Abstract:
Interrogation of a phase based transducer is performed by comparing the state of the transducer at two points in time to determine the rate of change with time of the measurand represented as a phase change. The rate of change, or derivative of the phase change typically has a much smaller amplitude than the signal itself, and the derivative measurement can therefore be thought of as a low sensitivity measurement to be used instead of or in combination with the normal signal measurement having higher sensitivity. In this way, large amplitude signals which might otherwise be subject to overscaling effects can be measured more effectively. For a signal with the majority of its energy centered at approximately 800 Hz, for example, the derivative of that signal will typically be attenuated by 60 dB with a period between the two measurement times of 200 ns.

Description:
BACKGROUND OF THE INVENTION 
     (1) Field of the Invention 
     The present invention relates to sensors which exploit a change in phase of an interrogation signal to determine a sensed parameter, and particularly, but not exclusively to fibre optic interferometric sensing. The present invention finds particular application in the filed of seismic surveying. 
     (2) Description of the Art 
     Fibre optic sensors employ a length of optic fibre arranged in such a way that a sensed parameter causes a strain to be imposed on the fibre. Typically the fibre is arranged in a coil, although other arrangements are possible. Such strain causes a change in phase of optical signal propagation in that fibre, which change can be detected by interferometric techniques. A variety of different arrangements for this type of transducer have previously been proposed, many of which have the coil of optic fibre wound on a deformable core or mandrel, which undergoes radial expansion or contraction in response to the sensed parameter, such as sensed vibration. 
     Such fibre optic sensors can exhibit extremely high sensitivities, and have the advantage of being completely passive, employing no power at the sensing transducer. Such sensors have also proved popular in applications where large arrays of sensors are required, on account of the relative ease with which they can be multiplexed. 
     An example of such an application is seismic surveying in the oil and gas exploration industry, where large time multiplexed arrays comprising hundreds or even thousands of vibration sensors and/or hydrophones can be used to sense reflections of an incident pulse from geological formations beneath the sea bed. Sampling such an array at regular periods provides 3D time lapsed data on existing or potential new reserves. 
     A problem experienced with this approach to sensing is that, for a given sampling rate, signals above a certain amplitude threshold cause the phase based sensed information to become distorted, and can cause failure of the demodulation process. This effect, commonly referred to as overloading or overscaling is dependent on the frequency of the measured signal. In seismic systems this can cause a particular problem with the direct arrival of the incident pulse, especially when that pulse has been generated close to the sensors (usually by an airgun towed from a surface vessel as it passes over the array). It is desirable to be able to record this incident pulse without the distortion that overscale can produce. 
     SUMMARY OF THE INVENTION 
     It is therefore an object of the present invention to provide improved sensing methods and apparatus. 
     According to a first aspect of the present invention there is provided a method of interrogating a phase based transducer, said transducer providing a change in phase of signal propagation in response to a sensed parameter, said method comprising receiving a signal propagated through said transducer, comparing the signal at a point representing the state of the transducer at a first time, and at a point representing the state of the transducer at a second time, and determining from said comparison a measure of the rate of change of phase with time of said signal. 
     The rate of change, or derivative of the phase change typically has a much smaller amplitude than the signal itself since the difference between the two times at which the signal is measured will usually be much less than the period of the signal being measured. Thus even if the transducer experiences a stimulus which causes the normal signal to overscale, the derivative signal is likely to be unaffected. The derivative signal or measurement can therefore be thought of as a low sensitivity measurement, which can be obtained severally and independently, to be used instead of or in combination with the normal signal measurement having higher sensitivity. For a signal with the majority of its energy centred at approximately 800 Hz, for example, the derivative of that signal will typically be attenuated by 60 dB with a period between the two measurement times of 200 ns. 
     In one embodiment, comparing the signal comprises combining a delayed version and an undelayed version of the received signal, preferably using an output interferometer. Alternatively the signal could be sampled at multiple different times and an algorithmic or signal processing approach employed to determine the derivative of the sensed parameter. 
     The actual value of the sensed parameter can be reconstructed by integrating the measured derivative value. However, if the noise floor is determined by system noise, then the noise floor is substantially the same for both the phase information and its derivative, the derivative signal suffers from a lower SNR. Considering then that overscaling may only occur infrequently in response to certain high amplitude inputs, such as the first break of an airgun used in geophysical surveying, it is beneficial in certain embodiments to measure both the derivative, and the actual value of the sensed parameter directly. Examples of such embodiments are described below and it will be appreciated that an adaptive system could be employed which measures the signal directly in a default state, and reverts to a signal integrated from a derivative measurement on detecting an overscale condition. For example, it would be possible to use a threshold value for the amplitude of the derivative signal to identify periods in which the normal signal was overloaded. 
     It has been found however, that by careful consideration of overscaling and its effect on the phase information produced by the transducer, that during overscaling, although a direct measurement of the phase value representative of the sensed parameter may be distorted, such a phase value can be used in conjunction with the derivative signal to produce a re-constructed value. In certain embodiments therefore a measure of phase from said received signal is derived in addition to the measure of the rate of change of phase. In particular embodiments the two measures (phase, and rate of change of phase) are obtained severally, in that each can be obtained without recourse to the other, and can be obtained substantially simultaneously. 
     As will be described below in greater detail, overloading occurs when the instantaneous frequency of the output of the transducer (which depends on the rate of change of phase) falls outside of the Nyquist frequency range determined by the rate at which this signal is sampled. Any instantaneous frequency that falls outside the Nyquist range will be folded about the limits of the range back into it. Depending on the amplitude and frequency of the sensed signal, the information may be folded or wrapped about the Nyquist frequency limits multiple times. The present inventors have found that the derivative information measured in embodiments of the present information can be used to determine how many times the information has been wrapped, or the factor by which the information exceeds the Nyquist limit. This then allows the directly measured parameter value to be corrected to provide a signal having an improved SNR to that provided by integrating the measured derivative signal. 
     The invention extends to methods, apparatus and/or use substantially as herein described with reference to the accompanying drawings. 
     Any feature in one aspect of the invention may be applied to other aspects of the invention, in any appropriate combination. In particular, method aspects may be applied to apparatus aspects, and vice versa. 
    
    
     
       DESCRIPTION OF THE DRAWINGS 
       Preferred features of the present invention will now be described, purely by way of example, with reference to the accompanying drawings, in which: 
         FIG. 1  shows a known type of fibre optic sensor package; 
         FIG. 2  is an interrogating waveform suitable for the package of  FIG. 1 ; 
         FIG. 3  illustrates a typical response from a package of the type shown in  FIG. 1 ; 
         FIG. 4  shows a system for interrogating a fibre optic package according to an aspect of the invention; 
         FIG. 5  illustrates an output obtainable from the system of  FIG. 4 ; 
         FIG. 6  shows a further output obtainable from the system of  FIG. 4 ; 
         FIG. 7  illustrates an arrangement where two output interferometers are used; 
         FIG. 8  shows the output obtainable from the system in  FIG. 7 ; 
         FIG. 9  illustrates multiple sampling points on a pulse of the type shown in  FIG. 3 ; 
         FIGS. 10 and 11  show time separations between samples on the same and different pulses. 
     
    
    
     DESCRIPTION OF THE INVENTION 
     Referring to  FIG. 1 , there is shown schematically a known type of fibre-optic sensor package, indicated generally  102 , comprising four individual fibre-optic sensing coils  104 ,  106 ,  108 ,  110  formed from a single length of optical fibre  13 , and arranged in series. A portion of the optical fibre  112  serves as the package input/output (i/o) fibre. Fibre-coupled mirrors  114 ,  116 ,  118 ,  120 ,  122  are coupled to the optical fibre  13  at respective locations along it such that each of the coils has a fibre-coupled-mirror coupled at each end of it. Other means of reflecting a portion of light from before and after each sensor such as in fibre Bragg gratings could be used instead of the fibre coupled mirrors. In practice for example, three of the coils could be arranged to form three orthogonal fibre optic accelerometers, with the fourth coil forming part of a hydrophone to form a four-component package suitable for seismic surveying applications. The physical arrangement of the coil in each transducer is not material to the present invention, and is not discussed here, however a range of possible arrangements will be known to the skilled reader. A large scale array of such packages can be coupled together, and interrogated periodically using multiplexing to provide time lapsed seismic imagery for example. 
     Referring to  FIG. 2 , an interrogation of the package  102  of  FIG. 1  may be carried out by introducing a pair of interrogating optical pulses  202 ,  204  into the package i/o fibre  112 . Pulses  202 ,  204  have respective frequencies ω 1 , ω 2  and pulse  202  is delayed by τ=2 L/c with respect to pulse  20 , L being the length of coil in the sensor and c being the speed of an optical pulse in the fibre. 
       FIG. 3  illustrates the optical output response of the package by considering the output formed by each of the pair of input pulses. In  FIG. 3   a  the first pulse  202  to arrive at the package is reflected off each of the 5 fibre-coupled mirrors to produce five output pulses  301 ,  302 ,  303 ,  304  and  305 , measured relative to an arbitrary time reference. Similarly, looking at  FIG. 3   b , pulse  204  produces five time delayed output pulses  322 ,  323 ,  324 ,  325  and  326  relative to the same arbitrary time reference. Because the input pulses are delayed by twice the time of flight through a single coil, and because the pulses exist on the same fibre, the two sets of outputs are superposed to produce six pulses  331 ,  332 ,  333 ,  334 ,  335  and  336  shown in  FIG. 3   c . Pulses  331  and  336  represent only a single reflection of a single pulse, however it will be understood that pulses  332  to  335  (shown shaded) each correspond to the combination of two pulses reflected by adjacent fibre coupled mirrors. It will be understood that these pulses therefore represent the combination of a pulse which has passed (twice) through the coil between the two adjacent mirrors, and a pulse which has not. Phase detection can therefore be used to determine the phase change imposed by that coil, and hence a measure of the sensed parameter is obtained as is known in the art. 
     If φ(t) is the sensed parameter, then the signal obtained from a photodetector used to measure a series of pulses returning from a sensor of the type described above can be written as cos(ω c t+φ(t)) i.e. the sensed information is represented as a phase change superimposed on a carrier signal of frequency ω c . Techniques that are well known to those skilled in the art can then be used to demodulate the phase signal from the carrier. The carrier frequency is typically chosen to be half of the Nyquist frequency, which is in turn half of the sampling frequency. It is usual for one sample to be made in each returning optical pulse and so the sampling frequency is the rate at which pulse pairs are transmitted into the array. By way of an example, the sampling frequency could be approximately 320 KHz, giving a Nyquist frequency of approximately 160 KHz and a carrier frequency of approximately 80 KHz. The sampling frequency will typically have a practical upper limit dependent upon the type and arrangement of sensor or sensors, amongst other factors. 
     An overscale condition occurs when the instantaneous frequency of the phase modulated carrier falls outside the Nyquist band i.e. when 
                 ⅆ     φ   ⁡     (   t   )           ⅆ   t       ≥       ω   N     -     ω   c             
or when
 
                   ⅆ     φ   ⁡     (   t   )           ⅆ   t       ≤     -     ω   c         ,         
where ω N  and ω c  are the Nyquist and carrier frequencies respectively. In practice this results in aliasing of instantaneous frequency back into the Nyquist band by folding or wrapping around one of its limits in frequency space. Depending on the magnitude and frequency of the sensed parameter, the instantaneous frequency can be wrapped back multiple times. If the sensed parameter is modelled approximately as φ(t)=φ 0  cos ω m t, then the condition for overscale not occurring, for the usual condition of ω N =2ω c  is sometimes expressed as
 
     
       
         
           
             
               φ 
               0 
             
             ≤ 
             
               
                 
                   ω 
                   c 
                 
                 
                   ω 
                   m 
                 
               
               . 
             
           
         
       
     
     Referring now to  FIG. 4 , there is shown a sensor package  402 , substantially as illustrated in  FIG. 1 . The package is interrogated by a pair of pulses produced by acousto-optic modulator  404 . 
     The output series of pulses is tapped off at junction  406 , passed through an isolator  408 , and to output interferometer designated by  410 . In the scheme of  FIG. 4 , the delay between input pulses need not be twice the time of flight of light through a sensing coil of package  402 , but is instead arranged to be twice the time of flight of light through delay coil  412  of the output interferometer. While the described embodiment employs a Michelson interferometer, the skilled reader would recognise that a Mach-Zehnder type interferometer with a delay coil in one of the arms could equally be used. In this case, arranging for the separation between the input pulses to be just the time of flight through the delay coil in one arm of the interferometer would allow equivalent measurements to be made. 
       FIG. 5  illustrates component pulse trains output from interferometer  410 . Pulse train  502  represents the output of the leading input pulse (designated by subscript 1) from mirrors B to E, resulting from the delay arm of the interferometer (designated Y). Pulse train  504  represents the output from the lagging input pulse (designated by subscript 2) from mirrors B to E, resulting from the undelayed arm of the interferometer (designated X). It can be seen that, in this way, interferometer  410  temporally aligns and interferes pairs of pulses, both of which have passed through the same sensing coil(s) of package  402 , but at different times (as indicated by the differing subscripts of temporally aligned pulses). In other words, each pulse reflected off fibre coupled mirrors B to E (pulses reflected off mirror A have not passed through a sensing coil), and gathering information on the associated sensing coil, is combined with a pulse having undergone the same optical path, gathering the same information, but at a later time. The output of the interferometer therefore represents the derivative of the phase value, in contrast to the actual value of phase which would usually be measured directly in the prior art arrangements of  FIGS. 1 to 3 . Thus using the terminology above, if the signal returned from the transducer is cos(ω c t+φ(t)) with φ(t) being a measure of the sensed parameter, the system depicted in  FIG. 5  derives a value representative of 
                 ⅆ     φ   ⁡     (   t   )           ⅆ   t       ,         
or the instantaneous frequency of the returned signal.
 
     Considering the combined output pulse centred at t=1, it will be understood that this represents the combination of two pulse having been reflected from mirror B, ie having passed through sensing loop AB, at two different times. The derivative of the parameter sensed by coil AB is therefore contained within and can be determined from this pulse. In a similar way, the pulse output from the interferometer at t=2 will be a combination of pulses, both of which have made double passes of sensing loops AB and BC. Once the derivative value is extracted from this pulse then, by subtracting the derivative value of sensing loop AB (obtained above) the derivative value of sensing loop BC is obtained. In this way, the derivative values for each of the sensing loops in package  402  can be obtained. 
     It is noted that the reflection from fibre coupled mirror A is not affected by the derivative, or rate of change of phase of any of the sensing loops of the arrangement explained above with respect to  FIGS. 4 and 5 . However the pulses reflected off mirror B (t=1) actually contain a signal that is the combination of the derivative of the phase modulation in coil AB and any of the downlead fibres (i.e. fibre between coupler  406  and the AOM  404 , between the coupler  406  and reflector A, and between the coupler  406  and the photodetector on the output of interferometer  410 ). In some applications the downlead fibre can be many kilometers long and so signal that its pickups up due to ambient noise can be significant. If the derivative signal from reflector A is measured and then subtracted from the signal from reflector B it will remove any signal from the downlead from that obtained for coil AB. 
     Turning to  FIG. 6 , there are illustrated component pulse trains representing the output components of a particularly preferred sensor interrogation arrangement. The sensor arrangement is substantially the same as that of  FIG. 4 , but with each sensor coil arranged to be twice the length of the delay coil of the interferometer. In one example, each sensor coil is 40 m in length, the interferometer delay coil is 20 m long, and the delay between the input pulse pair is approximately 200 ns. 
     It can be seen that pulse trains  604  and  606  are substantially the same as trains  502  and  504  of  FIG. 5 , that is, they represent the derivative or ‘low sensitivity’ information for each sensor coil (cumulatively). The output pulse at the time point indicated by  610  is caused by reflections off mirror A and carries no sensed derivative information (except that picked up by the downlead) and is omitted from  FIG. 5  to improve clarity. The output equivalent to that indicated at  614  of  FIG. 6  is the first considered pulse of  FIG. 5 . Another benefit of mirror A however can be seen by considering pulse trains  602  and  608  which result respectively from reflections of the first pulse via mirror X in the output interferometer ( FIG. 4 ,  410 ) and reflections of the second pulse from mirror Y in said interferometer. These are essentially of the same form as shown in  FIG. 3 , and combine to form output pulses carrying the direct parameter values, each pair of pulses representing a reflection from adjacent mirrors. For example, the output pulse at time  612  will be a combination of the leading input pulse reflection from mirror B and the lagging input pulse reflection from mirror A, and therefore contains direct, or ‘high sensitivity’ information sensed by coil AB which can be extracted directly in the known fashion. The fact that the two pulses have passed through different arms of interferometer  410  does not substantially affect the information contained within the pulse. 
     Therefore, in the arrangement described, both directly sensed (high sensitivity) values and derivative (low sensitivity) values can be obtained independently from interleaved sets of output pulses, resulting from the same input pulse pair. It can be seen that in the example of  FIG. 6 , this is achieved by arranging for the input pulse separation to be approximately half of that which is used for the interrogation of a given sensor package in the way described with reference to  FIG. 3 . 
       FIG. 7  illustrates an alternative arrangement for the output or readout interferometer  410  of  FIG. 4 . The response  702  from a package of sensors generated by an interrogation signal is split and passed to two interferometers  704  and  706 . In use, the interrogation signal is similar in form to that of  FIG. 2 , but here the two pulses have a delay equal to the total time of flight through the sensor package, so that output pulse trains resulting from each of the interrogation pulses are not interleaved. With four sensing coils in a package, each coil having a length of 40 m, a delay of 2 μs between interrogating pulses could be used for example. The delay loop  714  of interferometer  704  has a value equal to the delay of the interrogating pulses, while the delay loop  716  in interferometer  706  has a value less than the delay of  714 , by a double pass through one sensor. 
     The operation of the arrangement of  FIG. 7  will now be described with reference to  FIG. 8 . The series of pulses returning from the package as a result of the first input pulse are shown at  802 . The equivalent series resulting from the second input pulse are shown at  804 . Because of the delay of 2(N+1)L/c (where N=number of sensors in a package, L=length of a single sensor, c=speed of flight through sensor) between input pulses, the series  802  and  804  are not interleaved, and combine in the i/o fibre of the sensor package as shown at  806 , and it is this signal which is received at  702  of  FIG. 7 . The first interferometer  704  of  FIG. 7  has a delay equal to the input pulse delay, and therefore the signal returning from the delay arm of that interferometer is shown at  808 . It will be understood that the signal  708  of  FIG. 7  will be a combination of signals  806  and  808 , each pulse being made up of two pulses having been reflected off the same mirror in the sensor package, but at different points in time denoted by subscripts 1 and 2 in  FIG. 8 , and which correspond to first and second pulses respectively. A detector receiving signal  708  can therefore derive derivative phase or instantaneous frequency measures, as explained above. 
     Interferometer  706  of  FIG. 7  has a delay of 2 NL/c, and therefore the signal returning from the delay arm of that interferometer is shown at  810 . Signal  710  of  FIG. 7  is therefore a combination of signals  806  and  810 , with each pulse being made up of two pulses having been reflected off adjacent mirrors of the sensor package. A detector receiving signal  710  can therefore obtain direct measurements of the phase imposed by the sensor coils, as explained above. 
     It will be noted that the component pulses making up signals  708  and  710  will have passed through the sensor package with a greater time separation than in previously described embodiments. This results in the derivative signal being more sensitive than for a shorter input pulse separation. For a 2 μs input pulse delay, approximately 10 times more phase is accumulated over the derivative sampling time than if a 200 ns delay had been employed. Considering the ‘direct’ measurement of phase, since the two samples being combined in interferometer  706  represent instances at the sensor having a relatively large time separation, then the measure of the derivative of phase obtained from interferometer  704  can be used to provide a correction if necessary. 
     In the embodiment of  FIG. 4 , since only one interferometer output is employed, both ‘derivative’ and ‘direct phase’ signals appear on the same pulse train thereby using up twice as many time slots as that described in the prior art of  FIGS. 1-3 . By deriving the two sets of measurements (phase and derivative) independently, using two interferometers, restoration of substantially the full time domain bandwidth is achieved. 
     An alternative method to achieve an increase in the time separation between the pulses that generate the derivative signal is to keep the pulse separation at 200 ns but then mix reflections that were from different transmitted pulse pairs. In such an embodiment the time separation between respective pairs of pulses is determined by the length of fibre in the sensors in one segment of the array, but a value of around 5 μs would be typical.  FIG. 9  shows the pulse pattern that would be achieved using the same arrangement as in  FIG. 4  but with the coil  412  causing a delay of 5.2 μs. Pulse sequences  902 ,  904  result from the reflections of the first and second transmitted pulses respectively, while sequences  906  and  908  are those from the first and second pulses respectively that have passed through the delay coil of the interferometer. Reflections in sequences  906  and  908  overlap with reflections in sequences  902  and  904  which were from the subsequent pulse pair. 
     Since the delay imposed by the delay coil (5.2 μs) is 200 ns longer than the time separation between pulse pairs, the combination of pulses takes the same general form as if the delay imposed by the coil had been 200 ns (as per  FIG. 6 ) and so both normal and derivative signals are can be obtained from interleaved output pulses. In this case though, the pulses that combine to form the derivative signals pass through each sensor 5.2 μs apart and so the signal generated will be 26 times longer than if a 200 ns delay coil was used. 
     It can be shown that it is possible (in the embodiment described above) to use any delay time of the form [5N R +2(N D +1)0.2] μs, where the 5 μs is the time between pulse pairs and N R  is a positive integer. The number 0.2 corresponds to the time in μs between pulses in a pair and is dictated by the sensor length. N D  is also a positive integer. The derivative of a time varying signal will tend to be proportional to the frequency of the signal. Actual interferometer response observes this relationship at low frequencies, but tails off slightly at high frequencies. In the low frequency range of approximately 0-250 Hz (useful in seismic applications), the amplitude of the derivative signal can therefore be correspondingly low, and can suffer from low SNR. Using a longer delay however causes the interferometer sensitivity to be greatly increased, and to produce correspondingly increased output amplitudes. 
     Although the arrangement shown in  FIG. 9  does give a normal signal the large pulse delays may cause an increase in cross talk between different sensors and may increase the system noise level. These issues could be overcome by using the arrangement shown in  FIG. 7  and having the two interferometers  704  and  706  causing delays of 200 ns and 5.2 μs respectively. Thus interferometer  704  could be used to produce the normal signals, and if required short separation derivative signals, while interferometer  706  could be used to produce the longer separation derivative signals. 
     Referring to  FIG. 10 , there is shown in greater detail the profile of a pulse of the type shown in  FIG. 3   c , that is a pulse which represents the interference of a two pulses reflected from adjacent mirrors of a sensor package, and containing phase information on the parameter sensed by the coil located between those two mirrors (as opposed to its derivative). As noted above, this information is susceptible to being ‘lost’ when overscaling occurs. It has been appreciated by the present inventors however, that whereas previously a sample of the optical signal of such a pulse has been taken only once, it is possible to make two or more such measurements within each pulse. Using the sampled values, a two point phase algorithm, for example, can be employed to obtain a measure of the derivative of the phase information. 
     The pulse of  FIG. 10 , as represented by the idealised square wave profile shown dashed line, is nominally 100 ns in duration. As shown by the solid line however, the actual form of the pulse includes non-zero rise and fall times, and might typically result in a plateau of 90% of the maximum intensity value having a duration of 60 ns from which samples can be taken. In order to take two samples then, at a temporal spacing of 50 ns say, a sampling rate of 20 MHz is required. More realistically, to be able to take multiple samples from such a pulse, such as those indicated at  1006 ,  1008  and  1010  for example, it is desirable to use sampling rates greater than or equal to 50 MHz, 80 MHz or even 100 MHz. Such rates are currently achievable using commercially available technology. 
     Considering  FIG. 11 ; the time separation between each pulse is t p  and the sampling spacing within the pulse is τ s . Let the carrier frequency be ω c  then the single sampling signal S 1  obtained at a position  1006  at rate Fp (=1/t p ) can be modelled by
 
 S   1   =A  cos(ω c   nt   p +φ 1 )
 
where A is the signal amplitude into the demodulation system, n is the nth sample after some arbitrary start, and φ 1  is the phase associated with the sample at the first sampling position S 1 . The second sample S 2  is simply modelled by
 
 S   2   =A  cos(ω c ( nt   p +τ s )+φ 2 )
 
where φ 2  is the phase associated with the signal at the second sampling position.
 
     It is possible to employ an algorithm using the two samples S 1  and S 2 , to extract the instantaneous phase change Δφ=(φ 2 −φ 1 ) which can take the form: 
     
       
         
           
             
               Δ 
               ⁢ 
               
                   
               
               ⁢ 
               φ 
             
             = 
             
               
                 tan 
                 
                   - 
                   1 
                 
               
               ⁡ 
               
                 [ 
                 
                   
                     
                       
                         S 
                         2 
                       
                       ⁢ 
                       
                         
                           
                             A 
                             2 
                           
                           - 
                           
                             S 
                             1 
                             2 
                           
                         
                       
                     
                     - 
                     
                       
                         S 
                         1 
                       
                       ⁢ 
                       
                         
                           
                             A 
                             2 
                           
                           - 
                           
                             S 
                             2 
                             2 
                           
                         
                       
                     
                   
                   
                     
                       
                         S 
                         1 
                       
                       ⁢ 
                       
                         S 
                         2 
                       
                     
                     + 
                     
                       
                         
                           
                             A 
                             2 
                           
                           - 
                           
                             S 
                             1 
                             2 
                           
                         
                       
                       ⁢ 
                       
                         
                           
                             A 
                             2 
                           
                           - 
                           
                             S 
                             
                               2 
                               ⁢ 
                               
                                   
                               
                             
                             2 
                           
                         
                       
                     
                   
                 
                 ] 
               
             
           
         
       
     
     This is the differential phase acquired over the sampling period τ s  within a pulse. Assuming τ s  is significantly smaller than the carrier period t p , the change in phase Δφ is small, but this can correspond to a very large instantaneous frequency over the said sampling time. 
     The above algorithm which operates on two sampling positions on the pulse may provide an adequate solution in certain applications, but has been found to suffer from a degree of data dropout under certain conditions. While a limited amount of data dropout may be acceptable, it is possible to use a third sample S 3  (labelled as  1010 ) from the pulse to improve the estimate of Δφ. 
     By taking a third sample, a second phase change value Δφ 2  (corresponding to the phase change over time τ s  between S 2  and S 3 ) is obtained. The difference between the two phase changes Δφ 1  and Δφ 2  is assumed to be small in comparison to either one value, and by taking the maximum of the absolute value between the two phase measurements i.e. max (|Δφ 1 |, |Δφ 2 |) data dropout can be substantially eliminated up to a maximum ratio of F s /F p  (F s =1/τ s ) allowing an accurate reconstruction of a wrapped instantaneous frequency signal. The reconstruction process is typically carried out at the sampling rate F p , but may be decimated to other rates after reconstruction. 
     Using this technique, signals having instantaneous frequency values up to 500 times greater than the ‘normal’ overscaling limit can be reconstructed over a wide range of frequencies, for typical sampling frequencies of 160 kHz. 
     Where the change in phase Δφ is itself wrapped, this will happen between F p /2 and dc, bearing in mind that the centre of the instantaneous frequency change is at the carrier frequency. 
     A frequency unwrapping algorithm to unfold the instantaneous frequency operates by finding the time positions where the instantaneous frequency reaches zero and changing the sign of the signal between every other pair of zero crossings. The instantaneous frequency is finally integrated or used to reconstruct the lost phase information in the normal sensitive signal. 
     It will be understood that the present invention has been described above purely by way of example, and modification of detail can be made within the scope of the invention. 
     Although a fibre optic sensor package suitable for seismic surveying has been described, it will be appreciated by the skilled person that the invention is equally applicable to other types of phase based transducers employed in alternative applications. Examples include uses of fibre optic hydrophones in active sonar systems and measurements of surface vibration using a free space optical interferometer. 
     Each feature disclosed in the description, and (where appropriate) the claims and drawings may be provided independently or in any appropriate combination.