Inverse Problems in Geophysics

Transcription

1 σ ( Inverse Probles in Geophysics What is an inverse proble? - Illustrative Exaple - Exact inverse probles - Linear(ize inverse probles - Nonlinear inverse probles Exaples in Geophysics - Traveltie inverse probles - Seisic Toography - Location of Earthquakes - Reflection Seisology

2 σ ( What is an inverse proble? Forwar Proble Moel Data Inverse Proble

3 σ ( Treasure Hunt Gravieter?

4 σ ( Treasure Hunt Forwar Proble We have observe soe values: 10, 23, 35, 45, 56 µgals How can we relate the observe gravity values to the subsurface properties? Gravieter? We know how to o the forwar proble: G r' Φ( r = V ' r r' This equation relates the (observe gravitational potential to the subsurface ensity. -> given a ensity oel we can preict the gravity fiel at the surface!

5 σ ( Treasure Hunt Trial an Error What else o we know? Density san: 2,2 g/c 3 Density gol: 19,3 g/c 3 Do we know these values exactly? How can we fin out? Where is the box with gol? Gravieter? One approach: Use the forwar solution to calculate any oels for a rectangular box situate soewhere in the groun an copare the theoretical (synthetic ata to the observations. ->Trial an error etho

6 σ ( Treasure Hunt Moel Space But we have to efine plausible oels for the beach. We have to soehow escribe the oel geoetrically. Gravieter? -> Let us surface - ivie the subsurface into rectangles with variable ensity - Let us assue a flat surface x x x x x san gol

7 σ ( Treasure Hunt Non-uniqueness Coul we go through all possible oels an copare the synthetic ata with the observations? - at every rectangle two possibilities (san or gol ~ possible oels Gravieter - Too any oels! -We have possible oels but only 5 observations! - It is likely that two or ore oels will fit the ata (possibly perfectly well -> Nonuniqueness of the proble!

8 σ ( Treasure Hunt A priori inforation Is there anything we know about the treasure? - How large is the box? - Is it still intact? - Has it possibly isintegrate? - What was the shape of the box? - Has soeone alreay foun it? Gravieter This is inepenent inforation that we ay have which is as iportant an relevant as the observe ata. This is calle a priori (or prior inforation. It will allow us to efine plausible, possible, an unlikely oels: plausible possible unlikely

9 σ ( Treasure Hunt Uncertainties (Errors Do we have errors in the ata? - Di the instruents work correctly? - Do have to correct for anything? (e.g. topography, ties,... Gravieter Are we using the right theory? - Do we have to use 3-D oels? - Do we nee to inclue the topography? - Are there other aterials in the groun apart fro gol an san? - Are there ajacent asses which coul influence the observations? How (on Earth can we quantify these probles?

10 , (, (, (, ( k µ θ ρ σ = Treasure Hunt - Exaple Treasure Hunt - Exaple Gravieter Moels with less than 2% error.

11 , (, (, (, ( k µ θ ρ σ = Treasure Hunt - Exaple Treasure Hunt - Exaple Gravieter Moels with less than 1% error.

12 σ ( Treasure Hunt Exercise Exercise: Now let us assue that we know the box has not isintegrate into less than two pieces. Change the calculations of the synthetic ata an try to fin the box, oes it ake a ifference? Gravieter Paraetrization of the box with two pieces

13 σ ( Inverse Probles - Suary Inverse probles inference about physical systes fro ata Gravieter - Data usually contain errors (ata uncertainties - Physical theories are continuous - infinitely any oels will fit the ata (non-uniqueness - Our physical theory ay be inaccurate (theoretical uncertainties - Our forwar proble ay be highly nonlinear - We always have a finite aount of ata The funaental questions are: How accurate are our ata? How well can we solve the forwar proble? What inepenent inforation o we have on the oel space (a priori inforation?

14 σ ( Correcte schee for the real worl Forwar Proble True Moel Appraisal Proble Data Estiate Moel ~ Inverse Proble

15 σ ( Exact Inverse Probles Exaples for exact inverse probles: 1. Mass ensity of a string, when all eigenfrequencies are known 2. Construction of spherically syetric quantu echanical potentials (no local inia 3. Abel proble: fin the shape of a hill fro the tie it takes for a ball to go up an own a hill for a given initial velocity. 4. Seisic velocity eterination of layere eia given ray traveltie inforation (no low-velocity layers.

16 σ ( Abel s Proble (1826 z z P(x,z s Fin the shape of the hill! x For a given initial velocity an easure tie of the ball to coe back to the origin.

17 σ ( The Proble At any point: gz = 1 2 v 2 0 At z-z : = 1 g( z z' ( s / t 2 z After integration: t z s z ( = 2 g z 0 z 2 / ' z' ( z' P(x,z z s x

18 σ ( The solution of the Inverse Proble t( z = z s / z' 0 2 g( z z' z' z z P(x,z s x After change of variable an integration, an... f ( z ' = 1 π z ' a z ' t ( z z z z '

19 σ ( The seiological equivalent

20 σ ( Wiechert-Herglotz Metho

21 σ ( Distance an Travel Ties

22 σ ( Solution to the Inverse Proble

23 σ ( Wiechert-Herglotz Inversion The solution to the inverse proble can be obtaine after soe anipulation of the integral : T r r / c ( z p r = p 2 r ln = cosh 2 r r1 π 0 r 1 0 forwar proble 1 inverse proble p ξ The integral of the inverse proble contains only ters which can be obtaine fro observe T( plots. The quantity ξ 1 =p 1 =(T/ 1 is the slope of T( at istance 1. The integral is nuerically evaluate with iscrete values of p( for all fro 0 to 1. We obtain a value for r 1 an the corresponing velocity at epth r 1 is obtaine through ξ 1 =r 1 /v 1.

24 σ ( Conitions for Velocity Moel

25 σ ( Linear(ize Inverse Probles Let us try an forulate the inverse proble atheatically: Our goal is to eterine the paraeters of a (iscrete oel i, i=1,..., fro a set of observe ata j j=1,...,n. Moel an ata are functionally relate (physical theory such that 1 2 n = = = A ( 1 A A 2 n 1 ( (,..., 1 1,...,,..., This is the nonlinear forulation. Note that i nee not be oel paraeters at particular points in space but they coul also be expansion coefficients of orthogonal functions (e.g. Fourier coefficients, Chebyshev coefficients etc..

26 σ ( Linear(ize Inverse Probles If the functions A i ( j between oel an ata are linear we obtain = i A or ij = A in atrix for. If the functions A i ( j between oel an ata are illy non-linear we can consier the behavior of the syste aroun soe known (e.g. initial oel j0 : j i 0 Ai = Al ( j + j j 0 j +...

27 , (, (, (, ( k µ θ ρ σ = Linear(ize Inverse Probles Linear(ize Inverse Probles We will now ake the following efinitions:... ( = j j i j l i A A j ( ( 0 0 j i i i i j i i A A = + = Then we can write a linear(ize proble for the nonlinear forwar proble aroun soe (e.g. initial oel 0 neglecting higher orer ters: j j i i A j = 0 j ij i A = 0 j j i ij A A = A =

28 σ ( Linear(ize Inverse Probles = A Interpretation of this result: 1. 0 ay be an initial guess for our physical oel 2. We ay calculate (e.g. in a nonlinear way the synthetic ata =f( We can now calculate the ata isfit, =- 0, where 0 are the observe ata. 4. Using soe foral inverse operator A -1 we can calculate the corresponing oel perturbation. This is also calle the graient of the isfit function. 5. We can now calculate a new oel = 0 + which will by efinition is a better fit to the ata. We can start the proceure again in an iterative way.

29 σ ( Nonlinear Inverse Probles Assue we have a willy nonlinear functional relationship between oel an ata = g( The only option we have here is to try an go in a sensible way through the whole oel space an calculate the isfit function L = g( an fin the oel(s which have the inial isfit.

30 σ ( Moel Search The way how to explore a oel space is a science itself! Soe key ethos are: 1. Monte Carlo Metho: Search in a rano way through the oel space an collect oels with goo fit. 2. Siulate Annealing. In analogy to a heat bath, or the generation of crystal one optiizes the quality (iproves the isfit of an enseble of oels. Decreasing the teperature woul be equivalent to reucing the isfit (energy. 3. Genetic Algoriths. A pool of oels recobines an cobines inforation, every generation only the fittest survive an give on the successful properties. 4. Evolutionary Prograing. A foral generalization of the ieas of genetic algoriths.

31 σ ( Exaples: Seisic Toography Data vector : Travelties of phases observe at stations of the worl wie seisograph network Moel : 3-D seisic velocity oel in the Earth s antle. Discretization using splines, spherical haronics, Chebyshev polynoials or siply blocks. Soeties 10000s of travel ties an a large nuber of oel blocks: unereterine syste

32 σ ( Exaples: Earthquake location Seisoeters Data vector : Travelties observe at various (at least 3 stations above the earthquake Moel : 3 coorinates of the earthquake location (x,y,z. Usually uch ore ata than unknowns: overeterine syste

33 σ ( Exaples: Reflection Seisology Air gun Data vector : receivers ns seisogras with nt saples -> vector length ns*nt Moel : the seisic velocities of the subsurface, ipeances, Poisson s ratio, ensity, reflection coefficients, etc.

34 σ ( Inversion: Suary We nee to evelop foral ways of 1. calculating an inverse operator for =A -> =A -1 (linear or linearize probles 2. escribing errors in the ata an theory (linear an nonlinear probles 3. searching a huge oel space for goo oels (nonlinear inverse probles 4. escribing the quality of goo oels with respect to the real worl (appraisal.