Geology and Geophysics 612: Structural Geology Section
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Geology and Geophysics 612: Structural Geology Section

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  • cours magistral
  • cours magistral - matière : geology
GG612 Lecture1 1 Geology and Geophysics 612: Structural Geology Section Instructor: Steve Martel, POST 805, 956-7797, Class Themes The crust of the earth is deformed at many scales, locations, and times; this deformation produces identifiable structures in the crust such as fractures and folds. An appreciation of earth structures has both enormous practical value and profound intellectual implications for how we view this planet. This class deals with ways to recognize and characterize major structures in the earth's crust and ways to gain insight into how these structures form.
  • horizontal examples
  • n0°e n45°e n90°e s45°e s0°e s45°w s90°w n45°w
  • science in science
  • science science
  • geologic structures
  • bodies
  • time constraints
  • structural geology
  • knowledge
  • line

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D A N S I M O N
Kalman Filtering
bedded Systems Programming
Originally developed for use in spacecraft navigation, the Kalman filter turns out to be useful for many applications. It is mainly used to estimate system states that can only be observed indirectly or inaccurately by the
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Linear systems In order to use a Kalman filter to remove noise from a signal, the process that we are measuring must be able to be described by a linear system. Many physical processes, such as a vehicle driving along a road, a satellite orbiting the earth, a motor shaft driven by winding currents, or a sinusoidal
radio-frequency carrier signal, can be approximated as linear systems. A lin-ear system is simply a process that can be described by the following two equations:
State equation: x=Ax+Bu+w k+1kk k
Output equation: y=Cx+z k k k
In the above equationsA,B, andC are matrices;kis the time index;xis called the state of the system;uis a known input to the system;yis the measured output; andwandzare the noise. The variablewis called the process noise, andzis called the mea-surement noise. Each of these quanti-ties are (in general) vectors and there-fore contain more than one element. The vectorxcontains all of the infor-mation about the present state of the system, but we cannot measurex directly. Instead we measurey, which is a function ofxthat is corrupted by the noisez. We can useyto help us obtain an estimate ofx, but we cannot neces-sarily take the information fromyat face value because it is corrupted by noise. The measurement is like a politi-cian. We can use the information that it presents to a certain extent, but we cannot afford to grant it our total trust. For example, suppose we want to model a vehicle going in a straight line. We can say that the state consists of the vehicle positionpand velocityv. The inputuis the commanded accel-eration and the outputyis the mea-sured position. LetÕs say that we are able to change the acceleration and measure the position everyTseconds. In this case, elementary laws of physics say that the velocityvwill be governed by the following equation:
The Kalman filter not only works well in practice, but it is theoretical-ly attractive because it can be shown that of all possible filters, it is the one that minimizes the variance of the estimation error.
v=v+Tu k+1k k
That is, the velocity one time-step from now (Tseconds from now) will be equal to the present velocity plus the commanded acceleration multi-plied byT. But the previous equation does not give a precise value forv. k+1 Instead, the velocity will be perturbed by noise due to gusts of wind, pot-holes, and other unfortunate realities. The velocity noise is a random variable that changes with time. So a more real-istic equation forvwould be:
~ v+=v+Tu+vk k1k k
~ v wherekis the velocity noise. A simi-lar equation can be derived for the positionp:
12 ~ p=p+Tv+T u+p k+1k kk k 2
~ wherepkis the position noise. Now we can define a state vectorxthat con-sists of position and velocity:
p k x=   k v k
Finally, knowing that the measured output is equal to the position, we can write our linear system equations as follows:
2 1T T2 x=x+u+w     k+1k k k 0 1T     y=[1 0]x+z k k k
need a way to estimate the statex. This is where the Kalman filter comes in.
The Kalman filter theory and algorithm Suppose we have a linear system model as described previously. We want to use the available measurementsyto esti-mate the state of the systemx. We know how the system behaves according to the state equation, and we have mea-surements of the position, so how can we determine the best estimate of the statex? We want an estimator that gives an accurate estimate of the true state even though we cannot directly mea-sure it. What criteria should our esti-mator satisfy? Two obvious require-ments come to mind. First, we want the average value of our state estimate to be equal to the average value of the true state. That is, we donÕt want our estimate to be biased one way or another. Mathematically, we would say that the expected value of the estimate should be equal to the expected value of the state. Second, we want a state estimate that varies from the true state as little as possible. That is, not only do we want the average of the state estimate to be equal to the average of the true state, but we also want an estimator that results in the smallest possible variation of the state estimate. Mathematically, we would say that we
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Process noise covariance
T S=E(w w) w k k
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spose ctors, and E(á) means the expected value. Now we are finally in a position to look at the Kalman filter equations. There are many alternative but equiva-
press the equations. One tions is given as follows:
1 T + PkC Sz) ˆ uk)+Kk(yk+1Cxk) T1T SAP C AS CP w k z k
Kalman filter. It consists ations, each involving ulation. In the above 1superscript indicates on and aTsuperscript ix transposition. TheK d the Kalman gain, and is called the estimation ce. stimate (xˆ) equation is . The first term used to e estimate at timek+ 1 is e state estimate at timek, e known input at timek.  the state estimate if we measurement. In other e estimate would propa-st like the state vector in del. The second term in n is called thecorrection presents the amount by ct the propagated state o our measurement. of theKequation shows surement noise is large, , soKwill be small and much credibility to the ywhen computing the e other hand, if the mea-se is small,Swill be z ill be large and we will edibility to the measure-mputing the nextxˆ
Vehicle navigation Now consider the vehicle navi problem that we looked at earl vehicle is traveling along a road position is measured with an er 10 feet (one standard deviation) commanded acceleration is a co 2 1 foot/sec . The acceleration n 2 0.2 feet/sec (one standard tion). The position is measur times per second (T= 0.1). Ho
we best estimate the position of the moving vehicle? In vie measurement noise, we better than just taking ments at face value. SinceT= 0.1, the line represents our system c from the system model lier in this article as foll
1 0.1 0.005 = + x+1x u     k k 0 1 0.1     y=[1 0]x+z k k k
Because the standard de measurement noise is 1 matrix is simply equal to Now we need to d matrix. Since the positi tional to 0.005 times the and the acceleration 2 feet/sec , the variance o 2 noise is (0.005) (0. Similarly, since the velo tional to 0.1 times the the variance of the vel 2 2 -4 (0.1) (0.2) = 4 10 . ∙ ∙ covariance of the positi velocity noise is equal to deviation of the positio the standard deviation noise, which can be (0.005 0.2) (0.1 0.2) ∙ ∙ ∙ can combine all of thes to obtain the following
p  T S=E(xx)=E[p v]    w v   265 p pv 10 2×10 =E= 2 54 vp v2×10 4×10    
Now we just initializexˆ 0 initial estimate of position and veloci-ty, and we initializePas the uncer-0 tainty in our initial estimate. Then we execute the Kalman filter equations once per time step and we are off and running. I simulated the Kalman filter for this problem using Matlab. The results are shown in the accompanying fig-ures. Figure 1 shows the true position
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