Detecting gravitational waves from inspiraling binaries with a network of geographically separated detectors : coherent versus coincident strategies
Abstract
We compare two strategies of multidetector detection of compact binary inspiral signals, namely, the coincidence and the coherent for the realistic case of geographically separated detectors. The naive coincident strategy treats the detectors as if they are isolated  compares individual detector statistics with their respective thresholds while the coherent strategy combines the detector network data coherently to obtain a single detection statistic which is then compared with a single threshold. We also consider an enhanced coincidence strategy which is intermediate in the sense that though the individual statistics are added in quadrature and the sum compared with a single threshold, the estimated parameters are also checked for consistency. For simplicity, we consider detector pairs having the same power spectral density of noise, as that of initial LIGO and also assume the noise to be stationary and Gaussian. Further, since we consider the detectors to be widely separated on Earth, we take the instrumental noises to be uncorrelated; the wide separation implicitly means that since the detector arms must lie parallel to the Earth’s surface, the detectors necessarily have different orientations. We compare the performances of the methods by plotting the receiver operating characteristics (ROC) for the strategies. Several results are derived analytically in order to gain insight. Simulations are performed in order to plot the ROC curves. A single astrophysical source as well as a distribution of sources is considered. We find that the coherent strategy is superior to the two coincident strategies that we consider. Remarkably, the detection probability of the coherent strategy is 50% better than the naive coincident strategy. One the other hand, difference in performance between the coherent strategy and enhanced coincident strategy is not very large. Even in this situation, it is not difficult to perform the real data analysis with the coherent strategy. The bottom line is that the coherent strategy is a good detection strategy.
pacs:
95.85.Sz,04.80.Nn,07.05.Kf,95.55.YmI Introduction
Inspiraling binaries are one of the most promising candidates for first detection of gravitational waves (GW). The compact objects can be treated essentially as point particles leading to sufficiently adequate description of the system in terms of the postNewtonian formalism blanchet . The great accuracy of the post Newtonian approximation of the phase, about a cycle for a wave train cycles long, renders it amenable for matched filtering analysis arun . Inspiraling binaries are astrophysically important, because they will not only carry detailed information about the binary system, but also general relativistic deviations from Newtonian gravity in their orbit can experimentally be measured binary1 ; binary2 . The best available estimates suggest that at false alarm probability the expected number of neutron star(NS)NS binary coalescence seen per year by ground based interferometers is for initial detectors and for advanced detectors kalogera . In recent years, a number of ground based detectors are taking quality science data and are collaborating together, thus the time is ripe to consider analysis of network data for the detection of inspiraling binaries. The advantages of multidetector search for the binary inspiral is that, not only does it improve the confidence of detection, but it also provides information about the direction and polarization state of the source.
Two strategies currently exist in searching for inspiraling binary sources with a network of detectors: the coherent and the coincident. The coherent strategy involves combining data from different detectors phase coherently, appropriately correcting for timedelays and polarization phases and obtaining a single statistic for the full network, that is optimized in the maximum likelihood sense. On the other hand, the coincident strategy matches the candidate event lists of individual detectors for consistency of the estimated parameters of the GW signal. A coincidence search with real data has been carried out by several group pastcoin and data analysis on S5 data of the LIGO detectors has recently been performed LIGOCBC .
There is a long standing debate as to which strategy performs better. In an earlier paper himan , hereafter referred to as paperI, we compared the performances for the simple case of coaligned detectors located in the same place. The situation of two coaligned detectors located at the same place with correlated noise was first considered in paperI and improved in tagoshi , is applicable in the case of existing detector pairs H1H2. The coincident strategy has the advantage of reduction of false alarm, however this is at the price of reduced detection efficiency. On the other hand, in the coherent strategy, the false alarm rate is not reduced, the sensitivity is enhanced which in turn results in higher detection efficiency. Which of these two competing effects wins was determined by looking at the detection efficiencies of the two strategies at the same false alarm rate. Receiver operating characteristics (ROC) curve, which is the plot of detection efficiency versus false alarm rate, was drawn for both the strategies and from those curves it was inferred that for the viable false alarm regime the coherent strategy performs much better than the coincident strategy.
In this paper we consider the general case of widely separated pairs of detectors. Since the detectors are situated on the globe and have their arms lying parallel to the Earth’s surface, they necessarily have different orientations; we however consider the general problem of detectors with arbitrary locations and orientations, since this does not greatly add to the mathematical complexity of the problem. The coherent statistic for nonaligned detectors is completely different from that of the aligned case PDB . We cannot extrapolate the results of paperI to incorporate the more general case that we consider here. We compare the performance of the two strategies for the generalized case by obtaining the relevant ROC curves. We then investigate an enhanced coincidence strategy which is basically an improvement on the simple coincidence strategy for it is seen that when the detectors are not aligned, the performance of the simple coincidence strategy is very poor. In enhanced coincidence, although the two detectors are considered in isolation and the candidate event lists compared for consistency in the estimated signal parameters, the two statistics from individual detectors are added in quadrature and a single threshold is placed upon the resulting statistic. This strategy, for the case of two detectors only, renders a statistic identical in form to that of the coherent strategy. The details of this strategy are given in section III B.2. It should be noted that it is different from the coherent strategy.
The plan of the paper is as follows. In section II, we discuss the different coordinate systems required to describe the problem, the signal and the response of the detectors and the network. In section III, we discuss the false alarm and detection probabilities and derive the analytical formulae for the coincidence and coherent strategies. In section IV we perform simulations and then use these to plot the ROC curves for different detector pairs. We first briefly present the simulation method and then we determine the parameter windows required for coincidence analysis. We perform simulations for arbitrarily oriented detector pairs and obtain the false alarm and detection probabilities for each pair and use these results to plot the ROC curves. In section V, we summarize our results and discuss future directions.
Ii The signal and the response
In this section we discuss the GW signal and the response of individual detectors as well as of the network and introduce a normalization scheme and notations. We follow the conventions and notations of PDB . In order to make the paper self contained we briefly review the formalism. It provides us with an efficient framework for the analysis here in the context of widely separated detectors with arbitrary orientations.
ii.1 Reference frames and Euler angles
To understand the response to the GW signal of the detectors and the network, the first step is to identify the different coordinate systems naturally associated with the problem and the interrelations between them. The frames are related via rotations described by Euler angles which then appear in the network response in the form of beam pattern functions.
Wave frame  : The gravitational wave travels along the positive direction and and denote the axes of polarization  form a righthanded Cartesian coordinate system.
Frame of detector  : This is the frame of the detector having its origin at the intersection of the arms of the detector. The arms of the detector lie in the plane, which is the plane tangent to the surface of the Earth with the axis bisecting the angle between the two arms and is chosen such that the frame forms a righthanded coordinate system with the axis pointing radially out of the surface of the Earth. Note that the Earth is assumed to be a sphere for the purposes of this analysis.
Earth frame  : This is the Earth frame with its origin at the centre of the Earth. The axis points due north, the axis points to the intersection of the equator and the Greenwich meridian and the axis is chosen such that the coordinate system is right handed. This frame is usually chosen as the fiducial frame of reference, with respect to which the orientations and locations of each detector can be specified. In table 1 we specify the locations and orientations only of those detectors which we will use in our analysis. This is clearly not an exhaustive list of groundbased detectors existing or planned and the results obtained in this paper can also be obtained similarly for other detectors.
Detector  latitude()  longitude()  xarm()  yarm 

H1,H2  N  E  
L1  N  E  
V1  N  E  
K1  N  E 
Let be the Euler angles through which the Earth frame must be rotated in order to align with the wave frame. The angles and are closely related to spherical polar coordinates and is the polarization angle. We opt for the Goldstein convention Goldstein . The Euler angles that rotate the Earth frame to the frame of the th detector are then .
ii.2 Signal at a detector
Having described the coordinate system, we write down the response of the th detector to the GW signal in the Fourier domain. We adopt the following convention for the Fourier Transform,
(1) 
as in paperI.
In the stationary phase approximation the spinless, restricted postNewtonian inspiral signal is given by,
(2)  
where and are respectively the coalescence time and the coalescence phase of the binary. is the extended antenna pattern function which depends on seven angles  two angles specifying the location of the source, two angles specifying the orientation of the source and three Euler angles specifying the orientation of the detector with respect to the wave frame.
(3) 
where the are the usual antenna pattern functions depending on PDB . Note that is a complex quantity. Thus the phase of , say , where, represents the polarization phase and plays an important role in coherent detection. On the other hand we observe that simply adds on to the coalescence phase and therefore for the purposes of matched filtering in a single detector can be absorbed in . In the coincidence strategy each detector is treated separately, just like a single detector, and therefore, for matched filtering when using coincidence, can be absorbed in .
The factor depends on the distance to the binaries, the total mass and the reduced mass ratio , where and are the individual masses of the binary. In the units of it is given by,
(4) 
Instead of the total mass and the chirp mass of the binary system, it is customary to use the dynamical parameters and for in these parameters the template placing is approximately uniform. These parameters are defined as:
(5) 
where is the fiducial frequency usually chosen to be the seismic cutoff frequency  the lowest frequency of the detection bandwidth.
The phase of the signal in the th detector frame relates to the phase in the fiducial frame as,
(6) 
where is the timedelay between the th detector and the fiducial detector. The fiducial detector may be taken to coincide with the origin of the Earth frame, that is, a detector situated at the centre of the Earth or it could be chosen as one of the detectors in the network. For the phase in the fiducial frame we adopt the same 3PN formula we took in paperI (equations (2.7) and (2.8) in paperI) given by arun .
ii.3 The matched filtering paradigm for a network of detectors
By virtue of (2), we can represent the th template for the th detector as,
(7)  
where,
(8) 
We require that both the templates and have have unit norm; i.e. the scalar products . The scalar product of two real functions and for the th detector is defined as,
(9) 
where, we use the Hermitian property of Fourier transforms of real functions. is the one sided power spectral density (PSD) of the noise. Although in this paper we do not consider the case of different PSDs for different detectors, we still keep the discussion general so that it could also be used for the general case of different PSDs. This does not complicate the discussion much. It also follows from this definition that when the templates are computed using the stationary approximation, , i.e., they are orthonormal. Then, it is evident that is the amplitude of the waveform, i.e., . is the lower cutoff frequency which is normally taken to be the seismic cutoff  40 Hz for initial LIGO  and is the the upper cutoff frequency usually taken to be about 1 kHz when the signal power for inspirals within the usual mass range, say, and initial LIGO PSD, falls of below a fraction of a percent. In most of the cases we have taken the noise PSD of the detectors to be identical and the index can be dropped in Eq. (9). However, since for one case we consider different noise PSDs, we keep the subscript in this discussion to retain the generic nature.
The statistic for the th detector is,
(10) 
which is to be compared with the threshold.
Instead of two templates and , we find it more convenient to use a single complex template which combines the two together:
(11) 
The normalization factor is chosen such that . It is given by,
(12) 
In terms of , the amplitude of the waveform is .
In addition to the above normalization scheme followed for the construction of the template bank, particularly for coherent detection, where the network is treated as a whole, we also need the concept of network normalization. The total energy accessible to a network of detectors is a scalar and is given by,
(13)  
The quantity is the norm of in . The quantity has the significance of the maximum possible energy accessible by the network i.e., the energy accessible when all the detectors are optimally oriented; so denotes the ratio of the total energy received by the system to the maximum possible energy accessible.
This suggests the definition of the networknormalised signal to be
(14) 
where,
(15) 
The network vector lies in and has unit norm. This is the vector of polarization phases, which effectively ‘brings all the detectors to the same orientation’.
For the two detector network, the data consists of two data trains, where data is taken in the time interval . Assuming additive noise in each detector we have:
(16) 
The noise random variables satisfy the statistical property:
(17) 
where, the angular brackets denote ensemble average. In this paper we take the noise in the two detectors to be uncorrelated, i.e,
(18) 
which is not unjustified if the detectors happen to be geographically separated. For the sake of simplicity of the analysis, we assume the noise to be both stationary and Gaussian. This is assumed in the analysis in section III and for the simulations in section IV. However, analysis of real data suggests it is neither  the SNRs and the are correlated TAMA.
We conclude this section by introducing the definition of the complex correlations which will be particularly needed for construction of the coherent statistic. We retain the notation and definition as in paperI which was originally introduced in PDB . The complex conjugate of is denoted by . It is given by:
(19) 
where, and are the real and imaginary parts of ; they are obtained by taking the scalar products of the data with and respectively; that is,
(20) 
Iii False alarm rate and detection probability
Since in this paper we will be considering the case of two geographically separated detectors, we will consider pairs among the detectors H1, H2, L1, Virgo and LCGT, except for the pair H1 and H2 which happen to be in the same location. In this section we will derive and present analytical formulae for the false alarm rate and detection efficiency for the coherent and coincidence strategies. In coincidence detection we consider two sub categories, one of straight forward coincidence which we call naive coincidence and a more sophisticated coincidence strategy which we call enhanced coincidence. We therefore organise our discussion as follows:
 A.

Coherent detection
 B.

Coincidence detection
 1.

Naive coincidence detection
 2.

Enhanced coincidence detection
In the analytical formulae that we derive, we do not include all procedural steps of the numerical simulation; instead, we take a simplified route. The derived formulae are thus slightly different from those obtained from the results of the simulation. Nevertheless, the analytic formulae provide us guidelines by giving us functional forms for the false alarm and detection probabilities in terms of parameters which then can be determined from simulations. Even in such a simplified situation, as encountered in paperI, where we deal with aligned detectors in the same location, the formulae depend on the unknown parameters  number of statistically independent templates and number of statistically independent templates within the error window. These quantities are difficult, if not impossible, to evaluate analytically; we therefore determine them through simulations. We find here in the case of geographically separated detectors, the quantity is significantly different from that in paperI, because here the timedelay window becomes larger by the maximum timedelay between the two detectors. In this section, we obtain the relevant formulae in which the parameters and appear implicitly. The situation will be clear from the sections which follow.
iii.1 Coherent detection
Coherent detection involves combining data streams in a phase coherent manner so as to effectively construct a single, more sensitive detector. The maximum likelihood network statistic for two arbitrarily oriented geographically separated detectors has already been found. For the case of two arbitrarily oriented detectors the network statistic is given by PDB :
(21)  
where is the complex correlation of the th detector (=1,2). ^{1}^{1}1Note that in PDB the square root of is used. But here we use because it has the statistics for Gaussian noise which has a simpler mathematical expression and thus easier to implement. This is quite different from the coherent statistics for two coaligned detectors as it does not contain the terms involving crosscorrelation between the two detectors. So, the false alarm rate and the detection efficiency changes considerably even for slightly nonaligned detectors. The comparison between the two strategies therefore must be drawn separately for nonaligned detectors, for it is not only a more general situation, it is different altogether. In mohanty the two detector paradox regarding this abrupt change of detection statistic has been discussed and the improvement of the coherent strategy by suitably incorporating contributions from cross correlation terms has been advocated. We, however, restrict ourselves to the formalism of coherent detection as in PDB . Any improvement to this basic formalism will enhance the performance of the coherent search and will further strengthen our results obtained in section IV.
Secondly, the correlations in the two detectors are computed at the same mass parameters , and at timelags which differ at most by the light travel time between the detectors, where is the distance between the geographically separated detectors. This is matched filtering in which the network template is matched to the network data. The statistic is the maximum taken over the permitted timelags and polarization phases which make up the vector . We will see later that for enhanced coincidence, the same expression for the statistic appears, but the correlations in the two detectors are permitted to be evaluated at different points in the parameter space, constrained by a certain window or error box (which will be specified later in the text). We therefore note that this is not matched filtering.
To obtain the false alarm rate and the detection efficiency, first, we concentrate on a single template. When there is no signal, are Gaussian random variables with zero mean and unit variance. So, the probability distribution of LLR can be shown to be
(22) 
The rate of false alarm for a threshold of is,
(23)  
We argued in paperI that when we have closely packed correlated templates, we can treat the template bank effectively as having statistically independent templates. is smaller than the actual number of total templates in the bank  it will turn out that is substantially smaller. We will also make a simplifying assumption that the noise PSDs of the detectors are identical. With this assumption, the template placement is identical and so also is . In which case, for the entire template bank, the false alarm rate is,
(24) 
On the other hand, when, there is a signal of amplitude in the network data, the expectation value of the squared statistic is and the probability distribution of the statistic is:
(25) 
where is the modified Bessel function of the first kind. False dismissal occurs when in spite of the presence of the signal, the statistic falls below the threshold . The false dismissal probability is given by,
(26)  
The detection efficiency or detection probability is then just .
iii.2 Coincidence detection
In coincidence detection, the two detectors are treated essentially in isolation. Separate lists of candidate events are prepared; a candidate event occurs in a given detector (where ), when the statistic computed for the detector crosses the threshold set for the detector . This procedure produces two event lists, each for one detector. In absence of prior knowledge of the signal, we may choose the same threshold for both detectors. The next step involves matching the lists of candidate events and obtaining pairs of events. The matching is performed by ascertaining whether the estimated parameters for a pair of candidate events, each event chosen from a separate event list, lie in a predetermined parameter window. For a real astrophysical event, the estimated signal parameters must be consistent: ideally, the coalescence times must at most differ by the light travel time , where is the distance between the detectors, and the dynamical parameters must be identical. This ideal situation may only be realised in the limit of infinite SNR. However, for realistic SNRs, as are to be expected from astrophysical considerations, these constraints must be made less stringent, because the presence of noise introduces error in each parameter. The estimated parameters therefore will differ from their actual values; they must lie within an errorwindow . The determination of the size of this window is pivotal to this strategy.
The window is determined as follows. Consider first the parameter . For two geographically separated detectors, separated by distance , the maximum timedelay is . The estimated values of in the two detectors can differ in addition to from errors due to noise. We denote the error boxes in due to noise by and in detectors 1 and 2 respectively. The error boxes depend on and for the respective events and also on the probability we can tolerate in losing an event. We choose the probability of not losing the event to be 99 for each of the three parameters, giving a final probability of not losing an event to be for the error window. The error box is determined numerically by carrying out simulations (see next section). We assume that the errors in the two detectors are independent and so the total error is then realised as a quadratic sum. We have:
(27) 
Similar considerations hold for the dynamical parameters and their corresponding error boxes:
(28) 
and
(29) 
Let be the differences in the estimated parameters for the candidate pair of events corresponding to the parameters respectively, then we say that the parameters match if,
(30) 
Note that the dimensions of the window are symmetric under the interchange of the detector labellings and . This is important for consistency: if a trigger in detector has a window which includes a trigger in detector , then the trigger in detector has a window of the same size, albeit translated, which then includes trigger . Let be the number of templates in the window now determined by both and . Note that because of the symmetry in definition of the window size, we have .
Recently, a geometrical method for determining using Fisher information matrix has been proposed which uses ellipsoidal windows instead of a rectangular ones robinson . This scheme takes into consideration the correlation between the parameters which then reduces the false alarm rate. Incorporating ellipsoidal windows would have the effect of shifting the ROC curve for coincidence strategy a little to the left. We may estimate by how much the curve would shift. From the figure 1 shown in himan of the ellipse in the plane describing a template one can see that the area of the ellipse is about a quarter of the corresponding rectangle. However, for the parameter the window must take into account the timedelay for geographically separated detectors. This is done by the translating the error ellipsoid (due to noise only) around a trigger in one detector by the amount and check whether the so translated ellipsoid intersects the error ellipsoid of the second detector. If is reasonably larger than the errors in only due to noise, then there is almost no advantage to be gained in this parameter in using ellipsoidal windows. Thus the false alarm may be reduced by about a factor of 5 which is on the logarithmic scale (the logarithm is to base 10). Although the procedure would produce better results, for simplicity of implementation we perform the conventional coincidence search with rectangular windows and introduce a fudge factor in which takes into account the effect of using ellipsoidal windows in reducing the false alarm.
iii.2.1 Naive coincidence
The procedure for implementing naive coincidence for two arbitrarily located and oriented detectors is as follows:

Choose the same threshold for the two detectors. Let be the individual statistics of the two detectors, prepare two candidate event lists such that .

Look for pairs of candidate events, each candidate event coming from a different list, such that the sets of estimated parameters match, that is, the condition (30) is satisfied for the two events.
If the above requirements are satisfied, announce detection. We will now use these conditions to obtain expressions for the false alarm rate and false dismissal.
In paper I we have shown that for a network of two detectors, the expression for the probability of false alarm was obtained assuming a fixed window size dependent only on the threshold . Here we have a variable window size now depending on the SNRs (or equivalently ) of the two events and hence the derivation is more involved. However, we normally need the result at high value of the threshold in which case the expression for the false alarm probability assumes a simple form.
Let the statistics cross the threshold . The probability that lies in the infinitismal interval of size for some given template is . Now this is the approximation we make: we assume that there is exactly one false alarm in detector 2 lying within the window; in general we could have more than one false alarm in the window, but if we assume a high value of the threshold , then it is unlikely that one has more than one false alarm occurring within the window and thus this probability can be neglected (usually, in the literature the probability of at least one false alarm is calculated, but finally it is usually approximated to the probability of exactly one false alarm; here instead we directly compute this probability). Thus in this approximation, the probability of lying in an interval for the templates in the window is . This probability must be multiplied by the probability to obtain the probability in the rectangle and finally integrated from to . Thus for the probability density we obtain the expression:
(31) 
The false alarm probability is then given by integrating this probability density over the acceptable region. Thus,
(32) 
For high threshold and also using the fact that falls off rapidly with increasing , it is easily seen that this expression approximates to,
(33) 
As remarked before, depends crucially on the threshold , because the size of the error window strongly depends upon the SNR as suggested by both Fisher information matrix considerations as well as by the simulations we perform. It also depends weakly on the location of the signal in the parameter space. However, we choose to ignore this weak dependence and treat as a function of the threshold only as in paperI. Exact value of must lie between zero and the total number of templates within the parameter window. We choose to be the total number of templates within the parameter window following the practice adopted in paperI and such a choice leads to good agreement between theoretical estimates and numerical simulations.
We now turn to the detection probability. When a signal of network amplitude falls on the detectors, the expected value of the statistic in each detector will be different because of differing orientations of the detectors. This information is encoded in the extended antenna pattern functions which depend on the detector orientations. Since the noise in the detectors is uncorrelated, the detection efficiency is just the product of the detection efficiencies of the individual detectors:
(34) 
where, are the detection efficiencies of the individual detectors. In earlier literature, these have been calculated and here we merely state the result:
(35)  
Note that each is proportional to the extended antenna pattern functions respectively. The presence of the individual functions in the expression of detection efficiency demonstrates the weak point in this type of coincidence detection. In general, the response of the two detectors will be different because of different orientations, but since one has no a priori knowledge of the direction of the source, one sets the same threshold for the two detectors which leads to a lot of false dismissal. In fact, for the case when the configuration is such that the response of one detector is close to zero for some particular direction to the source, actual candidate events would be missed even if the noise level is low.
The main reason behind the poor performance of the naive coincidence strategy has its roots in the sky coverage. For a particular source at a fixed distance, the intrinsic SNR of the GW emitted depends crucially on the location and orientation of the source (the angles and ). For a single detector, and for a given threshold (minimum SNR), such a source would be ”visible” for certain directions and orientations. When the detectors are aligned the sky coverage of the coincident detector is the same as any one of the single detectors although the coincident detector still performs better than the single detector only because of low false alarm rate. For the nonaligned case, since different parts of the sky are covered by each detector, for naive coincidence, it is the intersection of the sky coverage of the two individual detectors which decides detection. This crucially hampers the naive coincidence strategy. Just to get a quantitative idea of the performance of this strategy, we consider a binary source at a distance 15 Mpc. The mass of each individual star is 1.4. The binary is taken to be optimally oriented, i.e., . The location of the source in the sky is varied and the intrinsic SNR is calculated for L1, VIRGO orientations and LIGO I design sensitivity. In a simplistic scheme, if the intrinsic SNR in a given detector is above 7 for a particular direction, we take that direction of the sky as covered. We find that the coverage of a single detector is while in naive coincidence the detectors cover only of the sky. The corresponding number for the coherent strategy is of sky coverage.
In order to counter the disadvantage of low sky coverage of this simplistic strategy, we consider a more sophisticated coincidence strategy which overcomes this problem. We describe this strategy in the next subsection.
iii.2.2 Enhanced coincidence detection
As we have pointed out for nonaligned detectors, the performance of coincidence strategy is poor because of low sky coverage. We adopt a more sophisticated strategy which overcomes this problem. The procedure is as follows:

Choose a low threshold and if are the individual statistics of the two detectors, prepare two candidate event lists such that .

Look for a pair of candidate events, the events coming from separate lists, such that the sets of estimated parameters match within the errorwindow; the window has already been defined in Eq. (30).

Choose the final (high) threshold and construct the final statistic and register detection if .
This strategy in essence leads to increased detection efficiency. Further, this strategy also involves preparing separate candidate lists and matching candidate events, but for the case of two detectors, the detection statistic is of the same form as that of the coherent strategy. The detection statistic is,
(36)  
Note that although this statistic has the same form as in the coherent case, as remarked before, it is actually different: the mass parameters here do not have to be the same for the two detectors, but are only constrained to lie in the window described above. Also in , the error box is somewhat larger than in the coherent case, where the timelag must differ at most by ; in addition to there is error introduced by noise which increases the size of the error box.
We first obtain the false alarm probability. We first note that the first two steps in the procedure are identical to those of naive coincidence except with the final threshold for naive coincidence replaced by the low threshold applied in the first step. Therefore, identical derivation follows, and hence the probability density here is the same as before and is given by Eq. (31). However, the integration region is different essentially because of the third step. Define the regions (or events) in the plane:
Let the region be the intersection of all the three regions. Note that since is chosen greater than we have as a proper subset of . The region ‘cut out’ from is an isosceles right angled triangle, we call . Thus where denotes disjoint union of the sets. The false alarm probability is given by integrating the probability density given by Eq. (31) over the region . Thus:
(38) 
We may be able to approximate this expression by writing which leads to:
(39) 
If again is a rapidly decreasing function of both its arguments, we can again write:
(40) 
On the other hand, becomes
(41)  
Thus, we have
(42) 
We now turn to the detection probability. We assume a signal in the detectors with amplitudes and true signal parameters , where . The true signal parameters will be the same for the two detectors while the parameter will differ by the light travel time depending on the direction of the wave. Thus must be interpreted according to the context. However, because of noise in the detectors, the detector statistics and in general will be evaluated at points in the parameter space different from , say at, and , which will lie close to the true signal parameters , if the signal amplitude is sufficiently high. We now compute the probability density function (pdf) for the statistic in the limit of high amplitude for detector 1. Similar arguments will hold for the pdf for detector 2. In the limit of high amplitude we may assume the errors in the estimated parameters in detector 1 to be Gaussian distributed with a covariance matrix . Also the amplitude parameter of the signal will be be reduced by the ambiguity function , where we assume that the ambiguity function only depends on the difference in signal parameters . The ambiguity function is normalised to which is also its maximum value. (Since we have assumed identical noise PSD for the two detectors, the ambiguity functions for the two detectors are the same and accordingly we have omitted the detector index for the ambiguity function.) The joint pdf for the statistic and signal parameters is given by,
(43) 
where,
(44) 
and,
(45)  
A similar expression obtains for detector 2 with the index 1 replaced by the index 2.
We now note that the covariance matrix scales as the inverse of the square of the amplitude or and in the limit of , the function tends to a delta function centered at the signal parameters . Now we integrate over the signal parameters or marginalise over them. Assuming high amplitude , we expand up to the second order in around , and the result after integration is:
(46) 
where are constants, namely, the second derivatives of with respect to evaluated at . This means that the pdf for detector 1 is essentially given by the first term plus corrections of order . These corrections are small if . Or writing this result explicitly:
(47) 
For the detector 2 we obtain an identical result with replaced by on the RHS. Finally, the pdf for the two detectors is obtained by simply multiplying the pdfs of the individual detectors since the statistics and are independent. Thus the detection probability or efficiency is obtained by integrating the product of the pdfs over the region and is given by,
(48) 
In this strategy the sky coverage is better than the naive coincidence because is chosen much smaller than the threshold chosen in naive coincidence strategy. Thus we expect the performance to be better than the naive coincidence. In the next section we compare the performances of all the strategies.
Iv Comparing performances
In the previous section we have derived analytical formulae for plotting the ROC curves. However, the quantities, and , are unknown and we need to determine them by numerical simulation before we can actually plot the curves. On the other hand, as we shall see below, the false alarm probability and the detection efficiency from the numerical simulation do not necessary agree with theoretical formulae perfectly due to various practical reasons. Thus, we have to resort to numerical simulations for evaluating quantitatively the performances of each detection strategy. Nevertheless, the analytical formulae provide guidance towards deriving useful fitting formulae from the simulation which are then used to plot the ROC curves. In paperI we have already discussed this matter in detail. However, we include a description for making the paper self contained.
iv.1 Simulation method
In this paper, we take the noise PSDs of all the detectors to be identical. We use the design sensitivity of initial LIGO DIS for all the detectors.
(49)  
where, , Hz, Hz, and Hz. For computing the SNR (scalar product) we set . We take the upper cutoff frequency to be 1024 Hz and the data are sampled at 2048 Hz in accordance with the Nyquist theorem. This choice of the noise PSD is simply to save the computation time of the simulation. If we take the PSD of advanced LIGO, the lower cut off frequency becomes around Hz. The length of the signal in the bandwidth becomes longer which makes the computation time of the matched filtering longer. Even if we were to adopt the PSD of advanced LIGO, we do not expect much qualitative change in the results. We consider 4 detector pairs listed in Table 2.
Pair  Light travel time [ms] 

L1V1  26.42 
L1H1  10.02 
L1K1  32.47 
V1K1  29.18 
Using 3PN restricted waveform, we place the templates 25 ms apart in the direction and 19 ms apart in the direction. This corresponds to a maximum mismatch of . We have taken a rectangle in the parameter space with 25 templates along each of the directions, totally giving us 625 templates in the parameters. We take a data train 32 seconds long and sample it at 2048 Hz. We generate 32 seconds Gaussian data and repeat the coherent and coincident searches for number of times. We test for various values of , and we find that is sufficient to give reliable estimates of the false alarm rate. Therefore the total data length of the simulation is seconds.
In the coherent searches, we first compute the statistics from each detector, and (i.e., Eq.(21)), which are functions of and . We compute and take the maximum over the possible time delay and over the mass template space. We obtain the following statistic,
where is the distance between the detectors and the speed of light. Since the coalescence time is sampled at 2048 Hz, the samples (sampled at the same rate as ) are correlated. In order to remove the correlation, following the procedure in Paper I, we divide the data train into ms short data trains, where the th short train starts at time . In each short data train, we take the maximum of ,
(51) 
defines the trigger list of the simulation.
In the coincident analysis, we first compute the statistics of each detector and take the maximum over the mass template space. As in the coherent analysis, we divide the data train into ms short data trains, and take the maximum of the statistics in each data train. We obtain the triggers from each detector defined as
(52)  
(53) 
In the naive coincidence detection, for each , we select a maximum of which satisfies the coincident conditions, namely, Eq.(30) and secondly . If such a exists, the pair define the trigger list of the naive coincidence detection. From this trigger list, we compute a enhanced coincident statistic defined as
(54) 
iv.2 Coincidence windows
To determine the size of windows for coincident analysis, one way is to use the Fisher information matrix for this purpose. However, it has been shown that at low SNR, SNR , the Fisher information matrix grossly underestimates the size of the window BSD . So, we determine empirically the number of templates falling within the errorwindow such that the signal is detected of the time. Since we have three parameters, the coalescence time and the dynamical parameters and , on the average, the fraction of signals detected will be . The simulations are carried out for a single detector for SNR() ranging from 4 to 25. Determination of at low SNR is difficult because the signal tends to get overwhelmed by false alarm triggers  in most cases, instead of the signal we pick up false alarms. Since the false alarm triggers occur anywhere in the parameter space at random, it becomes impossible to decide on the window. So, we restrict ourselves only to those signals for which . For SNR, we take the same window size as SNR, since the parameter estimation accuracy becomes nearly constant at large SNR due to the finite mesh size of the mass template space and the finite sampling rate. This constant window size at SNR does not affect our analysis very much because we will use the false alarm probability and the detection probability corresponding . The parameter estimation errors from the simulation are summarized in Table 3. It is useful to deduce a fitting formula of Table 3. We observe that the estimated error scales roughly inversely as the SNR which we denote by . Thus, we may write: