mechanics of solidread the PDF file then complete the Word sheet, and make a report. instructions in Word sheet
Inversion TheoryAssignment 7Seismic acoustic probleLet us assume that the earth can be treated as an acoustic medium and theinuence of variations in density can be ignored. In this case the propagation of theseismic waves in the earth can be described by the acoustic wave equation:p(r; t)1V2(r)@2@t2p(r; t) = f(t)(r rs); (1)where p(r; t) is the pressure eld, f(t) is the strength of a point source of energy, rsis the position of the source.The pressure eld can be presented as a Fourier integral:p(r; t) = 12Z +11p(r; !)ei!td!Then in frequency domain equation (1) can be written as a Helmholtz equation,p(r; !) + !2V2(r)p(r; !) = f(!)(r rs): (2)Let us describe the velocity distribution in the form:1V2
(r)
1V2n(r)(1 + (r)); (3)where Vn(r) is some normal” distribution of the velocity, and (r) describes thevariations of the square slowness from its normal value. In this case the waveeld canbe separated into the normal and anomalous (primary and secondary) partsp = pn + pa: (4)By substituting eq. (4) into eq. (2), we can nd the equations for normal andanomalous parts of the waveeld:[pn(r; !) + pa(r; !)] + !2V 2n (r)[pn(r; !) + pa(r; !)] + !2V 2n (r)(r)[pn(r; !) + pa(r; !)]= f(!)(r randpn(r; rs; !) + !2V2n(r)pn(r; rs; !) = f(!)(r rs); (5)pa(r; r0; !) + !2V2n(r)(1 + (r))pa(r; r0; !) =!2V2n(r)(r)pn(r; r0; !): (6)At innity (z ! 1); the normal eld and its vertical derivative vanish (radiationcondition).Assignment 7a:Rewrite equations (2), (3), (5) and (6) for 1D medium.Assignment 7b: Use 1D Green’s Theorem , equations (5) and (6), radiation condition and the fact that (z) = 0 for z 0 to obtain the integral equation(equation 7) for the anomalous (scattered) eld pa(z; zs; !) which is generated by theimpulsive source (z zs), (f(!) = 1); located at the point zs:pa(z0; zs; !) = !2Z +10(z)V2n(z)[pa(z; z0; !) + pn(z; z0; !)]pn(z; zs; !)dz; (7)where z0 is the point of observation and z is an integration point.Directions:Green’s Theorem in one dimension has the form:Z +11fu(z)Hv(z) v(z)Hu(z)gdz = u(z)v0(z) j+11 v(z)u0(z) j+11; (8)where H is the Helmholtz operatorH =d2dz2+!2V2n(z);Tips: Assume u (z) = pn(z; zs; !) and v (z) = pa(z; z0; !) also, to make use of theproperty of (z) function:+Z11f (z) (z zs) dz = f (zs).Assignment 7c1D waveeld integral operator.Consider the integral operator for the anomalous (scattered) eld pa(z; !):pa(z0; zs; !) = As[pa(z; !)] = !2Z +10(z)V2n(z)[pa(z; z0; !) + pn(z; z0; !)]pn(z; zs; !)dz:(9)Find the conditions when As is a contraction operator.Describe the method of successive approximations for the determination of theanomalous (scattered) eld, based o
Inversion Theory (Lecture 9)Regularizing methodsMICHAEL ZHDANOV1 Tikhonov regularizing methodDifferent modifications of the least square solutions of the linear inverse problems have resulted from the straightforward minimization of the corresponding misfit functionals. However all these solutions have a lot of limitationsand are very sensitive to a small variations of the observed data. An obvious limitation occurs when the inverse matrices (ATA)−1 or (ATW2A)−1don’t exist. However even when the inverse matrices exist they can be illconditioned (become nearly singular). In this case our solution would beextremely unstable and unrealistic. To overcome these difficulties we have toapply the regularizing methods.Let us consider first the general approach based on Tikhonov regularizationtechnique. The corresponding parametric functional can be introduced inthe following form:P α(m, d) = nWdAm − Wddn2 + αnWmm − Wmmaprn2,where Wd and Wm are some weighting matrices of data and model (notnecessary diagonal); mapr is some a priori model and n…n denotes euclideannorm in the spaces of data and models.In the majority of practical applications we assume that Wm = I, but it alsocan be chosen arbitrary ( for example as a matrix of first order or secondorder finite-difference differentiation to obtain the smooth solution).We can rewrite also the functional P α(m, d) in matrix notationsP α(m, d)=(WdAm − Wdd)T (WdAm − Wdd)+1α(Wmm − Wmmapr)T (Wmm − Wmmapr).According to the basic principles of the regularization method we have tofind the quasi solution of the inverse problem as the model mα minimizingthe parametric functionalP α(mα, d) = min .The regularization parameter α is determined from the misfit condition:nWdAmα − Wddn = δ,where δ is some a priori estimation of the level of ”weighted” noise of thedata:nWdδdn = δ. (1)To solve this problem let us calculate the first variation of P α(m, d):δP α(m, d) = 2(WdAδm)T (WdAm − Wdd)+2α(Wmδm)T (Wmm − Wmmapr)=0.The last equation can be rewritten asδmT k(ATWTd WdA + αWTmWm)m−ATWTd Wdd + αWTmWmmaprl = 0,from which we obtain at oncemα = (ATWTd WdA + αWTmWm)−1(ATWTd Wdd + αWTmWmmapr). (2)Usually the weighting matrices Wd and Wm are selected to be symmetric,so the eq. (2) can be rewrittenmα = (ATW2dA + αW2m)−1(ATW2dd + αW2mmapr). (3)The last expression gives the regularized solution of the generalized leastsquare problem.22 The Marquardt – Levenberg methodLet us consider a special case when Wd = I and Wm = I. Then eq.(3) turnsout to bemα = (ATA + αI)−1(ATd + αmapr). (4)Assume now that mapr =0:mα = (ATA + αI)−1ATd. (5)The solution (5) describes the classical Marquardt – Levenberg methodof damped least squares where α plays the role of a ”damping factor”.For better understanding how does the regularization parameter or ”-damping factor” α works let us apply the singular vector decompositionmethod to matrix A:A = UQVT .Then we have:ATA = VQ2VT . (6)From the last equation we obtain at once(ATA + αI)−1 = (VQ2VT + αVIVT )−1 =(V[diag(α + Q2i)]VT )−1 = V[diag( 1α + Q2i)]VT . (7)We can clearly see from equation (7) how regularization makes nearlysingular matrix to be well conditioned: even if Qi → 0 division by zero doesnot occur.3 The maximum a posteriori estimation method(the Bayes estimation)Let us consider the regularization technique from the point of view of theprobability theory. First of all introduce the following (normally distributed)densities of probability:31) P(d/m) – the conditional density of probability for the data d, given themodel m. It means that it is a density of probability of theoretical data dto be expected from a given model m.2) P(m/d) – the conditional density of probability for a model m, given thedata d.According to the Bayes theorem the following equation takes place:P(m/d) = P(d/m)P(m)P(d) , (8)where P(d) and P(m) are unconditional densities of probability for data andmodel parameters. It is assumed thatkml = mapr,where mapr is a priori constrained expectation of the model, and[cov(mi, mj )] = σm.Thus, considering the normally distributed parameters we haveP(m) = | σm |−12(2π)L2exp[−12(m − mapr)Tσ−1m (m − mapr)]. (9)Analogously, it is assumed that[cov(di, dj )] = σdandP(d/m) = | σd |− 12(2π)N2exp−1 2 (d − Am) Tσ−1 d (d − Am)The maximum likelihood method can now be used to find the model m0which maximizes the conditional probability P(m/d):P(m/d) = | σd |−12(2π)N2exp[−12(d − Am)Tσ−1d (d − Am)]·| σm |− 12(2π)L2exp[−12(m − mapr)Tσ−1m (m − mapr)]P −1(d). (11)4It is evident that to maximize the P( ˆm/ ˆd) we have to minimize the sum ofthe expressions in the exponential in eq. (11):fBayes = (d − Am)Tσ−1d (d − Am)+(m − mapr)Tσ−1m (m − mapr). (12)Note that the minimization of the first term gives the classical maximumlikelihood or weighted least square method.Let us calculate the first variation of fBayes:δfBayes = −2(Aδm)Tσ−1d (d − Am) + 2(δm)Tσ−1m (m − mapr).From the last equation we have(δm)T [ATσ−1d (d − Am) − σ−1m (m − mapr)] = 0.Thus the normal system of equation for minimization of fBayes is equal:ATσ−1d (d − Am) − σ−1m (m − mapr)=0,from which we have at once(ATσ−1d A + σ−1m )m = ATσ−1d d + σ−1m mapr,andm0 = (ATσ−1d A + σ−1m )−1(ATσ−1d d + σ−1m mapr). (13)By comparing equation (13) and (3) we see thatσ−1m = αW2m, (14)so σ−1m playing the role of the regularization parameters.Let us assume now that we have uncorrelated data:σd = σ2dI,and the same for the a priori covariance of the model:σm = σ2mI.Then equation (13) turns out to bem0 = (ATA + kI)−1(ATd + kmapr), (15)wherek = σ2dσ2m= αplays the role of the regularization parameter.5
MECH207 2020SpLaboratory #2: Measurement and UncertaintyIntroduction:In today’s laboratory we will learn measurement techniques that will be used in laboratory over thesemester. We will also examine the use of dimensional analysis and significant figures in performingroutine calculations. Lastly we will look at how uncertainty relates to measurement, precision andaccuracy.PrecisionScientific measurements should be as precise as possible; this means that every measurement willinclude one uncertain or estimated digit. The number of significant figures you can report will dependon the measuring tool used. For example, the ruler shown in Figure 1 has large marks every 1 cm, andsmaller marks every mm. The length of the nail is certainly between 3.7 and 3.8 cm, thus indicating atleast 2 significant figures. However, you can add one more digit to your reported value by estimatingthe value between the lines. Thus, you could report the length of the nail as 3.75 cm, where the last digitrepresents your “estimated” value. The lesson is one should always try to “`read between the lines”‘. Asecond rule to remember when making measurements is that if the length of the nail was exactly onthe 4 cm line, you should indicate the precision to which you can make this measurement by addingzero’s to your value and record the length as 4.00 cm.The precision of a measurement is often quoted in terms of how inaccurate the estimated digit is. Forexample in Figure 1 since the sensitivity of the ruler is 0.1 cm the uncertainty would be stated as halfthat or +/- 0.05 cm. The +/- indicates that the measurement could be shorter or longer then the statedvalue. The 0.05 cm indicates two things, first the number of places past the decimal point it is possible toread (in this case 2) and the precision to which the last digit can be estimated (in this case you canprobably distinguish if a length was ½ way between the two millimeter marks, but cannot guess if it was0.01 or 0.02).Remember, ALWAYS read between the lines, and report measurements to as many decimal places aspossible, using zeros to indicate additional precision if needed.Figure 1: Using a ruler.➔ Measure the nail with your metal ruler and report the result in cm. Measure the width of apaper in inches and report the result in inches.AccuracyThe accuracy of a measurement or calculated quantity refers to its agreement with some known value.For example, if we measured the density of a sample and then compared the value measured to thatcontained in a reference such as the Handbook of Chemistry and Physics. High accuracy would indicatethat the experimental value is very close to the known value listed in the reference.Precision vs. AccuracyThe term precision and accuracy are often confused by students. A representation of the differences canbe seen in Figure 2. Precision is how close each X is to each other, while accuracy is a measure of howMECH207 2020Spclose the X is to the center of the target. The worst measurements are represented by the first targetwhich is neither precise nor accurate. The best measurements are represented by the fourth targetwhich is both precise and accurate. Targets two and three represent precision or accuracy but not bothat the same time and either target could be an acceptable measurement depending on therequirements of the measurement.Figure 2: Precision and Accuracy.Measuring Requested QuantitiesIn most laboratory exercises you will be required to make many different measurements of length,mass, and other quantities. Often, the precision required to produce a reliable results is quite differentthan the precision to which you can make the measurement. While you should always write down themeasured value as precisely as possible, you do not always need exactly the specified value. Forinstance, if the instructions say to measure out 1.0 gram of a substance it is implied that the neededamount is 1.0 +/- 0.1 grams, thus you should make sure your measurement is between 0.9 and 1.1grams. It is not necessary (nor desirable) to try to measure exactly 1.000 grams of solid out, and often itis a waste of time to do so.When in doubt about how precise a measurement should be, ask your laboratory instructor.Recording Measured ValuesAll measurements made in class should be written with the proper number of significant figures ANDunits. All calculations performed during a lab should show at least one sample calculation. The results ofany calculations must also contain the correct number of significant figures and units.Significant Figures: Rules• All non-zero digits are significant.• Zeros between non-zero digits are significant.• Leading zeros are never significant.• In a number with a decimal point, trailing zeros, those to the right of the last non-zero digit, aresignificant.• In a number without a decimal point, trailing zeros may or may not be significant. More informationthrough additional graphical symbols or explicit information on errors is needed to clarify thesignificance of trailing zeros.Example: 012.0560 has how many significant figures?MECH207 2020SpCalibrationMost mechanical devices used to make measurements in a laboratory are designed to be as precise aspossible (repeated measurement yields the same result). The accuracy of the device is established bymeasuring known quantities and correlating this with the values measured on the laboratoryequipment. This process is known as calibration. For example, glassware for measuring volume ispurchased already calibrated, thus we will only need to calibrate thermometers and balances (scales).Uncertainty of Measurement and Measurement ErrorThe result of any quantitative measurement has two essential components:• A numerical value (expressed in SI units as required by ISO 15189) which gives the best estimateof the quantity being measured (the measurand). This estimate may well be a singlemeasurement or the mean value of a series of measurements.• A measure of the uncertainty associated with this estimated value. In engineering this may wellbe the variability or dispersion of a series of similar measurements (for example, a series ofquality control specimens) expressed as a standard deviation or for a combined measurand ascombined standard uncertainty.By definition, the term error (or measurement error) is the difference between the true value and themeasured value. The most likely or ‘true’ value may thus be considered as the measured value includinga statement of uncertainty which characterizes the dispersion of possible measured values. As themeasured value and its uncertainty component are at best only estimates, it follows that the true valueis indeterminate (VIM, GUM). Uncertainty is caused by the interplay of errors which create dispersionaround the estimated value of the measurand; the smaller the dispersion, the smaller the uncertainty.Even if the terms error and uncertainty are used somewhat interchangeably in everyday descriptions,they actually have different meanings according to the definitions, so they should not be used assynonyms. The ± (plus or minus) symbol that often follows the reported value of a measurand and thenumerical quantity that follows this symbol, indicate the uncertainty associated with the particularmeasurand and not the error.If repeated measurements are made of the same quantity, statistical procedures can be used todetermine the uncertainties in the measurement process. This type of statistical analysis providesuncertainties which are determined from the data themselves without requiring further estimates. Theimportant variables in such analyses are the mean, the standard deviation and the standard uncertaintyof the mean (also referred to as the standard deviation of the mean or the standard error of the mean).MECH207 2020SpOther terms and Definitions:Approximation Error: The discrepancy between an exact value and some approximation to it. Forinstance, measurement of the data is not accurate due to the low sensitivity of the instrument (e.g. theaccurate width of a piece of paper is 2.5 cm but since the ruler does not show decimals, you round it to3 cm).Mean (x): arithmetic average or the sum of values (x) divided by the number of values (n),nx
x i
(in excel this is the function AVERAGE()).Standard Deviation: The standard deviation shows how much variation from the mean exists and isgiven by the equation below and in Excel as STDEV(). For a normal distribution of data (or “bell curve”asshown below), approximately 68.3% of the data values fall within one standard deviation from themean, and 98.4% fall within two standard deviations. Note that n is used in the denominator if you justhave a sampling of the population, n-1 in the denominator if you have the entire population (in excelthis is the function STDEV.P()).nx x
stdev − i
2( )Accuracy: In engineering, the accuracy of a measurement system is the degree of closeness ofmeasurements of a quantity to that quantity’s actual (true) value.Bias: Closely related to accuracy, bias is a quantitative term describing the difference between theaverage of measurements made on the same object and its true value.Precision: The reproducibility or repeatability of a measurement, as the degree to which repeatedmeasurements under unchanged conditions give the same results.Sensitivity: is an absolute quantity, the smallest absolute amount of change that can be detected by ameasurement.MECH207 2020SpUncertainty: Measurement uncertainties are classified into two categories, those contributing torandom uncertainty (or precision) and those contributing to systematic uncertainty (or bias). The typeof errors depends on the measurement process; random errors can be reduced by repeating themeasurement process and averaging the results, whereas the systematic errors will remain constant (forinstance if your measurement tool is improperly calibrated).Formulas for Standard Uncertainty:Source: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3387884/table/t2-cbr_33_2_49/.MECH207 2020SpLab 1: Uncertainty in Measurement ExerciseWhen an engineer performs a measurement, they will often do it multiple times. Why? Becauserepeated measurements allow for random measurement errors to cancel. The average or mean ofseveral measurements is almost always better than just one measurement. In addition, you shouldcheck whether there is systematic uncertainty (bias) in your measurement by checking the calibration ofthe measurement tool and method of the measurer.To get you started performing measurements we will begin with a simple exercise. Be sure to recorddata in the appropriate places in the results section as you complete each measurement. All resultsshould be written with the proper number of significant figures and units. ALL calculations andconversions should be shown. USE EXCEL or some spreadsheet or software for all routine calculations!Here you will measure dimensions (thickness, width and height) as well as mass for a standardcalibration block using different measurement tools to determine the density of the material. Compareyour results with your classmates and answer the questions below. Solutions should be TYPED andresults tabulated in standard units (cm, grams). EACH Individual should submit their OWN uniqueanalysis and results on Blackboard. Due 1/29.1) With your TEAM, measure the dimensions and mass of one standard block. First, using yourRULER, then each of the other tools (calipers and micrometer). Share this data with the classand be sure to pay attention to significant figures and unit conversions. Complete the tablegiven. EVERY TEAM MUST check their result with the instructor.2) What is the level of sensitivity given by each measurement technique? Give your response incentimeters. Was there a difference in sensitivity between the micrometer, calipers (coarse andfine), and the ruler measurements? If so, how much? Which technique is capable of thehighest sensitivity?3) Calculate the DENSITY of your block using your data for each tool. Calculate the standarduncertainty (see uncertainty formula attachment) for each result and show in a complete table.Which tool had the greatest uncertainty? Which tool had the lowest uncertainty? Was thissurprising?4) What is the difference between error and uncertainty?5) Assuming the “true value” of density for the steel block is given as 8.03 g/cm3, calculate theerror of the results and compare. Which technique was most accurate? Was this surprising?6) In your opinion, what is the best measurement tool for the density of the block material? Howdoes sensitivity (and uncertainty) relate to accuracy and precision? How could you improve theprecision and/or accuracy of this result? Describe the steps you would take. (Consider forexample mean and standard deviation, tool calibration, etc.).Bonus: Can a measurement be accurate but not precise? Answer in your own words.MECH207 2020Sp
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