reference [4], an approximated solution to (26) will imply a sufficiently smooth function, which is an extreme of Tikhonov's smoothing functional d~k 2 4)(k0k) = [[Ek0k fk[[+ (28)

c~k(poll~[12+pl-~z [)'

286

v. v. Fomin

where ak is the regularization parameter, and pj > 0 are the weighting multipliers. With the prescribed ak > 0, equation (26) proves to be strongly concave and its minimum is attained at a singular element k~ satisfying the Eulerian integral-differential equation 1

f~(z)dz + ak[po~(s) - Pl~Jd~k(s)] : fk

(29)

0

and the boundary conditions

o/~(0) = q~(O),

o/~(1) = q~(1).

(30)

The problem (24), (25), (29), and (30) must be complemented with the criterion of choosing ak. Function ~k(z) has been assumed monotonic. Therefore, the minimum ak, providing for the temperature profile's being monotonic at the moment of time t = tk, may be adopted as an optimal value of the regularization parameter d~ k d---~-< Ak'

z 9 [0, 11 .

(31)

Ultimately, the solution results in the determination of 3 x M unknown quantities: qok, 9 k, and ak from equations (24), (25), (29), and (30). With k being fixed, the problem is resolved through the method of subsequent approximations. The integral in equation (29) is replaced by the rectangular square formula. Equation (24) is approximated by the three-point difference scheme, resolved using the running technique. The regularization parameter is determined from condition (31) by assessing solutions with diverse a. The inequality, given below, provides a criterion for terminating the iterations at each timestep 1 (32)

fo ~k(z) dz - B k < 8 B ,

where 8B is the error in the value of B in (17), evaluated by the known errors of parameters measured by the gradient-distributed sensor. We will illustrate the validity of the algorithm. The measured parameters and the initial temperature profile are assumed to have the form TB = 21.5 + a(t), Tc =

where

T. = - A z

,(t)(~/6 + 1),

r(t) = (TB + TH)/2,

+ TB(O) + er(O)b(z),

Tn =

17.5 + a(t),

b(z) = (1 - z ) z ,

a(t) = 0.5 sin27rt sin47rt,

e = 0.03,

A = 4~

Then an exact solution to the problem (6)-(8) is provided by the relations T(z,t) = -Az

+ TB + erb,

w . ( z , t ) = o-~(TB - A z + erb)(A - e r ~ )

Isotherms corresponding to (33) axe designated in Fig. 1 by solid curves.

(33) (34)

Processingdata 0

287

!L/ /x,2/x

o.s

7.0

Figure 1. The isotherm field for the model problem (solid carves correspondto the accurate solution, and dots to the approximatedsolution at 61 = 0.2"C, 62 = 6a = 0.05*(]). We will attempt to assess the accuracy of the suggested methods and to examine the effect of the random noise in the input data on the error of calculating the isotherm position, using a model example. The problem will be handled with application of a grid having 100 grid points over t and 40 grid points over z. The occurrence of random noise 'mlitating measurement error will be modelled through substituting To, TB, and TH by To = To + 2~1(~1 - 0.5), TB = TB + 2~2(~2 - 0.5), and 7~H = TH + 2~3(~3 0.5), where 6j is indicative of the noise level, are the realizations of a random quantity with a uniform distribution of (~i E [0, 1]). The error of isotherm computing is estimated using the functional 6/~(T) = max [#(T) - #(~P)[, -

-

~e[o,2]

where #(T) and #(T) are the depths of the isotherms, derived by the accurate and approximated solution of equations (6)-(8). Figure 2 shows the dependences of 6# on T, corresponding to equations (33), (34), and data listed in Fig. 2 for several values of 61. For obviousness, the calculated data are submitted in a dimensional form (the gradient-distributed temperature sensor's length l -- 50 m). It is seen that the error of isotherm position reconstruction increases with the input data noise level. With 6i = 0 (curve 1), the value of 6# does not exceed 0.35 m (0.09l). This value of 6#, with the approximation indicated above and chosen grid parameters, characterizes the method's accuracy. It follows from the analysis of the behaviour of curves 2-7 depicting the calculated effect of 62 and 63 on 6# at 61 = 0.2"C that the error of calculation of the isotherm position is intimately associated with the error of prescribing parameters TB and TH (an insignificant increase of 62 and 63 from 0.05 to 0.09~ leads to the rising of 6# from 0.5-0.8 to 3-4 m, i.e. by a factor of 4-5). The noise in prescribing parameter To has a weaker effect on 6#. As the calculations (curves 2, 8-10) demonstrate, a seven-fold increase of 61 (0.2-1.5"C) at 62 = 63 = 0.05~ involves the growth of 6# merely by a factor of 3. Intercomparison of the accurate solution (32) with the approximated one shows that at b2 < 0.5~ an 62, 63 < 0.05~ there is a good

V. V. Fomin

288

(r)' L,''

7

s

o /

17.5

5"

;8 1~.S 1'9 liS 20 20.5 2iT,~C

Figure 2. Dependences of the error of isotherm position reconstruction in the model problem on the level of error of input information 6/.

qualitative agreement between them. This is confirmed by the calculation of isotherm depths for 81 = 0.2~ and 62 = 83 = 0.05~ (dots in Fig. 1). The present technique of treating temperature data was successfully applied to handle in situ data collected by the MGI-4206 probe SHLEIF-S having the following error of measured parameters [2]: 81 = 0.2~ 82 = 83 = 0.050C. In accord with the indicated 81, the value of error in the right-hand part of equation (13) was assumed to be equal to 0.25~ As the temperature profile measurement in the thermocline at the initial moment of time were unavailable, function fo(z) in (15) was approximated by the second degree polynom, whose coefficients were determined by the measured parameter values at t = 0. One of the examples of calculation of the field of vertical shifts in the thermocline, as observed by the 50 m long TDS in Guinea shelf waters (Cruise 48 of the R / V Mikhail Lomonosov) is illustrated in Fig. 3. The temperature field in Fig. 3b may be interpreted as a packet of short-period, low-mode internal waves with a ,-~ 10 m amplitude. The calculations point out to the possibility of using the numerical procedure to estimate internal wave amplitudes from GDS data that may raise the information content of the experiments without utilizing complementary sensors. An essential point in the suggested method involves an a priori assumption on the temperature profile's being monotonic within the thermocline and on one's being aware of the errors in the parameter measurements. The issue of the method's perspectiveness may be resolved only after the synchronously-obtained data, including GDTS measurements and the data on instantaneous temperature profile distribution in the thermocline will have been processed.

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289

Table 1.

Relationship between the curves and the values of 61 (0 C) Curve number,

F'~. 2

~

~2

1 2 3 4 5 6 7 8 9 10

0 0.2 0.2 0.2 0.2 0,2 0.2 0.7 1.0 1.5

0 0.05 0.07 0.075 0.08 0.09 0.10 0.05 0,O5 0.05

T*C 23

19 17 15

'

SO

~

t

q

v 'q rO ~, " 1

-,og 0

.,

ta

'

t'~ .

I~

[S si

/I

c Jt p

~JI 0

V

,

15

50

, #5

t,mr

c

Figure 3. Vertical displacements of the thermodine calculated from in situ data: (a) GDTS measurements; (b) the vertical displacement field recovered from solution of the problem (24)-(32); and (c) thermodine vertical displacements provided by the method discussed in ref. [2].

290

V. V. Fomin

REFERENCES

1. Ivanov, V.A., Kogan, B.S., Lisichenok, A.D. eta/. A towed complex for internal wave spatial parameter measurements. Okeanologia (1989) 29, 675-679. 2. Kuznetsov, A.S., Paramonov, A.N. and Stepanyants, Yu.A. Study of solitary internal waves in the tropical west Atlantic./zv. A/cad. Nauk SSSR, Fiz. Atmos. Oceana (1984) 20, 975-983. 3. Sabinin, K.D. High-frequency internal wave spectra in the equatorial Indian ocean. Okeanologia (1982) 22, 909-915. 4. Goncharsky, A.V., Cherepashcliuk, A.M. and Yagola, A.G. Numedcal Methods of Solving Inverse Astrophysical Problems. Moscow: Nauka (1978).