Modeling impulse response using Empirical Orthogonal Functions

Transcription

1 Modeling impulse response using Empirical Orthogonal Functions Trond Jenserud, Roald Otnes, Paul van Walree 1 1 Forsvarets forskningsinstitutt, P.O. Box 115, NO-3191 Horten, Norway, {trond.jenserud,roald.otnes,paul.vanwalree}@ffi.no

2 Modeling impulse response using Empirical Orthogonal Functions Trond Jenserud, Roald Otnes, Paul van Walree 1 1 Forsvarets forskningsinstitutt, P.O. Box 115, NO-3191 Horten, Norway, {trond.jenserud,roald.otnes,paul.vanwalree}@ffi.no There is a growing need for acoustic channel simulators. Among the reasons for this is the increased interest in underwater acoustic communications, but also the fact that measuring the characteristics of the underwater channel is very expensive. A channel simulator provides access to realistic channels, and allows us to compare algorithms and modulations under identical conditions. Channel simulation could be based on modeling or on measured channels. Model based simulation is difficult, but allows simulation under different conditions than measured. Horizontal underwater channels are characterized by time-varying multipath propagation. For simulations it is convenient to split the channel response into a deterministic and a random, time-variant part. In this paper we will consider the modeling of the deterministic part of the channel response. 1 Introduction The capability to model the impulse response of the acoustic channel is central to a model-based channel simulator. The impulse response (IR) can be considered to consist of a static and a time-varying part. The static part is the average impulse response including the mean surface loss due to waves. The time-varying part of the impulse response should in principle include all time-varying mechanisms (surface waves, internal waves, turbulence etc.), but in practice the moving sea surface will be the dominant effect in the frequency range of interest to acoustic communication. The modeling of the static impulse response (or power delay profile) is the topic for this paper. The experience from earlier measurements was that modeling the impulse response is challenging. Even for seemingly simple channels, such as waveguide propagation, it was difficult to mimic the main features of a measured IR. Typically, the modeled IR consisted of a relatively small number of arivals while the energy of the true IR was much more evenly distributed in delay. The paper starts with a brief description of the method of Empirical Ortogonal Functions, which is the method employed to analyze and synthesize the variability of sound speed fields. We then show that IR computations (at 5 khz) may be sensitive to even small details in the sound speed structure. The high sensitivity of the IR to oceanographic variability motivates a stochastic approach to IR computations, utilizing the variability of the measured SSPs. The method requires multiple measurements of the SSP. 2 Representation of sound speed profiles by Empirical Orthogonal Functions The method of Empirical Orthogonal Functions (EOFs) is useful for analyzing the spatial and temporal variability of geophysical fields [1]. Our application is the analysis of sound speed profiles. Assume we have multiple measurements (in time) of a sound speed profile, v(t, z). Then the sound speed profile can be represented by the following eigenfunction expansion v(t, z) = v(z) + N β k (t)c k (z) (1) k=1 where v(z) is the average sound speed profile, c k (z) are the EOFs and β k (t) are the corresponding (time varying) expansion coefficients. The EOFs are the eigenvectors of the (sample) covariance matrix of the data v(t, z). The expansion coefficients, β k, are the projections of the data on the respective EOFs. The EOFs describe the spatial patterns of variability, while the expansion coefficients give their time variation. The sound speed profiles can be reconstructed from the EOFs and the expansion coefficients according to Eq. (1). The strength of the EOF method is that usually very few EOFs is required to give a sufficiently good representation. The amount of variance explained by each EOF is determined by the magnitude of the corresponding eigenvalue λ k, i.e. the relative contribution from EOF number k is λ k / N i=1 λ i. An EOF representation requires multiple measurements of the sound speed profile. A common use of EOFs is to reconstruct a cleaner or smoother version of the data by truncating the sum in Eq. (1). The method of EOFs is also a useful tool for effective parametrization of sound speed profile or other oceanographic fields for inversion. We first apply the EOF method to SSPs measured in Vestfjorden in October The eigenvalues and eigenfunctions of the EOF expansion is shown in Fig. 1. The figure clearly shows the rapid decay of the eigenvalues indicating that only a few coefficients are necessary to reconstruct the profile with good accuracy. Note that the mean profile is subtracted from the data, which means that the EOFs represent the variability of the SSPs. The ability of the EOF method to reconstruct SSPs from few EOFs is demonstrated in Fig. 2, which shows the original profiles (left panel) and profiles reconstructed from the five largest EOFs (right panel).

3 35 Eigenvalues First eigenvectors plain Range of profiles (km) Figure 1: Eigenvalues and first three eigenfunctions of the EOF expansion. Figure 3: Range-dependent SSP from Vestfjorden, October Original ssp 5 From 5 EOFs 3 Sensitivity of the impulse response to oceanographic variability Figure 2: Original and reconstructed sound speed profiles using the five first EOFs. The reconstructed SSPs are slightly smoothed versions of the original profiles. Generating random SSPs In some applications it would be useful to generate a large number of SSPs with properties similar to some measured profiles. This could be achieved by reconstructing SSPs from EOFs using random expansion coefficients instead of the expansion coefficients obtained from data. The random coefficients need to have the same statistical properties as the original coefficients, and could be generated by simply drawing samples from a normal distribution with the correct variance or more correctly by drawing random samples with the correct expected spectrum (the expansion coefficients are time series). Modeling the impulse response of an acoustic channel is in principle easy; with perfect knowledge of the environment any propagation model of good quality will reproduce a measured IR. However, a synoptic range-dependent sound speed field is very seldom available for the modeling. To assess the effect of imperfect knowledge of the sound speed field, two cases will be considered through a model study: the effect of using range-independent SSP (measured profile) instead of the true range-dependent sound speed field and the effect of minor perturbations on the true range dependent profile. For the true sound speed field, SSPs measured in Vestfjorden during October 1993 will be used. The sound speed field, displayed in Fig. 3, shows a strong range dependence. Fig. 4 shows the error resulting from using a range independent SSP based on a single measured profile and the mean profile respectively, instead of the true range dependent profile. The source and receiver are located at a depth of 1 m in a region of relatively small variations in this case. The figure shows that measured and modeled IR differ significantly, particularly in the most energetic part of the response. This result is not unexpected and shows that in the case of strong range dependence a range independent profile is insufficient for IR modeling. Next we consider the effect of only slight perturbations (errors) of the true range dependent sound speed field. Employing the EOF method a smoothed version of the true sound speed field is constructed, as shown in Fig. 5. Fig. 6 compares IR computed from the true sound speed field with IR computed from the smoothed version. Although the sound speed fields are very similar, the IRs are uncomfortably different.

4 Figure 4: IR for full range dependent SSP (upper), mean profile (mid) and first profile (lower). Source and receiver are at 1 m depth. Figure 6: IR for true range dependent SSP (upper) and smoothed version (lower). Source and receiver is at 1 m depth. eof Range of profiles (km) Figure 5: Original SSP (b) and SSP reconstructed by five EOFs (r). 4 Stochastic modeling of the channel impulse response The results of the previous section indicate that IR is highly sensitive to the sound speed field. In this section we will consider how well real measurements of IR could be reproduced by modeling. For this purpose data from channel impulse response measurements in the North Sea will be used. (The measurements were performed as a part of the UCAC project [2].) In the measurement the receiver was stationary and the transmitter was towed away from the receiver up to a maximum range of 36 km. The IR has been estimated by using PRBS signals and LFM chirp signals. The PRBS measurements were only done every 48 seconds while the LFM chirps were done every 16 seconds. The bandwidth occupied by Figure 7: Power delay profile measured with PRBS signal. the signals were 3-7 khz for the PRBS signals and khz for the LFM signals. Twelve SSPs were taken at the receiver location during the measurement. A single profile at 4.6 km First a single IR 1 at a range of 4.6 km is considered. Fig. 7 shows the measured IR while Fig. 8 shows the modeled IR computed from a single sound speed profile. The modeling was performed with the ray trace program LYBIN [3], using high resolution bathymetry and bottom type 1.7. The modeling included loss due to surface roughness at a wind speed of 3 m/s. The center frequency for the PRBS signal, 5 khz, was used for the modeling. Source and receiver depths were 5 m and 9 m respectively. Comparison of measured and modeled IRs shows that many of the real-world paths are 1 More precisely, a 3-s average in the form of a power delay profile.

5 Figure 8: Power delay profile modeled with Lybin, using one measured SSP. present in the model, but not all. The spread on the individual paths is also too small, resulting in an IR that is too spiky. The measured IR has a noise floor at about 22 db. In order to obtain more realistic IRs we suggest to utilize the variability of the SSPs. This can be done in several ways; one is to generate a range dependent sound speed field. Another is to average (in power) over a number of IRs computed from range-independent sound speed fields. Fig. 9 shows the result of averaging over range-independent IR computations from 12 measured SSPs. No smoothing was applied to the SSPs in this case. This computation seems to result in too many paths, and is still spiky. It is believed that range-independent modeling using very detailed SSPs can give rise to artifacts, such as narrow waveguides, which may explain the unphysical paths. In an attempt to remedy this shortcoming smoothing was applied to the SSPs using the EOF method, i.e. smoothed SSPs were generated by reconstructing the sound speed profiles using only the five first EOFs. Using smoothed SSPs for the computation resulted in a modeled IR that more closely resembled the measured IR, as shown in Fig. 1. The EOF method could also be used to generate random SSPs as discussed above. Randomly generating 25 SSPs based on the EOF representation gives a result similar to Fig. 1, but even less spiky. IR versus range Figure 9: Power delay profile modeled with Lybin, averaged over 12 measured SSPs. Having established reasonable correspondence between measured of modeled IR at a single range, we will now compare measurements and modeling at all measured ranges. The data from the sea trial apparently has very exact sampling rate and good GPS information such that it is possible to synchronize different measured IRs without using data-based criteria such as strongest arrival. Assuming a group sound velocity of 1482 m/s gives well synchronized data. Fig. 11 shows the synchronized IR measurements using LFM chirp signals while Fig. 12 shows the corresponding model results. In the modeling, averaging over 25 randomly generated SSPs has been performed. The figures show the measurement and the modeling have several main features in common, even though they are less visible in the measurement. The agreement seems to be best at ranges below 1 km. This may be due to one or more of the following reasons: a) lower SNR in the measurements at longer ranges; b) accurate impulse response modeling being increasingly difficult as range increases; c) a range-independent SSP is assumed, based on measurements near the receiver. As the range increases, this assumption becomes less accurate. Figure 1: Power delay profile modeles with Lybin, averaged over 12 measured SSPs smoothed with EOF (5 coefficients) 5 Conclusions Our capability to model the static part of the impulse response of the acoustic channel has been investigated using real data from the North Sea. It was found that using range

6 [3] E. Dombestein, A. Gjersoe, and M. Bosseng, Lybincom 6. - description of the binary interface, Technical Report, FFI (21). Figure 11: Measured PDP vs range using LFM chirp signal, RX depth = 9 m. Figure 12: Modeled PDP vs range, averaged over 25 randomly generated SSPs using EOFs. independent non-smoothed (raw) SSPs gives generally poor results. The following remedies improve the realism in the modeled results. Smoothing the SSPs appears to remove artifacts such as unphysical paths. Averaging (in power) over several models (SSPs) results in more realistic IRs. The use of SSPs generated by either the true or random EOF coefficients gives similar results. The advantage of the latter method is that any number of SSPs could be generated allowing more averaging. References [1] H. Bjornsson and S. A. Venegas, A manual for EOF and SVD analyses of climatic data, McGill University (2). [2] M. Petterson, Ucac final report, Technical Report, Saab Underwater Systems (28).