CA2135433A1 - Method for fast kalman filtering in large dynamic systems - Google Patents

Abstract

The invention is based on the use of the principles of Lange's Fast Kalman Filtering (FKFTM) for large process control, prediction or warning systems where other computing methods are either too slow or fail because of truncation errors. The invented method makes it possible to exploit the FKF method for dynamic multiparameter systems that are governed by partial differential equations.
A notebook PC of Fig. 1 works as a supernavigator through exploiting the FKF method. The receiver concept comprises an integrated sensor, remote sensing, data processing and transmission system (3) of, say, a national environmental service and an off-the-shelf GPS receiver. The database unit (2) contains control (4) and performance information on the subsystems and geographical maps. The logic unit (1) yields real-time 3-dimensional visualizations on what is going on (5). Dependable accuracy information is provided when the stability conditions of the optimal Kalman filtering are be met by the overall system.

Description

2135~33 METHOD FOR FAST I~ALMAN FILTERING
IN LARGE DYNAMIC ~;Y~ IS

Technical Field This invention relates generally to all practical applications of the K~lm~n Filter and more partir,~ r1y to large dynamical systems with a special need for fast, colllpu~ationally stable and accurate results.

Background Art Prior to e~plaining the invention, it will be helpful to first understand the prior art of both the K~lm~n Filter (KF) and the Fast Kalman Filter (FKFTH) for calibrating a sensor system (WO 90/13794). The underlying Markov process is described by the equations from (1) to (3). The first equation tells how a measurement vector Yt depends on the state vector st at timepoint t, (t=0,1,2...). This is the linearized Measurement (or observation) equation:
Yt = Ht St + et (1) The design matrix Ht is typically composed of the partial derivatives of the actual Mea~urei~,ent equations. The second equation describes the time evolution of e.g. a weather balloon flight and is the System (or state) equation:
St = st l + Ut-l + at (2) t A St-l + B Ut-l + at more generally) which tells how the balloon position is composed of its previous position St 1 as well as of increments ut 1 and at. These increments are typically caused by a known uniform mohon and an unknown random acceleration, respectively.

The measurement errors, the acceleration term and the previous position usually are mutually uncorrelated and are briefly described here by the following covariance matrices:
Re = Cov(et) = E(etet') Ra = Cov(at) = E(atat') and - (3) A r A A
Pt l=Cov(st l)=Et(St_l-St-l)(St-l St-l),~
The Ralman forward recursion formulae give us the best linear unbiased estim~tes of the present state A A r A
St = St 1 +Ut_l + Kt~,~Yt-Ht(St- 1 + Ut- 1)~
and its covariance matrix Pt = Cov(st) = Pt l-KtHtPt-l where the Ralman gain matrix Kt is defined by Kt (Pt l+Ra?Ht{Ht(Pt-l+Ra?Ht+Re} (6) Let us now partition the estim~ted state vector st and its covariance matrix Pt as follows:

A --A - A A A
St bt ~ Pt=Cov(st)= Pb CV(bt~Ct)- (7) Ct Cov ( Ct ~b t ) c where bt tells us the estim~ted balloon position; and, Ct the estim~ted calibration parameters.
The respec~ive partitioning of the other quantities will then be as follows:
Ht= [Hb Hc ] = ~t Gl ~ ut= -ub-, at= -ab- ' uct act and, (8) Ra=~ Ra Cov(ab,ac ) t b t t Cov ( ac, ab ) Ra 213543~

The recursion formulae from (4) to (6) gives us now a filtered (based on updated calibration parameters) position vector A A r A
bt bt-l +Ub I +Kb tYt-Ht(St-l +Ut-l) ~ (9) and the updated calibration parameter vector A A ~
Ct Ct 1 +Uct l +Kct Yt Ht(St-l +Ut-l) (10) The K~lm~n gain matrices are respectively Kbt (Pbt 1 +Rab )Hbt{Ht(Pt-l +Ra?Ht+Re } + -and (1 1) Kc =(Pc +Ra )Hc {Ht(Pt 1 +Ra )Ht+Re } 1 + -The following modified form of the general State equation isintroduced Ast 1 +But 1 = I st +A(St-l St-l) t (12) where s represents an estim~ted value of a state vector s. Combine it with the Measurement equation (1) in order to obtain so-called Augmented Model:
Yt Ht et ~ St + A (13) Ast l+But 1 I A(St-l-St-l)-at i.e. Zt = Zt St + ~t The state parameters can now be computed by using the well-known solution of a Regression Analysis problem given below. Use it for Updating:

, -1 -1, -1 St = (Ztvt Zt) Ztvt Zt (14) The result is algebraically equivalent to use of the Kalman Recursions but not numerically. For the balloon tracking problem with a large number sensors with slipping calibration the matrix to be inverted in equations (6) or (11) is larger than that in formula (14).

The initi(7~ ti~n of the large Fast K~lm~n Filter (FKFT ) for solving the calibration problem of the balloon tracking sensors is done by Lange's High-pass Filter. It e~ploits an analytical sparse-matri~ inversion formula (Lange, 1988a) for solving regression models with the following so-called Canonical Block-angular matrix structure:

yl XlX G2 1 e2 (15) l,r --YK XKGK c eK

This is a matrix represçnt~tion of the Measurement equation- of an entire windfinding interco~ alison e~e~ ent or one balloon flight. The vectors bl,b2,...,bK typica11y refer to consec~tive position coordinates of a weather balloon but may also contain those calibration parameters that have a ~i nific~nt time or space variation. The vector c refers to the other calibration parameters that are constant over the sampling period.

The~ Regression Analytical approach of the Fast Kalman Filtering (FKFTM) for updating the state parameters including the calibration drifts in particular, is based on the same block-angular matrix structure as in equation (lS). The optimal estim~tes (A) of bl,b2,..,bK and c are obtained by m~kin~ the following logical insertions into formula (15) for each timepoint t, t=1,2,...:

,_ Yt,k .X ._ Xt,k Y1, ' A ~1, ;
bt 1 k+Ub ` I

Gt k Gk:= ~~~~'~~; bk:=bt k; and, ek:= ~ et,k ; for k=l,.. ,K;

(bt-l,k-bt-l,k) abtk and, (16) x r YK+l Ct-l+uct l; K+l L P ]
G :=[ I ]; c:=ct; and~ eK+l =(Ct-l-Ct-l) ct 2135~3~
s .

These insertions concluded the specification of the Fast Kalman Filter (FKFTM) algorithm for calibrating the upper-air wind tracking system.
Another application would be the Global Observing System of the World Weather Watch. Here, the vector Yk contains various observed inconsistencies and syst~m~tic errors of weather reports (e.g. mean day-night differences of pressure values which should be about zero) from a radiosonde system k or from a homogeneous cluster k of radiosonde stations of a country (Lange, 1988a/b). The calibration drift vector bk will then tell us what is wrong and to what extent. The calibration drift vector c refers to errors of a global nature or which are more or less common to all observing systems (e.g. biases in satellite radiances and in their vertical weighting functions or some atmospheric tide effects).

For all these large multiple sensor systems their design matrices H
typically are sparse. Thus, one can usually perform in one way or another the following sort of Partitioning:st= bY = ~'2 H = t,l Xt 2Gtt'2--Ct -Yt,K- - Xt K Gt K
(t7) Al - Bl A = . AK and, B = . BK
c c ~vhere ct typically r~,~.es~ls calibration parameters at time t bt k all other state parameters in the time and/or ~ volume A state tr~n~it;on matrix (bock-diagonal) at time t B matri~ (bock-diagonal) for state-independent effects ut at time t.

Consequently, two (or three) types of gigantic Regression Analysis problems Zt = Zt St + et (18) were faced as fol10ws:

2135~33 A~ ..ted model for a space volume case: see also equations (15) and (16).
e.g. for the data of an entire windtracking experiment with K consecutive balloon positions:

Y t ~1 t,l G ~ b ~ ~ t, 1 Albt l, l +Blubt 1 1 I bt 2 + Al-(bt 1, l-bt 1, l)-ab A2bt 1 2+B2Ub Xt 2 bt,KA2(bt l,2-bt-1 2)~ab t A Yt ,K .Xt,K Gt,K ~ et,K
AKbt 1 K+BKUb ~ AK(bt l,K-bt-l, K) abt K

AcCt-l C Ct_l . I AC(ct-l Ct-l ) act Augmented Model for a moving time volume (e.g. for "whitening" an observed "innovationsn sequence of residuals et over a moving sample of length L):

A tH t F t t e t ASt-l+But-l I . st l + A(St l-st-l) ~at A t 1 t-- 1 A t -- 1 Ast 2+But -2 I . st L+1 A(St-2~st-2) ~at-1 : Ct A Y t -L+l Ht L+l Ft-L+l ~ et-L+
ASt-L+But-L I A(St L-st L) -at-L
ACt 1 +Bu c t -1- - I A(Ct 1 -Ct- 1 ) ~ act Please observe that the matrix formula may take a nnested" Block-Angular structure. Fast semi-analytical solutions based on Updating: St {Z tVt t} t t t ( 19) for all these three cases were published in PCI /FI90/00122 (I,ange, ~ 990) .
WIPO, Geneva, Switzerland.

2135~33 The Fast K~lm~n Filter (FKF ) formulae for the recursion step at any timepoint t were as follows:

S l={Xt lVtllXt 1} Xt-lVt-~Yt-rGt-lCt) Ct = I~oGt-lRt-lGt-~ Gt IRt IYt I (20) where, for l=0,1,2,...,L l, D ~r-l ~ V V~ ~r-l ~ ~~ X~ ~r-l t-l t-r 1 At_~ At_l ~ t_lAt_l~ t-l t-V _ CV(et-l) t I-- Cov{A(st-l-l-st-l-l)-at-l}

y Yt-l t-l Ast l l + Bu Xt l= t-l Gt l t and, i.e. for l=L, Rt L= VtlL

Vt-L CV{A(Ct-l Ct-l)-act}

Yt L= ACt-l +BUCt 1 Gt-L I-2135~33 A major R & D project was initi~ted in 1988 which led to the start of cooperation between ECMWF and Meteo-France for the coding of a dynamical atmospheric model, an optimal interpolation, a variational data ~s~imil~tif~n and a K~lm~n Filter (FK), all in the same framework. The project is called IFS (Integrated Forecasting System), see Jean-Noel Thepaut and Philippe Courtier (1991): "Four-dimen~ional variational data ~simi1ation using the adjoint of a multilevel primitive-equation model", Quarterly Journal of the Royal Meteorological Society, Volume 117, pp.
1225-1254.
Similar K~1m~n Filter (KF) studies have recently been reported also by Roger Daley (1992): "The Lagged Innovation Covariance: A Performance Diagnostic for Atmospheric Data ~ssimi1ationn, Monthly Weather Review of the ~m~.ric~n Meteorological Society, Vol. 120, pp. 178-196, and Stephen E.
Cohn and David F. Parrish (1991): "The Behavior of Forecast Error Covariances for a K~lm~n Filter in Two Dimensions", Monthly Weather Review of the American Meteorological Society, Vol. 119, pp. 1757-1785.
Unfortunately, the ideal K~lm~n Filter systems described in the above reports have been out of reach at the present time. Dr. T. Gal-Chen of School of Meteorology, University of Oklahoma, reported in May 1988: "There is hope that the developl~.cnts of massively parallel super computers (e.g..
100~ desktop CRAYs working in tandem) could result in algorithms much closer to optimal...n, see nReport of the Critical Review Panel - Lower Tropospheric Profiling Symposium: Needs and Technologies N ~ Bulletin of the ~meric~n Meteorological Society, Vol. 71, No. 5, May 1990, page 684.
There exists a need for exploiting the principles of the Fast Kalman Fi1te.ring (FKFTM) method for a broad technical field (broader than just calibrating a sensor system in some narrow sense of word "calibration") with equal or better c~"~.~,ulational speed, reliability, accuracy, and cost benefits than other K~lm~n Filtering methods can do (see Cotton, Thompson &
Mielke, 1994: nReal-Time Mesoscale Prediction on Workstationsn, Bulletin of the ~meric~n Meteorological Society, Vol. 75, Number 3, March l 994, pp.
349-362).

2135~33 sl-mr ~ry of the Invention These needs are substantially met by provision of the generalized Fast K~lm~n Filtering (FKFTM) method for calibrating and adjusting the sensors and various model parameters of a dynamical system in real-time or in near real-time as described in this specification. Through the use of this method, the co.~lpu~ation results include the forecast error covariances that are absolute nt~cess~ry for warning, decision m~kin~ and control purposes.

Best Mode for Carrying out the Invention Prior to explaining the invention, it will be helpful to first understand prior art Kalm~n Filter (KF) theory e~ploited in the current e~ ental Nllmerie~l Weather Prediction (NWP) systems which themselves are much too complex to be described here. As previously, they make use of equation (l):

Measurement Equation: Yt = Ht St + et ...(linearized regression) where state vector st describes the state of the atmosphere at timepoint t.
Now, st usually represents all gridpoint values of atmospheric variables e.g. the geopotential heights of a number of different pressure levels.
The dynamics of the atmosphere is governed by a well-known set of partial dir~renlial equations ("plh~ ive" equations). By using of the tangent linear approximation of the NWP model the following expression of equation

(2) is obtained for the time evolution of the atmosphere at each time step:
tate Equation: St A st l + B Ut-l + at (the discret ized dyn-stoch model) The four--1im~--n~ional data ~s~imil~tion results (st) and the NWP forecasts ( s t)~ respectively, are obtained from the K~lm~n Filter system as follows:

St = St + Kt (Yt - Ht St) (2 1 ) St = A st l + B Ut-l 21~5~33 -where t t) Cov(st 1) A + Qt ... (prediction accuracy~
Qt CV(at) = E at at ... (system noise) Rt = Cov(et) = E et et ... (measurement noise) and the crucial Updating computations are based on the following Kalm~n Recursion:
Kt Pt Ht (Ht Pt Ht + Rt)-1 ...(Kalm~n Gain matrix) Cov(st) = Pt ~ Kt Ht Pt ...(estimation accuracy).

The matri~ inversion needed here for the compulation of the Kalman Gain matrix is e~ceeAingly difficult for any realistic NWP system because the data ~imil~tion system must be be able to digest several ten thousand data elements at a time.
The method of the invention will now be described. We start with the Augmented Model from equation (13):
Y t Ht et A = St + A
Ast l+But 1 I A(St 1-St l)-at i.e. Zt Zt St + ~t For the four-tlimen~ion~l data ~ imil~tion the following two equations are obtained for its Updating: --St = ~Ztvtlzt)-lztvtlzt ... (optimal estim a tion, b y Gauss - Markov) = {Ht RtlHt + Pt } (Ht Rt Yt + t t (22) or, = s t + Kt (Yt ~ Ht s t) ... (alternatively) and, Cov(st) = E(St--st)(st--St) = (ZtV Zt) (23) = {Ht Rt1Ht + Pt1} ... (estimation accuracy) , where, as previously, s t = A st l + B Ut-l ... (NWP "forecasting") Pt -- Cov( s t) = A CV(St 1) A' + Qt (24) but insteaf of t Pt H~ (Ht Pt Ht + Rt)-l ~ ... (K~lm;~n Gain matrix) we take Kt = Cov(st) Ht Rt (25) The ~l~gme-nted Model approach is superior to the use of the Kalman Recursion formulae for a large vector of input data Yt because the coml,ulation of the K~1m~n Gain matrix Kt required a huge matrix inversion when Cov(st) was unknown. Both methods are algebraically and statistically equivalent but certainly not numerically.
Unfortunately, the All~nented Model formulae above may still become much too difficlllt to handle numeric~lly if the number of the state parameters is overly large. This actually happens, firstly, if state vector st consists of enough gridpoint data for a realistic representation of the atmosphere.
A spectral decomposition (or empirical orthogonal functions) could be attempted here for the purpose of decreasing the number of state parameters. Secondly, there are many other state parameters that must be included in the state vector for a realistic NWP system. These are filrst of all related to systçm~tic (calibration) errors of observing systems as well as to the so-called physical parameterization schemes of small scale atmospheric processes.
Fortunately, all these problems are overcome by using the method of decoupling states through exploitation of the general Fast Kalman Filtering (FKFrM) method. For the large observing systems of the atmosphere their design matrices H typically are sparse. Thus, one can pelÇorm the following Partitioning: st= b Yt= ~t 2 Ht= ' Xt 2 Gtt~2--Ct -Yt,K- - Xt,K Gt,K

A = A and, B = B (26) Ac- Bc-where & typically ~e~ ..ts "calibration" parameters at time t bt k values of atmospheric parameters at grid point k (k=l,...K) A state transition matrix at time t (submatrices Al,...,AK,AC) B for state-independent effects (submatrices Bl,...,BK,BC).
Consequently, the following gigantic Regression Analysis problem is faced:

Y t, 1 t,l G ~ b ~ ~~ t, 1 bt 2 + A 1 (St l-St 1 ) -ab (27) A Yt ,2 Xt,2 Gt,2 A et~2 A2St-l+B2ut-l _ A2(St-l St-l)-abt 2 t A Y t ,K Xt K Gt,K ~ et,K
AKst 1 +BKUt-1 I AK(st l-st l ) -abt,K

A . A
Acst l+BCUt 1 I Ac(st l~St 1 )-ac The Fast Kalm~n Filter (FKFTM) formulae for the recursion step at any timepoint t are as follows:

bt,k {Xt,kVt,kXt,k} Xt,kVt ~Ytk-GtkCt) for k=l~2~ -~K
K , '-1 K (28) t Gt kRt kGt k~ ~ G kRt kYt k k=O ' ' ' k=O t~ , where, for k=1,2,...,K, R = V-l.~I X rX' V-l X ~-lX' V-l t,k t, k ~ t,kt t ,k t, k t,kJ t ,k t, k V Cov(et k) t,k Cov{Ak(St l-St-l) abt k}-Yt,k ^ Yt,k _AkSt_l + BkUt_l_ Xt k=
Gt k =- t ~ k and, i.e. for k=O, Rt o= Vt10 V { ^ }

Yt,o ACSt-l +
Gt,o= I-The data as~imil~tion accuracies are obtained from equation (23) as follows:
~ A A A
Cov(st)= CV(bt, 1~ bt,K~Ct) (29) Cl +DlSDi DlSD2 DlSDK -DlS

D2SDi C2+D2SD2 D2SDK D2S

DKSDi DKSD2 CK+DKSDK -DKS

-SDi ' -SD2 -SDK S
where Ck = {Xt,kVt,kXt,k} for k=1,2,.. ,K

Dk = {Xt kVt1kXt k} Xt,kVt,~;t,k for S ~ h~oGt,kRt,kGt,k~

Through these semi-analytical means all the matrices to be inverted for the solution of gigantic Regression Analysis models of type shown in equation (27) are kept reasonably small e.g. for the preferred embodiment of the invention shown in Fig. 1 and described below:
A supernavigator based on a notebook PC that performs the functions of a Kalm~n filtering logic unit (1) through exploiting the generalized Fast Kalman Filtering (FKF) method. The overall receiver concept comprises an integrated sensor, remote sen~ing, data proces.sing and transmission system

(3) of, say, a national atmospheric/oceanic service and, optionally, an off-the-shelf GPS receiver. The database unit (2) running on the notebook PC contains updated information on control (4) and performance aspects of the various subsystems as well as ~u~iliary information such as geo-gr~phir~l maps. Based upon all these inputs, the logic unit (1) provides real-time 3-~limçn~ional vi~u~li7ations (5) on what is going on by using the FKF recursions from equations (28) and on what will take place in the nearest future by using the predictions from equation (21). Dependable accuracy information is also provided when the well-known stability conditions of optimal Kalman filtering are be met by the observing system (3). These error variances and covariances are computed by using equations (29) and (24), respectively. The centralized data processing system (3) provides estim~tes of State Transition Matrix A for each time step t. These matrices are then adjusted locally (1) to take into account all observed small-scale transitions that occur in the atmospheric/oceanic environment.

The generalized Fast Kalm~n Filtering (FKF ) formulae specified by equations (28) and (29) are pursuant to the invented method.

Those skilled in the art will appreciate that many variations could be practiced with respect to the above described invention wi~:hout departing from the spirit of the invention. Therefore, it should be understood that the scope of the invention should not be considered as limited to the specific embodiment described, except in so far as the claims may specifically include such limitations.

2135~33 References (1) K~1m~n, R. E. (1960): "A new approach to linear filtering and prediction problems". Trans. ASME J. of Basic Eng. 82:35-45.

(2) Lange, A. A. (1982): "Multipath propagation of VLF Omega signals".
IEEE- PLANS '82 - Position Location and Navigation Symposium Record, December 1982, 302-309.

(3) Lange, A. A. (1984): "Integration, calibration and intercomparison of windfinding devices". WMO Instruments and Observing Methods Report No. 15.

(4) Lange, A. A. (1988a): "A high-pass filter for optimum calibration of observing systems with applications'Y. Siml~1ation and Optimi~tion of Large Systems, edited by A. J. Osiadacz, O~ford University Press/Clarendon Press, O~ford, 1988, 311-327.

(5) Lange, A. A. (1988b): "Determination of the radiosonde biases by using satPllite ra~ nce measurements". WMO Instruments and Observing Methods Report No. 33, 201-206.

(6) Lange, A. A. (1990): "Apparatus and method for calibrating a sensorsystem'r. Tn~ern~tional Application Published under the Patent Cooperation Treaty (PCT), World Intellectual Property Organization, International Bureau, WO 90/13794, PCT/FI90/00122, 15 November 1990.

(7) Lange, A. A. (1993): "Method for Fast K~lm~n Filtering in large dynamic systems". International Application Published under the Patent Cooperation Treaty (PCT), World Intellectual Property Organization, International Bureau, WO 93/22625, PCT/FI93/0192, 11 November 1993.

(8) Lange, A. A. (1994): "A surface-based hybrid upper-air sounding system". WMO Instruments and Observing Methods Report No. 57, 175-177.

Claims (4)

1. A method for optimal or suboptimal Fast Kalman Filtering (FKF), especially for calibrating or standardizing outputs from a multiple sensor system, where the Measurement and State Equations, in linearized forms, are as follows:
yt = Ht st + et (1) st = A st-1 + B ut-1 + at (2) where at time t yt = measurement vector of observed angles, distances, clouds, etc.
Ht = observations' gain or design matrix st = state vector, calibration parameters included et = measurement error vector Rt = Cov(et) = E et et' = covariance matrix of said errors ut = external forcing or control vector at = system noise vector Qt = Cov(at) = E at at' = covariance matrix of said system noise A = state transition matrix which typically depends on time t B = gain matrix of external forcing or controls, and where said vectors and matrices, due to physical or other structural relationships between their elements, are partitioned as follows:

yt= Ht= and, st = A = B = ut= at= (26) where ct = calibration (typically) parameters for a time/space volume bt,k = all other state parameters for said volume, and, Ab1, Ab2, ...,AbK and Ac = submatrices of said partitioned A
Bb1, Bb2, ...,BbK and Bc = submatrices of said partitioned B, wherein the improvement comprises the indicated new way of partitioning said state transition matrix A and/or said gain matrix B of said external forcing or said controls, and where the Fast Kalman Filter (FKF) formulae for a recursion step are:
?t,k = for k=1,2,...,K
(28) ?t = where for k=1,2,...,K, Rt,k = Vt,k = yt,k = Xt,k = Gt,k = and, for k=0, Rt,0 = Vt,0 = yt,0 = Gt,0 = I

where ?t,k and ?t are the optimal or suboptimal estimates (?) of said state parameters st and wherein the improvement comprises the possibility to compute their error variances and covariances from:

Cov(?t) = Cov(?t,1,...,?t,K,?t) (29) =

where Ck = for k=1,2,...,K

Dk = for k=1,2,...,K

S = and where optimal or suboptimal predictions ?t,k and ?t of said state parameters st can be obtained for prediction and process control purposes from:
?t = A ?t-1 + B ut-1 (21) where their error variances and covariances Pt are computed:
Pt = Cov(?t) = A Cov(?t-1) A' + Qt (24) where the " ~ "-notation indicates that said prediction (no observational data available) step takes place during a time interval (t-1,t).

2. The method of claim 1 wherein said logic means (1) operates in a decentralized or cascaded fashion but exploits in one way or another Kalman filtering wherein the improvement comprises the use of an algorithm obtained from the Fast Kalman Filter (FKF) formulae (28) and (29) of the description.

3. The method of claim 1 including the step of:
a) adapting by using Kalman filtering wherein the improvement comprises the use of an algorithm obtained from the Fast Kalman Filter (FKF) formulae (28) of the description, in said logic means (1), said information on said controls of or changes in said sensors or said external events as far as their true magnitudes are unknown.

4. The method of claim 2 including the step of:
a) adapting by using Kalman filtering wherein the improvement comprises the use of an algorithm obtained from the Fast Kalman Filter (FKF) formulae (28) of the description, in said logic means (1), said information on said controls of or changes in said sensors or said external events as far as their true magnitudes are unknown.