2 | | Still to write... |

| 2 | Last edited [[Timestamp]] |

| 3 | = ''' Tangent Linear Validation ''' = |

| 4 | |

| 5 | [[PageOutline]] |

| 6 | |

| 7 | == Principle == |

| 8 | The tangent validation test checks that the Tangent Linear model '''L''' is a first order approximation of the model '''M'''. We can then write a second order Taylor expansion as follows: |

| 9 | |

| 10 | {{{ |

| 11 | M(X0+p.dX) = M(X0) + L(p.dX) + e |

| 12 | }}} |

| 13 | with, |

| 14 | {{{ |

| 15 | X0: initial state vector |

| 16 | dX: perturbation state vector |

| 17 | p: scale factor |

| 18 | e: residu |

| 19 | }}} |

| 20 | when the factor '''p''' tends to zero we have: |

| 21 | {{{ |

| 22 | -1- M(X0+p.dX) - M(X0) tends to L(p.dX) |

| 23 | and |

| 24 | -2- e is equivalent to O(p^2), means Norm(e,.) <= K.p^2 (with K a constant) |

| 25 | }}} |

| 26 | |

| 27 | This means that for a given '''p0''', for p smaller than p0 the error of the Linear Tangent with respect to the model is decreasing as p^2. |

| 28 | |

| 29 | === First order validation === |

| 30 | Let's note |

| 31 | {{{ |

| 32 | Np = M(X0+p.dX) - M(X0) |

| 33 | Lp = L(p.dX) |

| 34 | }}} |

| 35 | the first order validation checks: |

| 36 | {{{ |

| 37 | for a given p0, for all p>p0: |

| 38 | Ep = Norm(Np,.)/Norm(Lp,.) tends to 1 |

| 39 | }}} |

| 40 | |

| 41 | === Second order validation === |

| 42 | the second order validation ensure that the residu is controlled. We check then: |

| 43 | {{{ |

| 44 | for a given p0, for all p>p0: |

| 45 | Rp = Norm(Np - Lp,.) behaves as p^2 |

| 46 | }}} |

| 47 | |

| 48 | note that we can also perform the following equivalent tests: |

| 49 | || test || expression || behaviour || |

| 50 | ||b ||Norm((Np - Lp)/Lp,.) || p || |

| 51 | ||c ||Norm((Ep - 1,.)/p || constant || |

| 52 | ||d ||Norm((Ep - Lp/(p.Lp),.) || constant || |

| 53 | |

| 54 | == Implementation == |

| 55 | |

| 56 | ... work in progress ... |