A model based system for simultaneously ... - Semantic Scholar

A Model Based System for Simultaneously Estimating Bathymetry and Sound Speed Profile Characteristics - Non-Isovelocity Simulation Results D. B. Cousins, Division Scientist,

J. H. Miller, Ph.D., Department of Ocean Engineering, University of Rhode Island, Narragansett R.1. 02882, USA [email protected] ]

BBN Technologies 127 John Clarke Rd. Middletown, RI, 02842, USA { dcousins~Z?bbn.coni) Absfract-

This paper presents further developments of an adaptive bathymetric estimation algorithm (ABE) for use with forward-looking bathymetric sonar in nun-isovelocity underwater acoustic environments. In addition to providing improved positional estimates of ocean bottom contacts in front of the host vehicle, it will automatically estimate and adapt to changes in the local sound speed profile (SSP). The ABE algorithm uses parametric SSP models of increasing complexity and tunes them “on the fly” to match the ohsewed refraction of the sonar bottom returns. The technique is an application of the Extended Kalman Filter (EKF) which fuses on-hoard navigational data of the vessel (from GPS or INS systems), multiple active forward-looking sonar returns from bottom contacts (time of arrival and angle of arrival estimates), and underwater sound propagation parameter estimates derived from an internal eigenray model. This paper shows simulation results for three scenarios using increasingly sophisticated ABE models: 1) discrete bottom sonar observations refracted by a linear sound speed gradients, 2) the same observations refracted by a 20d order gradient, and 3) tracking changes in a mean sound speed over a featureless (smooth) bottom model.

I. INTRODUCTION Traditional sonar-based bathymetric mapping relies upon accurate measurement of the Sound Speed Profile (SSP) in the area being surveyed, as errors in this measurement translate directly into erroneous bathymetry measurements. The SSP is usually measured directly using instrument (CTD) casts, a method that is quite time consuming. In order to be most effective, surveys must be able to measure the SSP in-situ and in real-time due the high variability of estuarine and coastal waters. This is not practical to do using just CTD casts. The adaptive bathymetric estimation algorithm (ABE), published previously (11 at Oceans 2000, provides a solution to the above problem by estimating and correcting for SSP characteristics on the fly. The work presented in this paper covers two extensions recently made to ABE. The first uses a discrete bottom model and increases the complexity of the SSP that can be modeled by the system. We will present simulation results of both n-linear and n2-linear SSP environments, so named for the dependency of sound speed vs. depth [2]. The second extension adapts the ABE algorithm to a different environment where the bottom is modeled as a featureless, constant slope with no discrete reference points to serve as sonar contacts (this will be hereafter referred to as the smooth bottom model). It will be seen that due to the

0-7803-7534-3/$10.0002002 IEEE

reduction of information resulting from the lack of discrete point contacts, the system cannot simultaneously estimate both sound speed and depth. However, given an initial calibration value for both mean sound speed and bottom depth (i.e. from a calibrated fathometer reading using a surveyed submerged structure), the system is able to correctly estimate depth and bottom slope while simultaneously tracking changes in the local sound speed. Again, simulation results will be presented. ABE uses an Extended Kalman Filter (EKF). A modelbased approach such as this provides several important advantages, and enables us to use the rich background of successful tracking and navigation applications. It has also been applied to estimating sound speed parameters using passive sources [3]. This discussion will now address the general EKF architecture of ABE and describe how it enables the following: Improve forward-looking bathymetric estimates (range and depth) over multiple pings. This is done much in the same manner that an EKF based tracker improves the position and velocity estimates of a sonar contactthat is by using a kinematic model of the ownship motion. Fuse dissimilar data sources such as active sonar returns (time and arrival angle measurements), navigation, “ground truth” information (kinematics), and acoustic propagation model parameters. Wis fusion is one of the reasons why the EKF has been applied so successhlly in inertial navigation systems. Estimate SSP characteristics. Embedded in the observation model of the ABE EKF is an actual model of high fiequency underwater acoustic propagation (in the form of a direct path eigenray generator). This is used to compute the expected propagation time along a ray path and the arrival angle of the ray. An easily parameterized SSP model is then chosen. The defining parameters of the profile are then augmented states which are added lo the EKF intemal state vector, and estimated along with the kinematic ownship and sonar contact model parameters. This estimation can be done even if the sound speed parameters are not directly observable and are only inferred by the effects of acoustic ray refraction. Correct noisy sonar measurements for the refraction caused by the unknown (but estimated) sound speed characteristics. This is a direct result of the predictorcorrector nature of the EKF.

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Provide online metrics for state estimation error and filter consistency (i.e. filter convergence, model mismatch and correct system operation). 11. MODELS FOR SOUND SPEED PROFILES

Three models for sound speed profile are used in this paper. .A. Slowly Varying Isovelocity. ~~

The sound c(xrz) has the following characteristics: 1) C is assumed isovelocity with depth 2) varies slowly enough over x as to be a constant for purposes o f computing observations (i.e. several ping cydes), and 3) is moheled as a perturbation around a nominal constant velocity C(Z) = c n o m +g(k).

The perturbation term g(k) is allowed to vary slowly from one ping cycle to the next in order to allow for gradual changes over time (due to crossing fronts, estuarine flows etc). This model was used in our prior work, and will be the one used in the Smooth Bottom section.

B. n-linear gradient The second model used allows the sound speed to vary linearly as a function of depth such that n(z) = C(O)/C(z) is linear:

C(4

= C(0) + gfi)

In this model the variable g = dC/dz is the parameter that is augmented into the state vector. Fig. l a shows a typical sound speed profile used for our simulations. Fig. Ib shows the slight perturbation from straight line (isovelocity) propagation, due to the fact that the raypaths

for this profile are arcs of circles. There is one issue that arises with this added complexity, and that is that given an arbitrary value for both C(0) and g there can be an infinite number of direct path eigenrays between the sonar and a bottom contact (this will be shown below as another way of saying that the EKF function h(Xk) is no longer a one to one transformation between the state and observation vectors). In order to overcome this problem, the sound speed at the sonar, C(O), is assumed to be known. This requires more hardware in the actual sonar system, but is still quite practical and easier than requiring CTD

c-"-linear gradient This profile is'slightly more complex than the previous one, but is still parameterized by only two variables C(0) andg. This profile has been used frequently in the literature [4][S] as it is easy to model and generates caustics quite readily.

C(z) = C(O){l+b(k)z}.'"

;where

b(k) = 2 g(k) / C(0).

Again, the variable g(k) is the parameter that is augmented into the state vector. We will use the values for g used by [SI. Fig. 2a shows a typical sound speed profile used for our simulations. Fig. 2b shows the perturbation from straight line (isovelocity) propagation, due to the fact that the raypaths for this profile are parabolic arcs. As with the n-linear case, C(0) must be measured at the array to eliminate an infinite number of direct path eigenray solutions. Io w _ t a n 4

~ l M

.

--sm

~ , m ~

.,OM Men

(A)

-A--(

~9w

llSP

(UV -,-I

>la6

-M

(A)

Ilea

,a90

0

Icm

AL-~ rim

m

-elm1

(B)

Fig. 1. n-linear sound speed model: (A) Sound Speed Profile as a function of depth (B) Deviation from straight line (red) raypaths for a 45" fan of beams with a 5'degree spacing (blue circular arcs).

raw

.

(B)

Fig. 2. n-linear sound speed model: (A) Typical Sound Speed Profileas a function of depth. Note this is not a realistic representation of typical downward refracting profiles at depths greater than a hundred meters. (B) Deviation from straight line (red) raypaths for a 45" fan beams with a 5' degree spacing (blue parabolic arcs).

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sonar array 81 x = 1,,=0, Conecled

Predicted

2

Actual Observations

-

0, velacity

VI

Observation

State

Model

Model slate Prcdiceon

Kalman

.

Observations

Fig. 3. The Extended Kalman Filter Algorithm: The ABE specific components are embodied in the Intemal State and Observation Models.

111. THE DISCRETE BOTTOM ABE ALGORITHM

Initial distance 2 !an, velocity = 2.5 ds,0.5 pingslsec n-linear SSP: C(0) = 1490 m i s initial gradienlg mismatch 0.0 s ~ assumed, ‘ 0.01 s’actual

:ig. 4. ABE Discrete Bottom Scenario Case 1: Two successive bottom contacts, sound speed gradient shown in Fig. 1.

A . EKF Algorithm Structure Fig. 3 shows the structure of the EKF used by ABE. The derivation of the traditional EKF can be found in many standard texts on the subject (for example see [ 6 ] ) . The EKF predicts the state of the modeled system for the next sonar observation interval by propagating the current state estimation through the lntemal State Model. Then, based upon the predicted state variables, it generates predictions of what the corresponding navigational and sonar observations should be. The difference between the predicted and the actual observations, known as the innovation, is then used to compute the Kalman Gain, which governs how the intemal model and the observations are weighted when used to correct the initial state estimation. The state variables include representation of the range and depth of the bottom contact(s), so this correction improves forward-looking bathymetric estimates by averaging over multiple sonar pings. The driving force for the convergence of the algorithm is generated by the mismatch in the estimated and actual (static) positions of the bonom contacts. The errors in contact position come from the mismatch in ray travel time and arrival angle. These arise from the mismatch in eigenrays generated with the estimated and the true sound velocity profiles. It is by this mechanism that ABE can estimate and adjust for the effects of sound speed profile characteristics. This remainder of the discussion will focus on the formulation of the Internal State Model and the Observation Model, since there is where the ABE-unique aspects lie. These models include the estimation and correction of bottom contact locations due to underwater acoustic propagation (as characterized by acoustic eigenrays).

E. Internal State (Augmented Kinematic) Model The internal state model consists of 1) a system of state space equations that describe the kinematics of how the ownship platform moves during the intervals between sonar pings, and 2) a set of state variables that encapsulate all relevant information about the system at an single point

in time. All of our simulations are done with onedimensional motion of the source sonar in the x direction, and with depth being negative down in the z direction (see Fig. 4). The definitions of the components in the model are listed in Table 1. Note that with the exception of the augmented sound speed profile parameters, this internal state model is the standard linear discrete random acceleration kinematic model used for decades in conventional tracking. The state equation for our system is given as follows: xk+,=Fxk+rvk where xk is the state vector: Xt

[W XdOlF) g@)kdk) (VI’,

F is the state transition matrix 0

0

0

0

F = T

1

0

0

0

0

0

1

0

0

0

0

0 1

0

0

0

0

1

I

0

(with T being the time period between sonar cycles), the state noise matrix: T*/2 0 0 0 T = T 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1

r is

and vt is the state driving noise vector h = [ V d k J V g ( V VI (k/ vc FI1’

which is white gaussian noise (WG) distributed as -N(O,Q) with Q being the process noise covariance matrix Q = diag[ 02 a ;

ot oc21.

The noise term Tvk accounts for the uncertainty in the system, and minor variations in the sonar platform’s course and position.

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TABLE 2.- OBSERVATION MODEL PARAMETERS FOR ABE ALLVARIABLES ARE GIVEN FOR ITERATION k

I ,- STATE MODEL PARAMETERS FOR AB€ ALLVARIABLES ARE GIVEN FOR ITERATION k The current kinematic state of the smsr platform x N I ray position TABLE

I 0:

system to converge to an estimate ofnfi/ same for estimate oft,@/

I

C. Discrete Bottom Nonlinear Observation Model The Observation Model is used to relate the state vector xk, with the observation vector (inputs) of the system. These observations (measurements) represent a "short-term integration" during which the value of the state vector is assumed to he constant. The Kalman filter can combine multiple observations types even if they are from completely dissimilar data sources. The definitions of the variables in the model are listed in Table 2. Note that for both cases simulated in this report, only a single beam was used. The observation equation takes the form:

Zk= h(Xk)+Wk where zk is the observation or measurement vector: zk= [Xcpsfi) xdolcpsfi)

Tip)

kr;fi)]'

h(xJ is a nonlinear vector transformation that will later be described in detail, and wk is the measurement WG vector Wk=

[ wxGPs(k)

W~dolcPsfi) W ~ f i )

w!&)

1'

which is distributed as -N(O,R) with R being the measurement noise covariance matrix

R

d j a g b XGPS

oxdotGps

02

eh2]'

Since the transformation between the position of the sonar contacts and the corresponding observed sonar parameters is non-linear, the EKF non-linear Observation model needs both the nonlinear vector transformation function 21,= h(xk)and its Jacobian

measurements of the corresponding kinematic state variables (or simple transforms thereof). The second component consists of the sonar observations, which are generated by an eigenray estimation algorithm that uses the state vector as inputs (i.e. all physical positions and SVP parameters). The Jacobian is not present in the observation model but is used in the EKF algorithm as the function that drives the convergence of the state estimates in the proper directions, allowing the ABE to correct sonar measurements for refraction and other sound speed effects. D. Numerical Direcf Path Eigenray computation of h(x) and H(x) Earlier versions of ABE used analytic formulas for both these functions. This approach had the advantage of rapid computation speed, but also had two major disadvantages. The first disadvantage was that the SSP used were limited to the class of those that have analytic formulas for the resulting eigenray paths. The second disadvantage was that one must derive the formulas for each entry of the Jacobian, which is rarely a straightforward task and usually requires approximations to provide functions that are numerically well behaved. We avoid these problems through the use of a numerical direct path eigenray generator, which allows for far easier implementation of different parameterized SSPs. The sonar components of h(xk)are generated numerically using the state vector as inputs. The partial derivatives of the Jacobian are also numerically evaluated by repeating the eigenray generation for each entry while individually perturbing the appropriate state input.

E. Simulation Results Case I : n-linear SSP 1) Scenario Description

Hx=ahi(xk)/ax,. The first function transforms the state vector xkrinto the observation vector, zk.This function consists of two major components. The first component are the navigation related observations that are usually direct, though noisy,

The discrete bottom ABE algorithm was simulated using the scenario presented in Fig. 4 previously, with N n time parameters enumerated in Table 3. Most of the values were chosen to reflect nominal operating conditions for sonar systems available for this study [7], while

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TABLE 3.- DISCRETE BOTIOM CASE N-LINEAR SlMULATlON PARAMETERS Initial itate vector conditions 10 xmq0 om I Initial sonar x position sonar velocity

I

ge I"

S""Ld(LL

Modelid rtatPnoisGtandard deviation (s.d.) a, 0.1 m i s Based on nominal x acceleration az I IO4 l i s I Arbitrary anificial process noise nk 1 IO4 m I Arbitrary arlificial process noise cL I I O * m I Arbitrary arlificial process noise Actual state noise s.d. used by the Simulation o ~ , . ~T, O .I d s Based on measured x acceleration om., I 0.0lis I x does not vary, mismatch to utc x pos. i s constant, mismatch to 0~ con I O.Om

I

I

1

~

1 I

I

~

7

1

I

.o,l~.-++-i-d~.L ...............

om

the above initial state vector condition XI for each Monte Carlo mn used to generate the statistics shown in Table 4. Note i s determined by final value ofx,,

-~ ... .

.A. . . . . . . . .

..................................... ...

....

............... ~ . , ,, ~ .,. ~~

...........

..........................

. .,. . .. .. ,. . .. .. .

6 [ 1 Depth deeper) I -2000 m of all contacts ( I Ox Modeled state noise standard deviations same as previous case Actual state noise standard deviation used by the simulation same as previous case Measurement noise standard deviation I 0.1 I lox bener resolution than previous case

.,.b .I.

(I

10

m

*

a

a

-,.,

Lo

L.L.-I 70

M

D)

(B)

I always the Same (2000111)

I

Fig. 8. ABE Discrete Bottom Scenario Case 2: Estimation ofgvs. time (A) first contact (B) sixth contact. Estimation converges three times as slowly as in Case I .

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I I

Estimation error

* Xdol error fluchlates about the achlal value, so maximum

I absolute error is given here.

I

TABLE 7.- MULTI-PASS PERCENT RELATIVE STATE ESTIMATION RESULTS FOR NI-LINEAR SSP: Same format as Table 6

0

, " m . 0 " 7 0 "

m /.I

(B) Fig. 9. ABE Discrete Bottom Scenario Case 1: Estimation of and vs. time (A) first contact (B) sixth contact. Initial errors in g take several pings 10 correct.

will discuss below why we do not also estimate the depth at x = 0 in the section on numerical eigenray computation. IV. THE SMOOTH BOTTOM ABE ALGORITHM A. Smooth Bortom State Model

The discrete bottom model bas been adapted to gently sloping (or flat), smooth bottoms. Such a model can be used for mud or sandy bottoms, certain portions of the deep ocean plain, or other such environments that do not have strongly localized contacts. In this formulation, the sonar uses a fixed set of steered beams, and tracks the detected bottom ensonification points. The measured points are used to generate a set of parameters characterizing the flat bottom using a least squares fit. The bottom is parameterized by z

= mfi) x

+ CO

The bottom model parameter mF). is then used as state variable and estimated along with the SSP parameters. We

B. Internal State (Augmented Kinematic) Model The state model is identical as that shown previously for the discrete bottom model, except that the state vector xk is now given by: Xk =

b(k/ xdotfi) g(k) mfi)]',

Note that the only changes between the discrete and smooth bottom models are that the smooth bottom slope parameter mfi) replaces the coordinates of the bottom contacts. F, the state transition matrix and r the state noise matrix are similar to the discrete case except they each loose the last row and column (and are now of constant size independent of the number of beams used) Variables related to m replace those for the bottom contact positions so that now vk = [ v,@) v&) v,,, fi) 1' which is distributed as -N(O,Q) with Q = diag[o,' a; am2].

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C. Smooth bottom Nonlinear Observation Model

to the current estimate of slope m and the initial calibrated depth CO The second complication arises from the weakly observable nature of the sound speed effects. Initially, the state model was formulated to estimate both m and io concurrently with g. It quickly became apparent that the EKF tends to adapt the bottom slope parameters rather than g to compensate for the errors due to sound speed mismatch Specifically, it moves the bottom depth to a point that corresponds to the measured T and the initial (erroneous) assumed sound speed parameter g=O. Given this behavior, we found that if we were able to fix the initial depth CO as well as the initial sound speed parameter g, that the system was then able to accurately estimate the bottom slope, and track g through significant changes in sound speed. Our rationalization for this selection of initialization parameters is that the system can use a fathometer reading of a well-surveyed bottom feature to calibrate the mean sound or artifact of known depth io speed correction g.

The Observation Model equations are also similar to the discrete bottom, but with the vertical wavenumber estimate of the beams omitted. Since the formulation uses a set of fixed beams, these are supplied as parameters into lhe eigenvector generation, so zk ,wk and R become Zk = [XGPsfk)

wk=

XdOtwsfi)

[ wrGP.s WxdorCPS wr

R = diag[o x~~~

Ti(k)

1'

1'

oXdotCPS2

a:]'

With the number o f rip)entries equal to the number of be'amsused. . .

D.Numerical Direct Puth Eigenruy computation of h(x) and H(x)

The numerical evaluations of h(x) and H(x) are performed in a similar manner as the discrete model. However, the introduction of fixed beam angles and the lack of discrete bottom contacts introduce complications. The first complication arises from the fact that the desired receive beam steering angles Bo, are computed at the sonar array using the current sound speed estimate ,C ,, hut the resultant beams actually point in the direction e,,, dictated by the actual sound speed C . , according to the relation

E. Simulation Results: Slowly Varying Isovelocity over a Sloping Bottom 1) Scenario Description

%,, = asin[ sin eo Wac/C,, 11 Eigenrays to .the bottom corresponding to the simulated observations are then computed using the intersections between the actual beam with launch angles and the smooth bottom with the actual slope m and initial depth b. However, the eigenrays generated by ABE to compute h(x) and H(x) are computed using C,, and Bo, and the intersection between the rays and the bottom corresponding

,e

ABE can use the Slowly Varying lsovelocity model (isovelocity over the maximum range of a ping), to track slowly varying changes in the SSP parameters, such as changes in the mean sound speed experienced as the ship moves through ocean fronts or estuarine outflows. The smooth bottom ABE algorithm was simulated using the scenario presented in Fig. IO, with run time parameters enumerated in Table 8. The calibration depth was chosen as -1000m, representing open ocean. The values of C(x) change slowly representing the transition

I I

.

1

-

fixed beams a t e 1.2.3

I

TABLE8.- SMOOTH B O l T O M SIMULATION PARAMETERS xwo rdota

R

Initial state vector conditions 10 1 Initial sonar x position 2.5 m i s Initial sonar velocity 10 m i s 1 Actual C=C(O) +g

I Om [

I

Bonom slope Modeled state noise standard deviation s.d. Based on nominal x acceleration 0.1 m i s Arbitrary artificialproeess noise Actual state noise r.d. used b the simulation

h BOUDm Ensonifications at 51.2.3 51.2.3

Initial velacity = 2.5 m i s . Initial calibration depth C0=-1000m

leovelocity SVP wilh a sound speed transition from: initial calibration at ping 0 of 15 IO ds,decreasing at ping 200 by 4 . 0 7 m i s per ping, reaching a constant 1490 m i s at ping 500. Bottomslopem= 0.017rod= I".

Fig. 10. ABE Smooth Bottom Scenario Showing typical geometry: three beams with a gently sloping bottom. Systcm was calibrated at x=O, CO(correct average sound speed computed from an object with calibrated depth).

Measurement noise s.d.

order of magnifude larger than a Carrier-

Sonar and environmental system parameters I -1000 m I Depth of calibration at x = 0 CO f I 89 kHz Modeled on AWARE system C(0) 1 i 5 O O m i s 1 CalibratedC(0) 7

NDo B,

1106

2 sec 1400

-40"-50-60"

Number of pings in run (2800 sec) Steenng angles of fixed beams (based upon estimated sound speed at array.

1

':r

Fig. 11. ARE Smooth Bottom Srrnario Estimation uf Sound Speed Currection 'Term g: A B t corrcctly tirllorr 5 rhnngcs dl mean sound spceJ. Dashed iines show ijtindard Jwiation of cstimdtc. tstiniation dcuadc.s near the 2nd of tlic mn hc&iu.c buat 1% in water rliat 1s too shallow to pnividc g o d sound ,ped estiiniltcs.

E

P .2 1

-

5I

(B)

Fig. 12. ABE Smooth Bonom Scenario Estimation of Bottom Slope ABE computes uncorrected bottom slope by doing a least squares fit of the detected bottom points along with the initial calibrated depth at x=O. The EKF direct estimation of bottom slope is far superior. from an open ocean to an area where the sound speed is influenced by estuarine outflow.

2) Single Run Characterization of Performance The system was run for a long series of 1400 pings. Fig. 11 shows that it was able to track the changes in g@)quite well. However, it required significant depth in order to do this. The dashed lines show f o ofthe estimation error. The estimation starts to degrade near the end of the run because the sonar is in water that is too shallow to provide measurable errors in q due to sound speed mismatch error. Fig, 12 show how well ABE is able to estimate the bottom slope m(k). The uncorrected values correspond to a bottom slope computed with a least squares fit and the assumption that g(k) = 0.

Fig. 13. ABE Smooth Bottom Scenario Estimation of Bottom Intercept Points.(A) error in range 6 (B) error in depth C Noisy range estimates in (A) are smoothed by the EKF, and erroneous depth estimates primarily due to sound speed mismatch are corrected even during the change in sound speed behveen times 400 s and 1000 s. Note that estimation starts to degrade after 2000 s because depth is insufficient to provide good correcting observations ( approx 4 8 0 m). Fig. 13a and b show the estimation emors in computing the bottom ensonification locations of the three beams vs. time In Fig. 13a, the 5 estimates are within 3m of the truth, and out perform the uncorrected observations by quite a large margin. Fig. 13b shows accurate depth estimates 6, even when the sound speed is in the process of changing. As noted earlier, when the depth reaches ahout -500m, the estimation of the sound speed correction term starts to degrade, resulting in equally degraded depth estimates. V. FUTURE WORK

Selected ABE algorithms described in this paper will next be driven by recorded sea-test data taken from a prototype forward-looking sonar [7].

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REFERENCES [I]

generated in a manner similar to the R-NEES. The underlying test statistic is the Normalized Innovation Squared (NIS), which uses the innovation vector and an innovation covariance matrix generated during each iteration of the EKF algorithm. Each instantiation of this statistic is chi-square distributed with No = length (23 degrees of freedom. Aggregating this statistic over N,, runs results in the R-NIS which is distributed as a chi squared with No N,, degrees of freedom. Again, a confidence test can be done to verify that the filter’s state estimation is consistent with the physical observations. Finally, the whiteness test of the innovations can be performed using the Run-average Autocorrelation Whiteness, statistic R-AW( 1) which is simply an averaged measure of the first lag autocorrelation of the innovation. When N,,,,is large, the central limit theory allows us to use a Gaussian confidence test. Time averaging over Np filter iterations (pings) may be used in place of N,, Monte Carlo ensemble averages for the computation of R-NIS and R-AW. Modifying R-NIS by time averaging produces the Time-average NIS (T-NIS) statistic, with Np replacing N,, The R-AW may be modified by using time averaging as well to produce the time-average autocorrelation whiteness test (T-AW) which is Gaussian distributed when Np. is large.

D.B. Cousins, J. H. Miller, “A Model Based System for Simultaneously Estimating Bathymetry and Sound Speed Characteristics-Simulation Results”, Oceans 2000 MTEIIEEE Proceedings pp, 213-219, September 2000.

[2] F. B. Jensen, W. A. Kuperman, M. B. Porter, H. Scbmldt, Computational Ocean Acoustics, New York: AIP Press, 1994. [3] J . V. Candy, E. J. Sullivan, ”Sound Velocity Profile Estimation: A System Theoretic Approach,” ZEEE Journal of Oceanic Engineering, Vol. 18 NO. 3: pp 240-252, July 1993 [4] M. A. Pedersen, D. F. Gordon, “Normal-Mode and Ray Theory Applied to Underwater Acoustic Conditions of Extreme Downward Refraction,” The Journal of the Acoustical Society of America, vol. 51, No. 1 (Pan 2), pp. 323-368, January 1972. [5] M. B. Porter, H. P. Bucker, “Gaussian Bean Tracing,” The Journal of the Acoustical Society of America, vol. 82, No. 4, pp. 1349-1359, October 1987. [6] Y.Bar-Shalom, X. R. Lee, Estimation and Tracking: Principles, Techniques, and Sofhyare. Artech House 1993 [71 D. B. Cousins, J. H. Miller, A. Tuttle, T. Weber, “The AWARE System - A Low-Cost Sonar for AntiGrounding and Collision Avoidance.” Presented at the 1999 Workshop on Underwater Signal Processing, IEEE Signal Processing Society, October 1999 APPENDIX: EKF STATISTICS

Several commonly used statistics have been derived for monitoring EKF filter performance [6].These statistics are based upon the following three properties of the EKF variables: 1) state errors should have zero mean and a magnitude commensurate with the filter generated state estimation error covariance matrix, 2) the innovations also have this property, and 3) the innovations are white. The first property must be tested in a Monte Carlo simulation, as it requires clairvoyant knowledge of the estimation error. This test is the Run-average Normalized Estimation Error Squared (R-NEES). It has a quadratic form containing the state estimation error vector and the associated covariance matrix generated by the EKF. Because of linear gaussian assumptions, each instantiation of the statistic is chi-square distributed with N,= length (xk) degrees of freedom. Aggregating the statistic over N , runs, each with independent initial values of xo, results in the RNEES which is chi squared distributed with N, N,, degrees of freedom and can be checked with a standard confidence test. The remaining two properties can be tested both by Monte Carlo and during a single real-time run. The Runaverage Normalized Innovation Squared (R-NIS) is

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