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\7 x \7 x G(1',1'o) - W2Il(E)G(1', 1'0) = Q(r)G(1',1'o) + 18(1' - 1'0) (4.1.1) 2fl,Ej(r). In (4.1.1). we have put the (E) part on the left-hand with Q(1') = W side of the equation while putting the E j part on the right-hand side. The solution to (4.1.1) can be expressed as an integral equation by using

the homogeneous dyadic Green's function G(O) (1',1'0)' which satisfies the wave equation with mean wavenumber k m = wJliJl) :

ActualHeight, ActualWidth, Background, BorderBrush, BorderThickness, Child, Clip, CornerRadius, Cursor, DataContext, DesiredSize, Dispatcher, Height, HorizontalAlignment, IsHitTestVisible, Language, Margin, MaxHeight, MaxWidth, MinHeight, MinWidth, Name, Opacity, OpacityMask, Padding, Parent, RenderSize, RenderTransform, RenderTransformOrigin, Resources, Style, Tag, Triggers, VerticalAlignment, Visibility, Width

=(0) _ _

(r,r o )

=(0)

Arrange, ArrangeOverride, CaptureMouse, CheckAccess, Equals, Finalize, FindName, GetAnimationBaseValue, GetHashCode, GetType, GetValue, HitTest, InvalidateArrange, InvalidateMeasure, Measure, MeasureOverride, MemberwiseClone, ReleaseMouseCapture, SetBinding, SetValue, ToString, TransformToVisual, UpdateLayout

2 =(0) _ _

(r',r o ) = 18(r - r' o )

= _ _

GotFocus, KeyDown, KeyUp, LayoutUpdated, Loaded, LostFocus, MouseEnter, MouseLeave, MouseLeftButtonDown, MouseLeftButtonUp, MouseMove, SizeChanged

(4.1.2) (4.1.3)

=(0)

(1',1't} Q(1'l)G(1'}, 1'0)

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ActualHeight, ActualWidth, Background, ClickMode, Clip, Content, ContentTemplate, Cursor, DataContext, DesiredSize, Dispatcher, FontFamily, FontSize, FontStretch, FontStyle, FontWeight, Foreground, Height, HorizontalAlignment, HorizontalContentAlignment, IsEnabled, IsFocused, IsHitTestVisible, IsMouseOver, IsPressed, IsTabStop, Language, Margin, MaxHeight, MaxWidth, MinHeight, MinWidth, Name, Opacity, OpacityMask, Padding, Parent, RenderSize, RenderTransform, RenderTransformOrigin, Resources, Style, TabIndex, TabNavigation, Tag, Template, TextAlignment, TextDecorations, TextWrapping, ToolTip, Triggers, VerticalAlignment, VerticalContentAlignment, Visibility, Width

Equation (4.1.3) may be solved formally by iteration to generate the Neumann series for the dyadic Green's functioll

= =(0) G(1',1'o) = G (1',1'0)

=(0)

ApplyTemplate, Arrange, ArrangeOverride, CaptureMouse, CheckAccess, Equals, Finalize, FindName, Focus, GetAnimationBaseValue, GetHashCode, GetTemplateChild, GetType, GetValue, HitTest, InitializeFromXaml, InvalidateArrange, InvalidateMeasure, Measure, MeasureOverride, MemberwiseClone, OnApplyTemplate, OnClick, OnContentChanged, OnContentTemplateChanged, OnGotFocus, OnIsEnabledChanged, OnIsPressedChanged, OnKeyDown, OnKeyUp, OnLostFocus, OnMouseEnter, OnMouseLeave, OnMouseLeftButtonDown, OnMouseLeftButtonUp, OnMouseMove, ReleaseMouseCapture, SetBinding, SetValue, ToString, TransformToVisual, UpdateLayout

(IV.2.6)

=(0)

=(0)

(1'1,1'0)

Click, GotFocus, KeyDown, KeyUp, LayoutUpdated, Loaded, LostFocus, MouseEnter, MouseLeave, MouseLeftButtonDown, MouseLeftButtonUp, MouseMove, SizeChanged

=(0)

=(0)

ActualHeight, ActualWidth, AreDatesInPastSelectable, Clip, Cursor, DataContext, DayStyle, DesiredSize, Dispatcher, DisplayDate, DisplayDateEnd, DisplayDateStart, DisplayMode, FirstDayOfWeek, Height, HorizontalAlignment, HorizontalContentAlignment, IsEnabled, IsHitTestVisible, IsTabStop, IsTodayHighlighted, Language, Margin, MaxHeight, MaxWidth,

=(0)

=(0)

(1'2, 1'0)

MinHeight, MinWidth, MonthStyle, Name, Opacity, OpacityMask, Padding, Parent, RenderSize, RenderTransform, RenderTransformOrigin, Resources, SelectableDateEnd, SelectableDateStart, SelectedDate, Style, TabIndex, TabNavigation, Tag, Template, Triggers, VerticalAlignment, VerticalContentAlignment, Visibility, Width

=(0)

=(0)

(1'1,1'2)Q(1'2)

ApplyTemplate, Arrange, ArrangeOverride, CaptureMouse, CheckAccess, Equals, Finalize, FindName, Focus, GetAnimationBaseValue, GetHashCode, GetTemplateChild, GetType, GetValue, HitTest, InitializeFromXaml, InvalidateArrange, InvalidateMeasure, Measure, MeasureOverride, MemberwiseClone, OnApplyTemplate, ReleaseMouseCapture, SetBinding, SetValue, ToString, TransformToVisual, UpdateLayout

(1'2,1'3)'''Q(1'n )G (1'n,1' o)+'" (4.1.4) The first term in the perturbation series (4.1.4) is the zeroth-ord(~r approximation and represents the solution to the homogeneous medium in which

there is no scattering. The next term is the first-order Born approximation. It has the physical interpretation of single scattering of the incident wave as shown in the figure below:

In general, the nth-order Born approximation represents the n-tuply scattered incident wave as represented below:

DateSelected, DisplayDateChanged, DisplayModeChanged, GotFocus, KeyDown, KeyUp, LayoutUpdated, Loaded, LostFocus, MouseEnter, MouseLeave, MouseLeftButtonDown, MouseLeftButtonUp, MouseMove, SizeChanged

For the statistics of the random fluctuations E jer), a common assumption made is that Ej (7') is a Gaussian stationary random process so that the first and second moments of E j(r) are sufficient to describe the statistics. Therefore, we have

(4.1.5)

(4.1.6)

ActualHeight, ActualWidth, Background, Children, Clip, Cursor, DataContext, DesiredSize, Dispatcher, Height, HorizontalAlignment, IsHitTestVisible, Language, Margin, MaxHeight, MaxWidth, MinHeight, MinWidth, Name, Opacity, OpacityMask, Parent, RenderSize, RenderTransform, RenderTransformOrigin, Resources, Style, Tag, Triggers, VerticalAlignment, Visibility, Width

(4.1.7)

where the summation extends over all possible distinct pairs of arguments to give (211, - 1)!! terms. For example, if 11, = 2, then we have

O(Z)=

(Ej(rI)Ej(7'2)Ej(7'3)Ej(7'4)) = (Ej(7't}Ej(7'2)) (E.r (7'3)Ej (7'4)) + (Ej(7'1)Ej(7'3))(Ej(7'2)Ej(7'4)) + (Ej(7'I)E.r(r4))(Ej(r2)Ej(r3)) (4.1.8)

Arrange, ArrangeOverride, CaptureMouse, CheckAccess, Equals, Finalize, FindName, GetAnimationBaseValue, GetHashCode, GetLeft, GetTop, GetType, GetValue, GetZIndex, HitTest, InvalidateArrange, InvalidateMeasure, Measure, MeasureOverride, MemberwiseClone, ReleaseMouseCapture, SetBinding, SetLeft, SetTop, SetValue, SetZIndex, ToString, TransformToVisual, UpdateLayout

In the case of non-Gaussian statistics, the moments of Ej(7') may still be cluster-expanded but not exclusively in terms of the two-point correlation function [Frisch, 1968]. Assuming Gaussian statistics and taking the ensemble average of (4.1.4), the perturbation series for the mean dyadic Green's function is obtained

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