Gravity Measurements at Sea by Use of the T.S.S.G. Part 1. Data Processing Method of the T.S.S.G.

Transcription

1 JOURNAL OF PHYSICS OF THE EARTH, Vol. 18, No. 1, Gravity Measurements at Sea by Use of the T.S.S.G. Part 1. Data Processing Method of the T.S.S.G. By Jiro SEGAWA Ocean Research Institute, University of Tokyo Abstract Gravity was measured at sea by use of the T.S.S.G. surface ship gravity meter. The mechanism of the T.S.S.G. and the method of data processings have already been dealt with in papers by Tomoda et al., (1962, 1968), but since then improvement has been made in both respects. In this paper methods of data processings are described with special emphasis on the 2nd order correction for finite sampling intervals and also on the effect of elasticity on the dynamic gravity meters. Effects of horizontal accelerations are also discussed in relation to the T.S.S.G., which have mattered with Vening-Meinesz submarine gravity meters, La Coste-Romberg gyrostabilized shipboard gravity meters or Graf-Askania surface ship gravity meters. The T.S.S.G. surface ship gravity meter was completed in 1961 and tested on board ship with success (C. Tsuboi, Y. Tomoda and H. Kanamori 1961, Y. Tomoda and H. Kanamori 1962). The instrument has been hitherto improved and used for world-wide measurements by Tomoda and his colleagues (Y. Tomoda, K. Ozawa and J. Segawa 1968, Y. Tomoda and J. Segawa 1966, J. Segawa 1967, Y. Tomoda 1968, Y. Tomoda 1967, A. Tokuhiro 1967). The T.S.S.G. is characterized by the sensor; a dynamic gravity meter of a string type with high vibration frequency. In this type of dynamic gravity meters the merits lie in their wide dynamic range, high sensitivity and in the convenience of direct connection to the digital processing. In the T.S.S.G. the frequency of chord vibration of a string modulated by the variation of vertical accelerations is digitally recorded. This is what has been kept invariable till now (1969) and will be hereafter. However several other points have been improved. Methods of data processing in the measurements by use of the T.S.S.G. have also been improved since Between 1961 and 1969 more than twenty survey cruises were performed and methods of data processing applied to each of the cruises are somewhat different with one another. So, in the following are listed those points in which different methods have been adopted.

2 20 Jiro SEGAWA cruise was finished. Processings of the data by use of an electronic computer were made on land afterwards. Sometimes the measurements proved to be in perfect failure because of unexpected accidents that occured but were not noticed during the measurements. data processing method in the previous list. The accidents came more often from punching The symbols are arranged from left to right in the order of items appeared in the previous hibitive to find out a single bit error among list. numerous number of the punched holes. In 1966 a small electronic computer was attached to the T.S.S.G. and used on the ship for real time processings though it was not enough in the accuracy. This computer had been originally designed for a commercial register, 2) July 31st-August 17th, 1961 Takuyo, Hydrographic office, Maritime Safety Agency West of Izu Islands-Shikoku Basin processings performed by the computer were A, A, A, A, A, A, in a somewhat simplified way on account of 3) October 30th, 1963-February 17th, 1964 small capacity of the computer. Umitaka-maru, Tokyo University of Fisheries (1450 tons) Indian Ocean-South China Sea A, A, A, A, A, A, 4) October 22nd, 1964-March 13th, 1965 Umitaka-maru, Tokyo University of Fisheries Southern Sea A, A, A, A, A, A, 5) November 1st, 1965-February 12th, Umitaka-maru, Tokyo University of ations caused by a ship's motion, corrections Fisheries for the 2nd order effect and others, which North Pacific Ocean are processed on real time basis. Eotvos A, A, A, A, A, A, correction, Bouguer correction and gravity 6) July 11th-August 25th, 1966 anomalies are calculated a little later than Umitaka-maru, Tokyo University of the former calculations, because these calcu- Fisheries lations need informations about the position and velocity of the ship and the water depth at the position. Automatic acquisition of those data is a problem which should be solved in the near future. The T.S.S.G. gravity meter is now being used by the Ocean Research Institute, University of Tokyo and the Hydrographic Office, Maritime Safety Agency. The survey cruises during which gravities were measured in the period between 1961 and 1969 are listed below, where the period of survey, the ship used, the area measured and the method of data processings are indicated. The alphabetical symbols A, B or C indicated in the following list correspond to those used to classify the 1) July 10th-13th, 1961 Takuyo, Hydrographic office, Maritime Safety Agency (770 tons) The Bay of Tokyo-Tateyama-Miyakejima Island A, A, A, A, A, A, the Sea near and around Japan A, A, A, A, A, A, 7) October 15th, 1966-March 3rd, 1967 Umitaka-maru, Tokyo University of Fisheries Southern Sea all around the Antarctica AB, AB, AB, AB, AB, A, 8) July 11th-August 25th, 1967 Umitaka-maru, Tokyo University of Fisheries the Sea near and around Japan

3 Gravity Measurements at Sea by Use of the T.S.S.G. Part B, B, B, B, B, B, 9) July 15th-August 18th, 1967 Hakuho-maru, Ocean Research Institute, University of Tokyo (3200 tons) Japan Trench-Kuril Trench A, A, A, A, A, A, 10) November 1st, 1967-February 23rd, 1968 Umitaka-maru, Tokyo University of Fisheries West Pacific Ocean-Coral Sea-Tasman Sea B, B, B, B, B, B, 11) December 1st, 1967-February 18th, 1968 Hakuho-maru, Ocean Research Institute, University of Tokyo West Pacific Ocean A, A, A, A, A, A, 12) July 14th-August 16th, 1968 Hakuho-maru, Ocean Research Institute, University of Tokyo Northwest Pacific Ocean-Emperor Seamount A, A, B, A, A, B, 13) November 14th, 1968-March 3rd, 1969 Hakuho-maru, Ocean Research Institute, University of Tokyo North Pacific Ocean-South Pacific Ocean-Southern Sea A, B, B, A, A, B, 14) April 26th-June 17th, 1969 Hakuho-maru, Ocean Research Institute, University of Tokyo Japan Trench-Japan Sea A, B, B, A, C, B, 15) May, 1965 Takuyo, Hydrographic office, Maritime Safety Agency, Japan Trench A, A, A, A, A, A, 16) May, 1966 Takuyo, Hydrographic office, Maritime Safety Agency Japan Sea, Yamato Bank A, A, A, A, A, A, 17) May, 1967 Takuyo, Hydrographic office, Maritime Safety Agency The sea near and around Japan AB, AB, AB, AB, AB, A, 18) June, 1968 Meiyo, Hydrographic office, Maritime Safety Agency (350 tons) Japan Trench-Continental shelf south of Hokkaido A, A, A, A, C, B, 19) October-November, 1968 Meiyo, Hydrographic office, Maritime Safety Agency Continental shelf in the Japan Sea A, A, A, A, C, B. In the cruises (1)-(6), (9), (11) and (15)-(19) the outputs from the gravity meters were all punched on perforating paper tapes. In (5) and (10) they were connected directly to a small computer on the ship and in (7) and (17) both paper punchings and on-line processings were used. In (12) the data once punched on tapes were immediately processed by use of a high speed computer equipped on the vessel. It was a transient method to be followed in (13) by an on-line real time processing. In (14) a comparison measurement was made on real time basis between two sets of T.S.S.G. with the same computer by making use of the function of process control. 2-1 Progress of the method in Data.Processing (1) Introduction to data processings In the T.S.S.G. the frequency or period of a dynamic gravity meter is measured. In order to record the vibration of a string digitally samplings of data with a certain sampling interval are necessary and therefore the fundamental data are given in an intermittent form. Nevertheless, for the sake of simplicity, we will at first formulate the relation between an output of the dynamic gravity meter and external accelerations as a continuous, time-dependent function, regardless how it is measured. Let the frequency and period of a dynamic gravity meter be defined as f(t) and T(t)= 1/f(t) respectively, as a function of time t, and vertical accelerations in the ship acting

4 22 Jiro SEGAWA second period caused by the ship's motion. The long period component includes the variation of earth's gravity as well as the variation of apparent gravity due to Eotvos effects or long period horizontal accelerations. Then the relation of the external accelerations to the frequency or period of the dynamic gravity meter is, in an ideal case, or 2-1 (1) providing a string used in the dynamic gravity meter be perfectly flexible and tensed exclusively by the external accelerations. The constant K is determined by comparing the frequency of the dynamic gravity meter with the gravity value at a calibration station. In the actual gravity meter the effect of elasticity intrinsically contained in the physical property of the string and the effect of tension caused by cross springs (ref. 2-2, 2-3) can not be neglected. The above relation thus leads to a more suitable relation or 2-1 (2) where K1, K2 and K3 are constants. In the beginning of the data processings of T.S.S.G. the second and third terms were neglected, but detailed investigations of measured gravity values have led to a conclusion that it was a wrong treatment, and the old data are now being checked and recalculated. As will be described in 2-2 and 2-3, when we treat the second and third terms as small corrections to the values obtained only from the first term, the second term plays mainly as if to increase the 2nd order correction co-efficient and the third term to modify the constant of proportion in the first term. In order to remove the vertical acceleration of ship's motion a digital low pass filter is applied to the measured accelerations by use of a weighted mean method. If we assume a weight function W(t) as a function of time and make a weighted mean as 2-1 (3) then power spectra of G(t') are equal to the product of those involved in vertical accelerations and those involved in the weight function. In order to remove wave accelerations a weight function W(t) is selected to be very small in the spectrum in which ship's motion predominates, as compared with the intensity in the spectrum of longer period. Intermittent samplings of the frequency or period of a dynamic gravity meter produces a decrease of averaged accelerations due to a non-linear relation shown in 2-1(1). Correction of this non-linear effect is called a 2nd order correction for finite sampling intervals. Corrections for temperature variation and for shorter vibrations of 2 to 100Hz are sometimes necessary particularly in the T.S.S.G. and are made according to the circumstances. Eotvos correction, Bouguer correction (or water correction), correction for drifts of the meter and calculation of gravity anomalies are similar to those with other gravity meters. (2) Actual methods of data processings The method of data processings in the T.S.S.G. has come through three stages of progress since In the first stage the period of a dynamic gravity meter is measured by electronic counters and recorded on perforating paper tapes. By the electronic counter is measured the time which it takes for a certain number of vibration to occur in the dynamic gravity meter. Let the number of vibration be a constant N0, and the time measured be Si. Then Si/N0 is an averaged period Ti for the i-th sampling interval Si. So the sampling interval in period measurements is variable and proportional to the very measured period. The period measurement is made successively so that it does not break as long as approximately 8-10 minutes with the aid of two electronic counters used interchangingly. Duration of the measurement is so determined that it may not be shorter than the spread of the weight function for low pass filtering

5 Gravity Measurements at Sea by Use of the T.S.S.G. Part and not be longer than the time in which the earth's gravity varies little when in that interval the ship proceeds at normal speed. A representative value of gravity is determined, for convenience, in this interval. Here we are going to describe the actual method of data processings executed by an electronic computer on the basis of the records punched on perforating paper tapes. The formulation is made according to the ideal relation indicated in equation 2-1 (1), and deviation from it will be discussed in other sections later. Now the program followed by the electronic computer is to apply the weighted mean expressed in equation 2-1 (3) to intermittent data. If the equation 2-1 (3) is rewritten in terms of T(t) we get 2-1 (4) The counters are gated on and off at every N0-th vibration of the dynamic gravity meter, and the period of each gate interval is counted by the number of clock pulses. The averaged period measured by the electronic counter is not the one averaged over "time" but the one averaged over "vibration number". So we can consider the period of a dynamic gravity meter to be a function of "vibration number" n, as T(n) instead of a function of time T(t), and the i-th averaged period Ti is given by the integration with respect to n as 2-1 (5) and the relation between the time t and the vibration number n is expressed as 2-1 (6) The weighted mean 2-1 (4), if expressed in terms of n, becomes 2-1 (7) and equation 2-1 (5) is also transformed as 2-1 (8) 2-1 (9) From equations 2-1 (8) and 2-1 (9) the weighted mean can be expressed approximately as a function of averaged period Ti's which are given intermittently, by applying the Taylor's expansion to the term 1/T(i) in the integral, as follows. 2-1 (10)

6 24 Jiro SEGAWA where the weight function W(i) is assumed cerned accelerations given intermittently. The first term in equation 2-1 (10) is a principal term and the second is a 2nd order correction for finite sampling intervals. Considering that the motion of a usual vessel is of a period of 5-7 seconds and that its amplitude seldom exceeds 100 gals, the magnitude of {T(i)-Ti}Ti will be smaller than 1/100 for the averaged sampling interval of 0.5sec. Therefore the terms of higher order than its third power influences the order of 1m gal at the most. The sampling interval is decided, though customarily, to be nearly 0.5sec, but it depends on the natural frequency Fig. 1. An example of the weight function which of dynamic gravity meters which varies a has been used since little with one another. G(i') obtained in equation 2-1 (10) is a In the second stage a small computer is smoothed function which usually varies slowly attached to the T.S.S.G. with a frequency with a small amplitude. So it is unnecessary counter specially designed for it. This is to calculate the value for every sampling named 'T.S.S.G.-D.P.U.'. The T.S.S.G.- interval. When the variable i' amounts to D.P.U. measures frequency of a dynamic 1000 by the measurement of about 0.5sec gravity meter and processes on real time sampling interval successively made for 8-10 basis. min, G(i') is calculated only for intermittent In the measurement of frequency we measure the number of vibration in a definite time interval, contrary to the previous measurement of period. When the time interval is 2-1 (11) As time t and a variable of the order i are connected by an equation 2-1 (12) the equation which corresponds to 2-1 (8) is 2-1 (13) and the equation corresponding to 2-1 (9) is 2-1 (14)

7 Gravity Measurements at Sea by Use of the T.S.S.G. Part 1. 25

8 26 Jiro SEGAWA From equation 2-1 (14) the weighted mean 2-1 (13) is expressed in terms of fi by the Taylor's expansion, similarly with equation 2-1 (10) as 2-1 (15) where gi* is defined as gi*=kfifi2. As in the case of equation 2-1 (10) the first term in equation 2-1 (15) is a principal term and the second is a 2nd order correction for finite sampling intervals. The digital processings are always ac- measurement carried out in the first stage it is negligible for the reason that frequency of clock signals to measure a period is made sufficiently high. On the other hand, as to the frequency measurement its error depends on the frequency of a dynamic gravity meter; remained in the actual case. This trouble has been solved by electronically multiplying the frequency of the dynamic gravity meter 10 times as large as its natural frequency with the aid of a frequency multiplier. By this In the processings by use of the T.S.S.G.- D.P.U. the method is the one somewhat simplified, because of limited capacity of the computer (Tomoda, 1967 and Segawa, 1968). One of the differences exists in the weight function. The weight function here in use is of a triangular form formulated by 2-1 (17) the natural frequency of a string now in use where i is a variable integer, I a constant integer. The constant I is in an actual case selected as 500 for a thousand data sampled at consecutive intervals of 0.5sec. Vertical accelerations due to a ship's motion of the period of 5-7sec can be reduced to 1/10000 by the weight function. Contrary to the first stage of data processings only a single weighted mean value is given in this case, which corresponds to a value representing a central value in the period of 500sec (that is, the present gravity meter is used. It is not i'=500 in G(i') in equation 2-1 (15)). The desirable to change the sampling interval error originating from this weighted mean is longer in spite of the fact that it enables one less than 10 mgals on condition that the ship's to make the error smaller, because the longer vertical acceleration does not exceed 50 gals. sampling interval causes the larger 2nd order The other difference lies in the way to correction. correct for the 2nd order effect, which we will discuss in the next section. A flow chart programmed in the second stage is shown in Fig. 3. In the third stage a real time processing was started for the first time with sufficient accuracy. A high speed computer designed for process control equipped on the ship has made it possible. Calculations are made according to equation 2-1 (10), but as to the

9 Gravity Measurements at Sea by Use of the T.S.S.G. Part 1. 27

10 28 Jiro SEGAWA

11 Gravity Measurements at Sea by Use of the T.S.S.G. Part nd order correction two different methods are used, which we will discuss also in the next section. A flow chart in a real time processing is shown in Fig nd order correction In this section methods of 2nd order correction which have been hitherto adopted will be explained, and criticism against it will be described in the next section. data obtained in the i'-th sampling interval. Then it is, from equation 2-1 (10) in case of period measurements 2-2 (1) and from equation 2-1 (15) in case of frequency measurements 2-2 (2) dispersions of vertical accelerations given by on condition that the former is linearly con- nected to the latter. Assume the vertical estimation of the correction within an accuracy of several mgals would be enough for our present purposes. There are two ways in such approximate estimations. One is the method dealt with in the paper of Tomoda et al. (1962), where the 2nd order correction is estimated from 2-2 (3) 2-2 (5) estimation of the 2nd order correction becomes In the case of a simple harmonic acceleration its dispersion is stationary and its long time average is represented simply by an average over a single period. If the weight function is supposed to be a constant and the summation is made over a single period in equation 2-2 (5). then where 2-2 (7) over a period. 2-2 (6) Hence, for a simple harmonic acceleration expressed by equation 2-2 (4), an approximate

12 30 Jiro SEGAWA the averaged 2nd order effect will be obtained from the dispersion of vertical accelerations by equation 2-2 (7), provided that the vertical acceleration is of a line spectrum and that its distribution is kept invariable with time. A precise analytical representation of an equation corresponding to equation 2-2 (7) for the case of variable sampling intervals is difficult, but it will be proved in the later section that it is not so different from the expression given as 2-2 (7). The 2nd order correction coefficient S is plotted in Fig. 5 in In Fig. 6, examples of the relation between 2nd order effects and dispersions of vertical accelerations are indicated. From the figures it is found that the linear relation is very well fitted to the data obtained from actual vertical accelerations, of which distribution of spectra is complex and variable with time. However slopes of the most fitted lines are far larger than are expected of theoretical co-efficients derived from equation 2-2 (7), and besides they vary much from a case to another. This has been a question to be solved for these several years, but for practical use a temporary 2nd order correction coefficient determined not from the theory but from experimental data treated statistically was adopted and applied, for instance, to the data of a certain cruise by a certain vessel. The experimentally determined co-efficient will cause not a small error in case when the ship's movement is extraordinarily large, but the error will not exceed 10 mgals as far as the amplitude of vertical accelerations is limited to 50 gals. The 2nd order correction made in T.S.S. G.- D.P.U. is also estimated from the dispersion of vertical accelerations. The dispersion is however calculated in a more simplified way. where f is an averaged frequency over a time of measurement (500sec). However in T.S.S. G-D.P.U. an estimation of the dispersion is made from the average of absolute 2-2 (9) This relation is valid within an accuracy of several per cent as far as a simple harmonic acceleration is concerned. For accelerations composed of complex spectra, however, the error originating from equation 2-2 (10) is not negligible. Let the vertical acceleration g be equated to Then the frequency relation can be derived from it as hence and the averaged absolute difference is 2-2 (11) 2-2 (12)

13 Gravity Measurements at Sea by Use of the T.S.S.G. Part Fig. 6. Examples of the relation between the 2nd order effects and dispersions of vertical accelerations (left: Umitaka-maru in the Indian ocean. right: Hakuho-maru in the west Pacific ocean. The abscissa is expressed by averaged dispersions of vertical accelerations in galgal2).

14 32 Jiro SEGAWA Therefore the right side of equation 2-2 (10) is expressed by an inequality 2-2 (13) Assuming the cross term in the right side to be very small compared with the first term this inequality becomes 2-2 (14) The right side in inequality 2-2 (14) is equal to a correct dispersion for accelerations of complex spectra. Now it could be noted that generally the 2nd order effect caused by finite sampling intervals does not depend, to the strict sense of the word, on either the dispersion or the power spectrum of the vertical acceleration, but depends on the form of its variation itself (that is, amplitudes and phases of the variations). So it would be more reliable to estimate the 2nd order effect from the variation within each sampling interval, which could be determined by an interpolation by use of the neighbouring mean values. Tokuhiro (1967) first tested an interpolation by 2nd and 4th order polynomials. This proved to be superior to the way mentioned above in the fidelity to follow the variation of ship's motion, and yet an apparent decrease of gravity caused by ship's vertical motion is too large to be compensated by a theoretically determined correction. An actual method taken by Tokuhiro is a modified interpolation method to meet the requirement that the corrected gravity within 10min should keep a steady value. The discrepancy between the theoretical 2nd order effect and the actual one will be considered in the next section. In a real time processing by use of a high speed electronic computer (Facom equipped on the Hakuho-maru) the method of 2nd order correction is either the one in which it is calculated from the dispersions of vertical accelerations determined strictly, or the one by interpolation which is almost the same as that taken by Tokuhiro except for a slight modification. 2-3 Re-examination of 2nd order correction Spectra of vertical accelerations observed on board ship Distributions of power spectra involved in ship's vertical motion have been calculated from the data sampled from those measured by the T.S.S.G. In Fig. 7-1, 2 are shown the spectral distributions observed on board the respectively. The numbers indicated in the figures are those serially given to each of the gravity values in the order of time, and the distributions of power spectrum are determined separately in relation to each gravity value at the interval of 12min in case of the Umitaka-maru and of 10min in case of the Hakuho-maru. The power spectra have been obtained from a successive data over sec with a block shift of every 30sec interval in order to know how its distribution changed. So about eleven to twelve sets of the power spectrum distributions can be obtained with each of the numbers. For convenience a series of distributions involved in a certain number are divided into two groups of the first half and of the second half, and plotted in the same figure. From these calculations it is concluded that the vertical motion with the period of 5-6sec predominates on the Umitaka-maru and with the period of 6-7sec on the Hakuho-maru. This difference may well be thought to be caused by the difference of ship's size; the Umitaka-maru is 1450 tons in gross and the Hakuho-maru is 3200 tons. Although no data are available the predominant spectrum of the vertical motion of the Takuyo (770 tons) is of the period of 4-5sec (Tokuhiro, Personal communication). The distribution of power spectra involved in vertical motions is wide-banded as has been thought previously. It ranges from 3 to 9sec, for instant, in case of the Hakuho-maru.

15 Gravity Measurements at Sea by Use of the T.S.S.G. Part Fig Distributions of power spectra in the vertical accelerations of the Umitaka-maru (the abscissa is expressed by period of the acceleration in second).

16 34 Jiro SEGAWA

17 Gravity Measurements at Sea by Use of the T.S.S.G. Part 1. 35

18 36 From these considerations we reach a conclusion that it is wrong to assume ship's motion to be mono-chromatic Distribution of the 2nd order correction co-efficients determined experimentally In Fig. 8 the distributions of slopes of the Fig. 8. (1) Distribution of slopes of the most fitted line in case of the cruise (3) in 1964 (Umitakamaru). (2) Distribution of slopes of the most fitted line in case of the cruise (6) in 1966 (Umitakamaru). The abscissa is indicated by mgal/(gal)(gal)2. most fitted line drawn by the least square method between decreases of apparent gravity and the dispersions of vertical accelerations are indicated. The slope of the most fitted line however can not directly be connected to the coefficient of 2nd order correction, although it might be probable that the 2nd order effect be the most responsible for determining the slope. The slope of the most fitted line to each of sampled data is very sensitive to a slight change of gravity values caused by a real variation of the earth's gravity or an error of measurement. Besides it is more confusing that the most frequent values of the slope are much larger than those estimated from theoretical formulas 2-2 (1) or 2-2 (2), even if the complexity of spectra in vertical motions of a ship is taken into account. For practical use, therefore, we take a most predominating slope as a temporary value of the co-efficient; mgals/(gal)(gal)2 in case of the cruise (3) in 1964 and mgal/(gal)(gal)2 in case of the cruise (6) in 1966 (the cruise numbers are indicated Numerical experiment of the 2nd order correction by use of model gravity data There are two assumptions involved in the process introducing equation 2-2 (7). They are 1) Sampling intervals of the period of a dynamic gravity meter is constant to be represented by an averaged interval. 2) Period of ship's vertical acceleration should be divisible by the averaged sampling interval. With the aid of these assumptions introduction of the equation to estimate the 2nd order correction coefficient for a simple harmonic acceleration has been simplified. It would otherwise be impossible to estimate it analytically. Justification of equation 2-2 (7) can be made fairly well by the following numerical experiment. In case of a simple harmonic acceleration the frequency f(t) of a dynamic gravity meter is related to vertical accelerations as

19 Gravity Measurements at Sea by Use of the T.S.S.G. Part following equation where Considering that the frequency f(t) is always positive and that an integral of the left-hand side, therefore, increases monotonically with time, the solution is uniquely determined as t=ti. By use of ti thus determined the averaged period Ti can be obtained as (5) (3) (4) Period measurements are so made as to measure a time over which the dynamic gravity meter steadily vibrates N times. So the time in which a dynamic gravity meter vibrates i.n times can be obtained from the Then we get the following equation taking into account.

20 38 Jiro SEGAWA (6) Table 1. Comparison of the 2nd order corrections obtained from model gravity data. data with various amplitude and period from which apparent gravities and decreases due to the 2nd order effect can be calculated (Table 1). Comparison of the 2nd order effect obtained from the model data to that estimated from equation 2-2 (7) shows that they agree well within an accuracy of 1 mgal. This will assure that the estimation of the 2nd order effect indicated in equation 2-2 (7) is right. It will, on the other hand, be an assuring that the above expansion is enough evidence that the 2nd order effect almost to be used as a substitute for actual data. depends as an averaged effect on the power Thus we have obtained some model gravity spectra of vertical accelerations as far as they

21 Gravity Measurements at Sea by Use of the T.S.S.G. Part are periodic in spite of the fact that it essentially depends on the amplitude and phase of the variation Treatment in case of complex spectra The existence of two different waves in vertical accelerations is considered. When Then an average of vertical accelerations in the i-th sampling interval is where (4) while an average of frequency in the same sampling interval is (3) The i-th apparent gravity obtained from equation (3) is (4) where (5) accelerations.s1 and S2 correspond to 2nd order correction coefficients for simple harmonic accelerations when they exist alone. An expression which applies to a more general case where more than two different waves exist can be deduced from the above considerations, as 2nd order correction (8) where Sm is a 2nd order correction co-efficient the m-th harmonic vertical acceleration Calculation of the 2nd order correction from distributions of power spectrum Actual 2nd order correction co-efficients seem to be much larger than those estimated by theoretical considerations based on ideal equation 2-1 (1). As is described in 2-3-2, the 2nd order correction co-efficient is for practical use determined from the most frequent coefficient that fits the relation between dispersions of vertical accelerations and observed decreases of apparent gravity due to the accelerations. In this section a test is made to know if it is useful to treat each of the power spectra separately and to combine each effect afterwards. Calculations of the 2nd order correction are made according to equation (8), but co-efficient Sm used is not the one directly

22 40 Jiro SEGAWA obtained from equation 2-2 (7), but the one multiplied by a certain factor uniformly over all the components: In the data measured by the Umitaka-maru [in cruise (3)] the aver- while the predominating period of vertical motions observed in the ship is from 5 to 6sec. So if the ratio of an averaged sampling interval to the period of the vertical accelerations is assumed to be 0.1 the 2nd order correction co-efficient deduced from equation 2-2 (7) is mgal/(gal)(gal)2. This is too small. Therefore, for practical use, multiplication by a factor 3 is made and a co-efficient of mgal/(gal)(gal)2 is used uniformly for every datum. Similarly, when we apply a different correction coefficient to each component of the power spectra it is assumed 3 times as large as originally obtained from equation 2-2 (7). Comparison between the corrections applied in two different ways is shown in Fig. 9, where the apparent variation of gravities, the 2nd order corrections deduced both from dispersions and from power spectra and the corrected gravities are demonstrated. From these figures it is found that no improvement is made by combining the 2nd order effect from each power spectrum which is a way to take the differences of the period into account nd order correction obtained by interpolation Interpolation is applied to the integral found in equation 2-2 (1) (1) by substituting a polynomial into the period Fig. 9. Comparison between two different ways of the 2nd order corrections.

23 Gravity Measurements at Sea by Use of the T.S.S.G. Part T(i) between i and i+1 as If the following notations are used as then (6) where (3) where (7) (4) (8) where I is a period expressed in terms of i. where When the sampling interval of T(i) is about 0.5sec and the period of variation of T(i) is about 5sec the corresponding value of I will be about 10. In such a case the value deduced Now applications of these interpolation to from equation (4) is about This actual data have proved that corrections are means that power spectra of T*(j) is by about almost determined by an interpolation by use 2% smaller than that of T(i) at the period of a linear equation, and that use of higher order polynomials results in only a slight modification. Fig. 10-1, 2. Examples of the 2nd order corrections by the interpolation method (5)

24 42 Jiro SEGAWA In Fig is shown an example of analysis made for gravity data of the Takuyo, the Hydrographic office by means of the 2nd and 4th order polynomials (Tokuhiro, 1967), and in Fig are shown results of interpolation by means of the 1st, 2nd and 4th order polynomials with respect to the data obtained by the Hakuho-maru (cruise (12) indicated From these results it is found that gravity values corrected for the 2nd order effects still indicate a variation which shows the same tendency as the 2nd order effects. If gravity values after being corrected for the 2nd order effect is assumed to be steady the corrections which have been made must be "undercorrection". A more appropriate correction (9) Fig. 11. A simplified structure of the dynamic gravity meter. connect it sideways to the supporting frame (see Fig. 11). Taking aside the constant tension caused by cross leaf springs, as it is clearly unrelated to the 2nd order effect, we consider here only about the influence of intrinsic elasticity of the string. Now a differential equation to represent the chord vibration of a string having elasticity is The same conclusion has been introduced also in sections 2-2 and from the relation between variations of apparent gravity and dispersions of vertical accelerations. However, expressions (6), (7) and (8) apply well to model gravity data: In Table 1 comparison of the 2nd order corrections between those required and those deduced from 4th order interpolation is made with respect to the model gravity data with different amplitudes and periods. From this it is assured that the agreement is well On the differences between theoretical 2nd order corrections and those required by actual data It has been concluded by the tests based on various standpoints that the theoretical 2nd order corrections deduced from an ideal relation 2-1 (1) does not apply to the actual data. This discrepancy will not be accounted for but the fundamental relation between frequency of a dynamic gravity meter and the external accelerations is changed. Two reasons will be suggested for which the dynamic gravity meter deviates from the ideal state; a metal string used intrinsically shows some elasticity of its own, and besides it is initially tensed by cross leaf springs to (1) distance along the string, displacement, time, bulk density, cross section, tension, Young's modulus of elasticity and radius of gyration in the direction of bending, respectively. If we solve the above equation on condition that the string is fixed at both ends and that the tension of the string is so large that the effect of elasticity is taken to be small compared with it, then we will obtain an expression of natural frequency of the string as (2) where l is length of the string. The second term in equation (2) is independent of the tension F, which means that the string vibrates at frequency of with If tension F is caused only by an external acceleration g acting on a weight m, then

25 Gravity Measurements at Sea by Use of the T.S.S.G. Part we get (4) Suppose that the string is a in width and b in thickness and that it vibrates in the direction of thickness. Then a radius of gyration of the string is expressed by (5) In an actual string, for example, parameters are chosen like; a=0.01cm, b=0.0015cm, l=2.5cm, (6) The first term is a principal term that fits to a string of ideal state, the second is that which is supposed to be responsible for the 2nd order effect and the last is a constant. Let the external acceleration g be equal to acceleration. Then, the averaged effect of the second term in (6) can be approximated as (7) equation (7) the second term is directly connected to the dispersion of external accelerations, and signed positive. Let it be (8) (9) A coefficient indicated in this expression corresponds to a 50 mgal positive correction which must be applied to the data calculated only from the principal term in case when dispersions amount to 10000(gal)(gal)2. This correction varies nearly in phase with a 2nd order correction for finite sampling interval originated from the principal term. An addition of to the theoretical 2nd order correction coefficient deduced from equation 2-2 (7) is not necessarily sufficient to compensate the decrease of gravity in the actual data. That is perhaps because the values of parameters taken at present do not meet the actual case Treatment of elastic tension A gravity meter string is initially tensed by cross springs. The tension is different according to the conditions under which the gravity meter is constructed. Let the initial tension expressed in terms of gravity unit be denoted by ac. Then we get a fundamental relation to represent a state of the dynamic gravity meter by combining equation (4) with ac, as (1) The most legitimate treatment of actual gravity data will be to follow expression (1) exactly, provided all the parameters are known. For the present case, however, it is more urgent to correct the already determined gravity data which have been calculated on the basis of an idealized equation 2-1 (1). Let frequencies be f1 and f2 at two different places where gravities are g1 and g2, respectively. Let the constant K be deter-

26 44 Jiro SEGAWA mined at the first place according to the idealized equation, as (2) Then gravity value g2' at the second place calculated also from the idealized equation is (3) By use of a real relation (1) applying to the dynamic gravity meter a true gravity g2 of the second place can be expressed as (4) Here we choose a quantity A which is This expression can be rewritten by an appropriate approximation by use of g1, g2' and ac, eliminating f1 and f2 as (9) value itself, but the variation is very small as the variation of gravity is merely several thousandth of its total value, causing an error of less than 1 mgal in the correction. Then expression (9) is modified as (10) (11) If g1, g2 and g2' are known, quantity A of the dimension of gravity can be estimated from (12) where (f is an average of f) (6) The second term in equation (6) is a correction for the initial tension caused by cross springs and the third term is a correction for residual frequency caused by elasticity of the string. If both terms are combined we get an expression or a more simplified expression (7) (8) We will name A 'an elastic tension' which includes both the effect of elasticity of a string and that of an initial tension exerted of A will be allowed if accuracy of 1 mgal is aimed at. In the present discussion the case when there is a large periodic acceleration acting on a string has not been taken into account, but it is separately dealt with in the previous section. The effect of elasticity of a string in the presence of periodic acceleration can be involved into the 2nd order effect by adjusting its co-efficient properly. The g2' found in the above expressions should be a value whose 2nd order correction has already been made. Thus we obtain a correction equation (11) applying not only to static external accelerations, but also to periodic acceler-

27 Gravity Measurements at Sea by Use of the T.S.S.G. Part ations, though in a way of an approximate treatment. 2-4 Accuracy of the vertical of a gravity meter axis and the effect of horizontal accelerations What have so far been problematical about shipborne gravity meters in connection to the accuracy of vertical and to horizontal accelerations are: (1) Off-leveling error (2) Browne's effect (3) Periodic stabilization error (4) Cross coupling effect. These errors are what have mattered with vertical. In this case the deflection should be a stationary one, fluctuations of the vertical being not taken into account because it is likely to be coupled with horizontal accelerations. As regards the T.S.S.G. such deflection is caused by its stable platform of the vertical gyroscope on which the dynamic gravity meter is directly mounted. A static accuracy of the vertical gyroscope used in T.S.S.G. is better than 4'. This is equal to about 0.7 mgal error in gravity. When the vertical gyroscope is subjected to the surge and sway accelerations it may be off level a little more than when it is kept still. A test on board ship has shown that deflections of the gyroscope are caused mainly by gyro erection devices in which horizontal accelerations are not linearly converted to electric outputs, giving rise to a deflection of zero point to which the gyroscope refers. The aspect of gyroscopes can be tested under horizontal accelerations through a level vial which is heavily damped, and it has been found that the amount of deflection due to horizontal accelerations is not so much changed by the increase of the acceleration; the differences, occur only between the state of stand-still and that of cruising. Therefore it is possible to adjust the gyro erection devices differently according to the circumstances. In an actual cruise a deflection of 20' which corresponds to a 20 mgal decrease of gravity was observed though it depends on the characteristics of the gyro erection devices used, but once it was corrected the gyroscope seemed to be kept vertical within an accuracy of several minutes. (2) Browne's effect If there is any correlation between deflections of a gravity meter axis and horizontal accelerations it causes a positive error in the measured gravity. or its time average is given by 2-4 (2) The second term is a time average of the off-levelling error, and the third is a coupling with horizontal accelerations. These correction terms were historically first pointed out by Browne (1937) applied to the Vening Meinesz submarine gravity meter. With the Vening Meinesz Pendulums which are free to swing in gimbals the effect of the second and third terms in equation 2-4 (2) is represented and higher order terms. This is the second order correction or Browne correction which has been further investigated by La Coste (1959) and La Coste and Harrison (1961). A gravity meter free to swing in gimbals in-

28 46 Jiro SEGAWA placed at a correct distance below the gimbal joint. The correct distance is equal to the equivalent length of the gimbal suspension system. When there is a resonance between gimbals and horizontal accelerations making too large the amplitude of swinging the 4th and higher order terms must be taken into account. Equation 2-4 (2) holds generally when there is a deflection of the measuring axis of a gravity meter no matter whether the meter is suspended in gimbals or mounted on a stabilized platform. Browne corrections however should be restricted to the case of a gravity meter which is free to swing in gimbals, having a meter with a stabilized platform treated separately. The data obtained by Vening Meinesz submarine gravity meters before 1937 would involve errors due to Browne effect (more or less than 20 mgal). With La Coste Romberg gravity meters designed before 1964 it was needed to correct for a large amount of Browne effect. (3) Periodic stabilization error As was pointed out by La Coste and Harrison (1961) deflection of a gravity meter axis if coupled with horizontal accelerations causes a tremendous error. A stabilized platform to keep the measuring axis vertical is required to be accurate by 4" in verticality if 1 mgal accuracy in gravity is aimed at under horizontal accelerations of 100gals. This error can be estimated from the third term in equation 2-4 (2) by supposing the 2-4 (3) Fig. 12. With the present instrument 4" accuracy in verticality is very hard to attain and furthermore no means are available which enable us to measure such deflections. In the T.S.S.G. a vertical gyroscope works at the same time as a reference of vertical and a stabilized platform, eliminating follow-up systems such as potentiometers, servo motors, gears and something like that. The gravity meter system consisting of a dynamic gravity meter, a spin motor and gimbals are carefully adjusted to be supported at the center of gravity so as to minimize disturbances to the precession. The vertical in a moving vessel is determined from a direction in which the horizontal component of accelerations is smoothed out. The horizontal component of acceleration is sensed by a pair of horizontal accelerometers attached to the meter. Erection torques applied to the vertical gyroscope must be as strong as it can keep up with the rotation of the earth but is not desired to be too strong because it causes short period motions owing to horizontal accelerations. We here consider behaviours of a vertical gyroscope. Let x-x' and y-y' be horizontal axis and let z-z' be a vertical axis fixed to the earth (Fig. 12). The gyroscope is deflected 2-4 (4) 2-4 (5) 2-4 (6)

29 Gravity Measurements at Sea by Use of the T.S.S.G. Part (7) 2-4 (8) and in such ideal case it is obvious that coupling of deflection with horizontal accelerations should be ignored. With the present T.S.S.G. however the proportionality of the output of erection torque to horizontal acceleration does not rigorously hold. Horizontal accelerometers first put to use in the T.S.S.G. (Tomoda and Kanamori 1962) were of the on-off control consisting of a mercury drop enclosed in a vacuum tube. This kinds of accelerometer will work no longer proportionally to the acceleration. However accelerometers now in use are of the proportional type consisting of differential transformers the sensitivity of which can be made practically sufficiently high. But the electric circuit accompanying the accelerometer cannot afford the wide dynamic range required in following the variation of horizontal accelerations, resulting in practically the same on-off control as in the case of mercury drops. When gyro erection devices are of the onoff control erection torque Fy is expressed as 2-4 (9) 2-4 (10) In this case integral 2-4 (8) can not necessarily be zero though the residual is expected to be small. There may of course exist deflections caused by other reasons such as those due to unbalancing of the gravity meter systems or to frictional forces. But it is by no means easy to measure those deflections in relation to the coupling with horizontal acceleration. La Coste et al., (1967) reported with the La Coste and Romberg stabilized platform shipboard gravity meter that the accuracy requirement for vertical errors in phase with horizontal accelerations is not easy to measure directly, that direct measurement would require a vertical reference that is known to be accurate to better than 4" at all times and that therefore this requirement was checked by subjecting the stabilized platform mounted gravity meter to various horizontal accelerations both in the laboratory and at sea. The requirement of 4" accuracy is too strict, but if the requirement is to be applied only to an instantaneous state it would not be so hard being attained by a careful adjustment of the meter, as is the case with La Coste's meter which is of the 'long term accuracy of 1" and of the 'short term accuracy of 8"' in phase with horizontal acceleration of 50gals. (4) Cross coupling effect The cross coupling effect is caused by coupling between vertical movements of the pivoted gravity meter beam and the component of horizontal acceleration in the direction of the beam. With a gravity meter which is intrinsically unmovable, to which the T.S.S.G. belongs, it is not so important. It is interesting to note that a gimbal supported gravity meter is not subject to cross coupling errors if the gravity meter is placed at the correct distance below the gimbal joint. This adjustment avoids cross coupling errors by eliminating all forces on the gravity meter except a force along its sensitive axis (La Coste 1959). The absence of cross coupling effects in gimbal supported gravity meter is a real advantage over stabilized platform gravity meters. Here we consider the relationship of cross coupling effect with special reference to Graf- Askania surface ship gravity meter and La Coste-Romberg ship-board gravity meter both of which are mounted on a stabilized platform. The expressions given below are those repro-