Code covered by the BSD License

# Okada: Surface deformation due to a finite rectangular source

### François Beauducel (view profile)

30 Nov 2009 (Updated )

Computes Okada's 1985 solution for displacements, tilts and strains due to fault dislocation.

%OKADA85 Surface deformation due to a finite rectangular source.
%	   E,N,DEPTH,STRIKE,DIP,LENGTH,WIDTH,RAKE,SLIP,OPEN)
%	computes displacements, tilts and strains at the surface of an elastic
%	half-space, due to a dislocation defined by RAKE, SLIP, and OPEN on a
%	rectangular fault defined by orientation STRIKE and DIP, and size LENGTH and
%	WIDTH. The fault centroid is located (0,0,-DEPTH).
%
%	   E,N    : coordinates of observation points in a geographic referential
%	            (East,North,Up) relative to fault centroid (units are described below)
%	   DEPTH  : depth of the fault centroid (DEPTH > 0)
%	   STRIKE : fault trace direction (0 to 360 relative to North), defined so
%	            that the fault dips to the right side of the trace
%	   DIP    : angle between the fault and a horizontal plane (0 to 90)
%	   LENGTH : fault length in the STRIKE direction (LENGTH > 0)
%	   WIDTH  : fault width in the DIP direction (WIDTH > 0)
%	   RAKE   : direction the hanging wall moves during rupture, measured relative
%	            to the fault STRIKE (-180 to 180).
%	   SLIP   : dislocation in RAKE direction (length unit)
%	   OPEN   : dislocation in tensile component (same unit as SLIP)
%
%	returns the following variables (same matrix size as E and N):
%	   uN,uE,uZ        : displacements (unit of SLIP and OPEN)
%	   uZE,uZN         : tilts (in rad * FACTOR)
%	   uNN,uNE,uEN,uEE : horizontal strains POSITIVE = COMPRESSION (unit of FACTOR)
%
%	Length unit consistency: E, N, DEPTH, LENGTH, and WIDTH must have the same
%	unit (e.g. km) which can be different from that of SLIP and OPEN (e.g. m) but
%	with a possible FACTOR on tilt and strain results (in this case, an
%	amplification of km/m = 1000). To have FACTOR = 1 (tilt in radians and
%	correct strain unit), use the same length unit for all aforesaid variables.
%
%	[...] = OKADA85(...,NU) specifies Poisson's ratio NU (default is 0.25 for
%	an isotropic medium).
%
%	Formulas and notations from Okada [1985] solution excepted for strain
%	convention (here positive strain means compression), and for the fault
%	parameters after Aki & Richards [1980], e.g.:
%	      DIP=90, RAKE=0   : left lateral (senestral) strike slip
%	      DIP=90, RAKE=180 : right lateral (dextral) strike slip
%	      DIP=70, RAKE=90  : reverse fault
%	      DIP=70, RAKE=-90 : normal fault
%
%	It is also possible to produce partial outputs, with following syntax:
%	   [uE,uN,uZ] = OKADA85(...) for displacements only;
%	   [uE,uN,uZ,uZE,uZN] = OKADA85(...) for displacements and tilts;
%	   [uE,uN,uZ,uNN,uNE,uEN,uEE] = OKADA85(...) for displacements and strains;
%	   [uZE,uZN] = OKADA85(...) for tilts only;
%	   [uZE,uZN,uNN,uNE,uEN,uEE] = OKADA85(...) for tilts and strains;
%	   [uNN,uNE,uEN,uEE] = OKADA85(...) for strains only.
%
%	Note that vertical strain components can be obtained with following equations:
%	   uNZ = -uZN;
%	   uEZ = -uZE;
%	   uZZ = -(uEE + uNN)*NU/(1-NU);
%
%	produces a 3-D figure with fault geometry and dislocation at scale (if
%	all of the fault parameters are scalar).
%
%	Equations are all vectorized excepted for argument DIP which must be
%	a scalar (beacause of a singularity in Okada's equations); all other
%	arguments can be scalar or matrix of the same size.
%
%	Example:
%
%	   [E,N] = meshgrid(linspace(-10,10,50));
%	   figure, surf(E,N,uN)
%
%	considers a 5x3 fault at depth 2, N30-strike, 70-dip, and unit dislocation
%	in all directions (reverse, senestral and open). Displacements are computed
%	on a regular grid from -10 to 10, and North displacements are plotted as a
%	surface.
%
%
%	Author: Franois Beauducel <beauducel@ipgp.fr>
%	   Institut de Physique du Globe de Paris
%	Created: 1997
%	Updated: 2014-05-24
%
%	References:
%	   Aki K., and P. G. Richards, Quantitative seismology, Freemann & Co,
%	      New York, 1980.
%	   Okada Y., Surface deformation due to shear and tensile faults in a
%	      half-space, Bull. Seismol. Soc. Am., 75:4, 1135-1154, 1985.
%
%	Acknowledgments: Dmitry Nicolsky, Qian Yao, Halldor Geirsson

%	Development history:
%	   [2014-05-24]: fixes a bug for tilt calculation (K1) when DIP=90.
%	   [2012-11-08]: solves partially mathematical singularities in
%	      specific cases like DIP=90, STRIKE=0, and fault reaching surface.
%	      Detected by Qian Yao.
%	   [2012-08-29]: allows vectorization of RAKE, SLIP and OPEN.
%	   [2011-03-08]: help review.
%	   [2011-03-06]: new optional argument to plot fault geometry with
%	      output arguments, and bug correction for the fault centroid position
%	      (in calculation and plot).
%	   [2010-11-29]: change coordinates and depth to fault centroid
%	      (instead of middle top edge).
%	   [2010-09-24]: bugs correction in the syntax of I1, K2 and uyy_tf
%	      functions, affecting some components. Detected by Dmitry Nicolsky.
%
%
%	Redistribution and use in source and binary forms, with or without
%	modification, are permitted provided that the following conditions are
%	met:
%
%	   * Redistributions of source code must retain the above copyright
%	     notice, this list of conditions and the following disclaimer.
%	   * Redistributions in binary form must reproduce the above copyright
%	     notice, this list of conditions and the following disclaimer in
%	     the documentation and/or other materials provided with the distribution
%
%	THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
%	AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
%	IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
%	ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE
%	LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR
%	CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF
%	SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS
%	INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN
%	CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
%	ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE
%	POSSIBILITY OF SUCH DAMAGE.

if nargin < 10
error('Not enough input arguments.')
end

if nargin > 12
error('Too many input arguments.')
end

if any(~cellfun(@isnumeric,varargin(1:10)))
error('Input arguments E,N,DEPTH,STRIKE,DIP,LENGTH,WIDTH,RAKE,SLIP,OPEN must be numeric.')
end

if ~isscalar(varargin{5})
error('DIP argument must be scalar.')
end

% Default values for optional input arguments
plotflag = 0;	% no plot
nu = 0.25;	% isotropic Poisson's ratio

% Assigns input arguments
e = varargin{1};
n = varargin{2};
depth = varargin{3};
strike = varargin{4}*pi/180;	% converting STRIKE in radian
dip = varargin{5}*pi/180;	% converting DIP in radian ('delta' in Okada's equations)
L = varargin{6};
W = varargin{7};
rake = varargin{8}*pi/180;	% converting RAKE in radian
slip = varargin{9};
U3 = varargin{10};

switch nargin
case 11
if isnumeric(varargin{11})
nu = varargin{11};
else
makeplot = varargin{11};
end
case 12
makeplot = varargin{12};
end

if exist('makeplot','var')
if strcmp(makeplot,'plot')
plotflag = 1;
else
error('Unknown last argument.')
end
end

if plotflag & any([numel(depth),numel(strike),numel(L),numel(W),numel(rake),numel(slip),numel(U3)]>1)
warning('Cannot make plot with fault geometry parameters other than scalars.')
plotflag = 0;
end

% Defines dislocation in the fault plane system
U1 = cos(rake).*slip;
U2 = sin(rake).*slip;

% Converts fault coordinates (E,N,DEPTH) relative to centroid
% into Okada's reference system (X,Y,D)
d = depth + sin(dip).*W/2;	% d is fault's top edge
ec = e + cos(strike).*cos(dip).*W/2;
nc = n - sin(strike).*cos(dip).*W/2;
x = cos(strike).*nc + sin(strike).*ec + L/2;
y = sin(strike).*nc - cos(strike).*ec + cos(dip).*W;

% Variable substitution (independent from xi and eta)
p = y.*cos(dip) + d.*sin(dip);
q = y.*sin(dip) - d.*cos(dip);

% Displacements
if any(nargout==[3, 5, 7, 9])
ux = -U1/(2*pi) .* chinnery(@ux_ss,x,p,L,W,q,dip,nu) ... % strike-slip
- U2/(2*pi) .* chinnery(@ux_ds,x,p,L,W,q,dip,nu) ... % dip-slip
+ U3/(2*pi) .* chinnery(@ux_tf,x,p,L,W,q,dip,nu); ... % tensile fault

uy = -U1/(2*pi) .* chinnery(@uy_ss,x,p,L,W,q,dip,nu) ... % strike-slip
- U2/(2*pi) .* chinnery(@uy_ds,x,p,L,W,q,dip,nu) ... % dip-slip
+ U3/(2*pi) .* chinnery(@uy_tf,x,p,L,W,q,dip,nu); ... % tensile fault

uz = -U1/(2*pi) .* chinnery(@uz_ss,x,p,L,W,q,dip,nu) ... % strike-slip
- U2/(2*pi) .* chinnery(@uz_ds,x,p,L,W,q,dip,nu) ... % dip-slip
+ U3/(2*pi) .* chinnery(@uz_tf,x,p,L,W,q,dip,nu); ... % tensile fault

% Rotation from Okada's axes to geographic
ue = sin(strike).*ux - cos(strike).*uy;
un = cos(strike).*ux + sin(strike).*uy;
end

% Tilt
if any(nargout==[2, 5, 6, 9])
uzx = -U1/(2*pi) .* chinnery(@uzx_ss,x,p,L,W,q,dip,nu) ... % strike-slip
- U2/(2*pi) .* chinnery(@uzx_ds,x,p,L,W,q,dip,nu) ... % dip-slip
+ U3/(2*pi) .* chinnery(@uzx_tf,x,p,L,W,q,dip,nu); ... % tensile fault

uzy = -U1/(2*pi) .* chinnery(@uzy_ss,x,p,L,W,q,dip,nu) ... % strike-slip
- U2/(2*pi) .* chinnery(@uzy_ds,x,p,L,W,q,dip,nu) ... % dip-slip
+ U3/(2*pi) .* chinnery(@uzy_tf,x,p,L,W,q,dip,nu); ... % tensile fault

% Rotation from Okada's axes to geographic
uze = -sin(strike).*uzx + cos(strike).*uzy;
uzn = -cos(strike).*uzx - sin(strike).*uzy;
end

% Strain
if any(nargout==[4, 6, 7, 9])
uxx = -U1/(2*pi) .* chinnery(@uxx_ss,x,p,L,W,q,dip,nu) ... % strike-slip
- U2/(2*pi) .* chinnery(@uxx_ds,x,p,L,W,q,dip,nu) ... % dip-slip
+ U3/(2*pi) .* chinnery(@uxx_tf,x,p,L,W,q,dip,nu); ... % tensile fault
uxy = -U1/(2*pi) .* chinnery(@uxy_ss,x,p,L,W,q,dip,nu) ... % strike-slip
- U2/(2*pi) .* chinnery(@uxy_ds,x,p,L,W,q,dip,nu) ... % dip-slip
+ U3/(2*pi) .* chinnery(@uxy_tf,x,p,L,W,q,dip,nu); ... % tensile fault
uyx = -U1/(2*pi) .* chinnery(@uyx_ss,x,p,L,W,q,dip,nu) ... % strike-slip
- U2/(2*pi) .* chinnery(@uyx_ds,x,p,L,W,q,dip,nu) ... % dip-slip
+ U3/(2*pi) .* chinnery(@uyx_tf,x,p,L,W,q,dip,nu); ... % tensile fault
uyy = -U1/(2*pi) .* chinnery(@uyy_ss,x,p,L,W,q,dip,nu) ... % strike-slip
- U2/(2*pi) .* chinnery(@uyy_ds,x,p,L,W,q,dip,nu) ... % dip-slip
+ U3/(2*pi) .* chinnery(@uyy_tf,x,p,L,W,q,dip,nu); ... % tensile fault

% Rotation from Okada's axes to geographic
unn = cos(strike).^2*uxx + sin(2*strike).*(uxy + uyx)/2 + sin(strike).^2.*uyy;
une = sin(2*strike).*(uxx - uyy)/2 + sin(strike).^2.*uyx - cos(strike).^2.*uxy;
uen = sin(2*strike).*(uxx - uyy)/2 - cos(strike).^2.*uyx + sin(strike).^2.*uxy;
uee = sin(strike).^2*uxx - sin(2*strike).*(uyx + uxy)/2 + cos(strike).^2.*uyy;
end

% Assigns output arguments
switch nargout
case 2
varargout = {uze, uzn};
case 3
varargout = {ue, un, uz};
case 4
varargout = {unn, une, uen, uee};
case 5
varargout = {ue, un, uz, uze, uzn};
case 6
varargout = {uze, ezn, unn, une, uen, uee};
case 7
varargout = {ue, un, uz, unn, une, uen, uee};
case 9
varargout = {ue, un, uz, uze, uzn, unn, une, uen, uee};
case 0
plotflag = 1;
otherwise
disp('Unvalid number of output arguments.')
end

% no output argument: plots geometry of the fault and dislocation
if plotflag
figure
plot(e,n,'.r','MarkerSize',.1)
alpha = pi/2 - strike;
x_fault = L/2*cos(alpha)*[-1,1,1,-1] + sin(alpha)*cos(dip)*W/2*[-1,-1,1,1];
y_fault = L/2*sin(alpha)*[-1,1,1,-1] + cos(alpha)*cos(dip)*W/2*[1,1,-1,-1];
z_fault = -d + sin(dip)*W*[1,1,0,0];
ddx = U1*cos(alpha) - U2*sin(alpha)*cos(dip) + U3*sin(alpha)*sin(dip);
ddy = U1*sin(alpha) + U2*cos(alpha)*cos(dip) - U3*cos(alpha)*sin(dip);
ddz = U2*sin(dip) + U3*cos(dip);
patch(x_fault,y_fault,z_fault,.3*[1,1,1],'EdgeColor','k','LineWidth',2)
patch(x_fault+ddx/2,y_fault+ddy/2,z_fault+ddz/2,.6*[1,1,1], ...
'EdgeColor','k','LineWidth',1,'FaceAlpha',.5)
patch(x_fault-ddx/2,y_fault-ddy/2,z_fault-ddz/2,.6*[1,1,1], ...
'EdgeColor','k','LineWidth',1,'FaceAlpha',.5)
xlabel('East'); ylabel('North'); zlabel('Vertical')
view(3); grid on; axis equal; rotate3d
end

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% Notes for I... and K... subfunctions:
%
%	1. original formulas use Lame's parameters as mu/(mu+lambda) which
%	   depends only on the Poisson's ratio = 1 - 2*nu
%	2. tests for cos(dip) == 0 are made with "cos(dip) > eps"
%	   because cos(90*pi/180) is not zero but = 6.1232e-17 (!)
%	   NOTE: don't use cosd and sind because of incompatibility
%	   with Matlab v6 and earlier...

% =================================================================
% Chinnery's notation [equation (24) p. 1143]

% -----------------------------------------------------------------
function u=chinnery(f,x,p,L,W,q,dip,nu)
u = feval(f,x,p,q,dip,nu) ...
- feval(f,x,p-W,q,dip,nu) ...
- feval(f,x-L,p,q,dip,nu) ...
+ feval(f,x-L,p-W,q,dip,nu);

% =================================================================
% Displacement subfunctions

% strike-slip displacement subfunctions [equation (25) p. 1144]

% -----------------------------------------------------------------
function u=ux_ss(xi,eta,q,dip,nu)
R = sqrt(xi.^2 + eta.^2 + q.^2);
u = xi.*q./(R.*(R + eta)) ...
+ I1(xi,eta,q,dip,nu,R).*sin(dip);
k = find(q~=0);
u(k) = u(k) + atan(xi(k).*eta(k)./(q(k).*R(k)));

% -----------------------------------------------------------------
function u=uy_ss(xi,eta,q,dip,nu)
R = sqrt(xi.^2 + eta.^2 + q.^2);
u = (eta.*cos(dip) + q.*sin(dip)).*q./(R.*(R + eta)) ...
+ q.*cos(dip)./(R + eta) ...
+ I2(eta,q,dip,nu,R).*sin(dip);

% -----------------------------------------------------------------
function u=uz_ss(xi,eta,q,dip,nu)
R = sqrt(xi.^2 + eta.^2 + q.^2);
db = eta.*sin(dip) - q.*cos(dip);
u = (eta.*sin(dip) - q.*cos(dip)).*q./(R.*(R + eta)) ...
+ q.*sin(dip)./(R + eta) ...
+ I4(db,eta,q,dip,nu,R).*sin(dip);

% dip-slip displacement subfunctions [equation (26) p. 1144]
% -----------------------------------------------------------------
function u=ux_ds(xi,eta,q,dip,nu)
R = sqrt(xi.^2 + eta.^2 + q.^2);
u = q./R ...
- I3(eta,q,dip,nu,R).*sin(dip).*cos(dip);

% -----------------------------------------------------------------
function u=uy_ds(xi,eta,q,dip,nu)
R = sqrt(xi.^2 + eta.^2 + q.^2);
u = (eta.*cos(dip) + q.*sin(dip)).*q./(R.*(R + xi)) ...
- I1(xi,eta,q,dip,nu,R).*sin(dip).*cos(dip);
k = find(q~=0);
u(k) = u(k) + cos(dip).*atan(xi(k).*eta(k)./(q(k).*R(k)));

% -----------------------------------------------------------------
function u=uz_ds(xi,eta,q,dip,nu)
R = sqrt(xi.^2 + eta.^2 + q.^2);
db = eta.*sin(dip) - q.*cos(dip);
u = db.*q./(R.*(R + xi)) ...
- I5(xi,eta,q,dip,nu,R,db).*sin(dip).*cos(dip);
k = find(q~=0);
u(k) = u(k) + sin(dip).*atan(xi(k).*eta(k)./(q(k).*R(k)));

% tensile fault displacement subfunctions [equation (27) p. 1144]
% -----------------------------------------------------------------
function u=ux_tf(xi,eta,q,dip,nu)
R = sqrt(xi.^2 + eta.^2 + q.^2);
u = q.^2 ./(R.*(R + eta)) ...
- I3(eta,q,dip,nu,R).*sin(dip).^2;

% -----------------------------------------------------------------
function u=uy_tf(xi,eta,q,dip,nu)
R = sqrt(xi.^2 + eta.^2 + q.^2);
u = -(eta.*sin(dip) - q.*cos(dip)).*q./(R.*(R + xi)) ...
- sin(dip).*xi.*q./(R.*(R + eta)) ...
- I1(xi,eta,q,dip,nu,R).*sin(dip).^2;
k = find(q~=0);
u(k) = u(k) + sin(dip).*atan(xi(k).*eta(k)./(q(k).*R(k)));

% -----------------------------------------------------------------
function u=uz_tf(xi,eta,q,dip,nu)
R = sqrt(xi.^2 + eta.^2 + q.^2);
db = eta.*sin(dip) - q.*cos(dip);
u = (eta.*cos(dip) + q.*sin(dip)).*q./(R.*(R + xi)) ...
+ cos(dip).*xi.*q./(R.*(R + eta)) ...
- I5(xi,eta,q,dip,nu,R,db).*sin(dip).^2;
k = find(q~=0);
u(k) = u(k) - cos(dip).*atan(xi(k).*eta(k)./(q(k).*R(k)));

% I... displacement subfunctions [equations (28) (29) p. 1144-1145]
% -----------------------------------------------------------------
function I=I1(xi,eta,q,dip,nu,R)
db = eta.*sin(dip) - q.*cos(dip);
if cos(dip) > eps
I = (1 - 2*nu) * (-xi./(cos(dip).*(R + db))) ...
- sin(dip)./cos(dip).*I5(xi,eta,q,dip,nu,R,db);
else
I = -(1 - 2*nu)/2 * xi.*q./(R + db).^2;
end

% -----------------------------------------------------------------
function I=I2(eta,q,dip,nu,R)
I = (1 - 2*nu) * (-log(R + eta)) - I3(eta,q,dip,nu,R);

% -----------------------------------------------------------------
function I=I3(eta,q,dip,nu,R)
yb = eta.*cos(dip) + q.*sin(dip);
db = eta.*sin(dip) - q.*cos(dip);
if cos(dip) > eps
I = (1 - 2*nu) * (yb./(cos(dip)*(R + db)) - log(R + eta)) ...
+ sin(dip)./cos(dip) * I4(db,eta,q,dip,nu,R);
else
I = (1 - 2*nu)/2 * (eta./(R + db) + yb.*q./(R + db).^2 - log(R + eta));
end

% -----------------------------------------------------------------
function I=I4(db,eta,q,dip,nu,R)
if cos(dip) > eps
I = (1 - 2*nu) * 1./cos(dip) * (log(R + db) - sin(dip).*log(R + eta));
else
I = -(1 - 2*nu) * q./(R + db);
end

% -----------------------------------------------------------------
function I=I5(xi,eta,q,dip,nu,R,db)
X = sqrt(xi.^2 + q.^2);
if cos(dip) > eps
I = (1 - 2*nu) * 2./cos(dip) ...
.* atan((eta.*(X + q.*cos(dip)) + X.*(R + X).*sin(dip)) ...
./(xi.*(R + X).*cos(dip)));
I(xi==0) = 0;
else
I = -(1 - 2*nu) * xi.*sin(dip)./(R + db);
end

% =================================================================
% Tilt subfunctions

% strike-slip tilt subfunctions [equation (37) p. 1147]

% -----------------------------------------------------------------
function u=uzx_ss(xi,eta,q,dip,nu)
R = sqrt(xi.^2 + eta.^2 + q.^2);
u = -xi.*q.^2.*A(eta,R).*cos(dip) ...
+ ((xi.*q)./R.^3 - K1(xi,eta,q,dip,nu,R)).*sin(dip);

% -----------------------------------------------------------------
function u=uzy_ss(xi,eta,q,dip,nu)
R = sqrt(xi.^2 + eta.^2 + q.^2);
db = eta.*sin(dip) - q.*cos(dip);
yb = eta.*cos(dip) + q.*sin(dip);
u = (db.*q./R.^3).*cos(dip) ...
+ (xi.^2.*q.*A(eta,R).*cos(dip) - sin(dip)./R + yb.*q./R.^3 ...
- K2(xi,eta,q,dip,nu,R)).*sin(dip);

% dip-slip tilt subfunctions [equation (38) p. 1147]

% -----------------------------------------------------------------
function u=uzx_ds(xi,eta,q,dip,nu)
R = sqrt(xi.^2 + eta.^2 + q.^2);
db = eta.*sin(dip) - q.*cos(dip);
u = db.*q./R.^3 ...
+ q.*sin(dip)./(R.*(R + eta)) ...
+ K3(xi,eta,q,dip,nu,R).*sin(dip).*cos(dip);

% -----------------------------------------------------------------
function u=uzy_ds(xi,eta,q,dip,nu)
R = sqrt(xi.^2 + eta.^2 + q.^2);
db = eta.*sin(dip) - q.*cos(dip);
yb = eta.*cos(dip) + q.*sin(dip);
u = yb.*db.*q.*A(xi,R) ...
- (2*db./(R.*(R + xi)) + xi.*sin(dip)./(R.*(R + eta))).*sin(dip) ...
+ K1(xi,eta,q,dip,nu,R).*sin(dip).*cos(dip);

% tensile fault tilt subfunctions [equation (39) p. 1147]

% -----------------------------------------------------------------
function u=uzx_tf(xi,eta,q,dip,nu)
R = sqrt(xi.^2 + eta.^2 + q.^2);
u = q.^2./R.^3.*sin(dip) ...
- q.^3.*A(eta,R).*cos(dip) ...
+ K3(xi,eta,q,dip,nu,R).*sin(dip).^2;

% -----------------------------------------------------------------
function u=uzy_tf(xi,eta,q,dip,nu)
R = sqrt(xi.^2 + eta.^2 + q.^2);
db = eta.*sin(dip) - q.*cos(dip);
yb = eta.*cos(dip) + q.*sin(dip);
u = (yb.*sin(dip) + db.*cos(dip)).*q.^2.*A(xi,R) ...
+ xi.*q.^2.*A(eta,R).*sin(dip).*cos(dip) ...
- (2*q./(R.*(R + xi)) - K1(xi,eta,q,dip,nu,R)).*sin(dip).^2;

% -----------------------------------------------------------------
function a=A(x,R)
a = (2*R + x)./(R.^3.*(R + x).^2);

% K... tilt subfunctions [equations (40) (41) p. 1148]
% -----------------------------------------------------------------
function K=K1(xi,eta,q,dip,nu,R)
db = eta.*sin(dip) - q.*cos(dip);
if cos(dip) > eps
K = (1 - 2*nu) * xi./cos(dip) .* (1./(R.*(R + db)) - sin(dip)./(R.*(R + eta)));
else
K = (1 - 2*nu) * xi.*q./(R.*(R + db).^2);
end

% -----------------------------------------------------------------
function K=K2(xi,eta,q,dip,nu,R)
K = (1 - 2*nu) * (-sin(dip)./R + q.*cos(dip)./(R.*(R + eta))) ...
- K3(xi,eta,q,dip,nu,R);

% -----------------------------------------------------------------
function K=K3(xi,eta,q,dip,nu,R)
db = eta.*sin(dip) - q.*cos(dip);
yb = eta.*cos(dip) + q.*sin(dip);
if cos(dip) > eps
K = (1 - 2*nu) * 1./cos(dip) .* (q./(R.*(R + eta)) - yb./(R.*(R + db)));
else
K = (1 - 2*nu) * sin(dip)./(R + db) .* (xi.^2./(R.*(R + db)) - 1);
end

% =================================================================
% Strain subfunctions

% strike-slip strain subfunctions [equation (31) p. 1145]

% -----------------------------------------------------------------
function u=uxx_ss(xi,eta,q,dip,nu)
R = sqrt(xi.^2 + eta.^2 + q.^2);
u = xi.^2.*q.*A(eta,R) ...
- J1(xi,eta,q,dip,nu,R).*sin(dip);

% -----------------------------------------------------------------
function u=uxy_ss(xi,eta,q,dip,nu)
R = sqrt(xi.^2 + eta.^2 + q.^2);
db = eta.*sin(dip) - q.*cos(dip);
u = xi.^3.*db./(R.^3.*(eta.^2 + q.^2)) ...
- (xi.^3.*A(eta,R) + J2(xi,eta,q,dip,nu,R)).*sin(dip);

% -----------------------------------------------------------------
function u=uyx_ss(xi,eta,q,dip,nu)
R = sqrt(xi.^2 + eta.^2 + q.^2);
u = xi.*q./R.^3.*cos(dip) ...
+ (xi.*q.^2.*A(eta,R) - J2(xi,eta,q,dip,nu,R)).*sin(dip);

% -----------------------------------------------------------------
function u=uyy_ss(xi,eta,q,dip,nu)
R = sqrt(xi.^2 + eta.^2 + q.^2);
yb = eta.*cos(dip) + q.*sin(dip);
u = yb.*q./R.^3.*cos(dip) ...
+ (q.^3.*A(eta,R).*sin(dip) - 2*q.*sin(dip)./(R.*(R + eta)) ...
- (xi.^2 + eta.^2)./R.^3.*cos(dip) - J4(xi,eta,q,dip,nu,R)).*sin(dip);

% dip-slip strain subfunctions [equation (32) p. 1146]

% -----------------------------------------------------------------
function u=uxx_ds(xi,eta,q,dip,nu)
R = sqrt(xi.^2 + eta.^2 + q.^2);
u = xi.*q./R.^3 ...
+ J3(xi,eta,q,dip,nu,R).*sin(dip).*cos(dip);

% -----------------------------------------------------------------
function u=uxy_ds(xi,eta,q,dip,nu)
R = sqrt(xi.^2 + eta.^2 + q.^2);
yb = eta.*cos(dip) + q.*sin(dip);
u = yb.*q./R.^3 ...
- sin(dip)./R ...
+ J1(xi,eta,q,dip,nu,R).*sin(dip).*cos(dip);

% -----------------------------------------------------------------
function u=uyx_ds(xi,eta,q,dip,nu)
R = sqrt(xi.^2 + eta.^2 + q.^2);
yb = eta.*cos(dip) + q.*sin(dip);
u = yb.*q./R.^3 ...
+ q.*cos(dip)./(R.*(R + eta)) ...
+ J1(xi,eta,q,dip,nu,R).*sin(dip).*cos(dip);

% -----------------------------------------------------------------
function u=uyy_ds(xi,eta,q,dip,nu)
R = sqrt(xi.^2 + eta.^2 + q.^2);
yb = eta.*cos(dip) + q.*sin(dip);
u = yb.^2.*q.*A(xi,R) ...
- (2*yb./(R.*(R + xi)) + xi.*cos(dip)./(R.*(R + eta))).*sin(dip) ...
+ J2(xi,eta,q,dip,nu,R).*sin(dip).*cos(dip);

% tensile fault strain subfunctions [equation (33) p. 1146]

% -----------------------------------------------------------------
function u=uxx_tf(xi,eta,q,dip,nu)
R = sqrt(xi.^2 + eta.^2 + q.^2);
u = xi.*q.^2.*A(eta,R) ...
+ J3(xi,eta,q,dip,nu,R).*sin(dip).^2;

% -----------------------------------------------------------------
function u=uxy_tf(xi,eta,q,dip,nu)
R = sqrt(xi.^2 + eta.^2 + q.^2);
db = eta.*sin(dip) - q.*cos(dip);
u = -db.*q./R.^3 ...
- xi.^2.*q.*A(eta,R).*sin(dip) ...
+ J1(xi,eta,q,dip,nu,R).*sin(dip).^2;

% -----------------------------------------------------------------
function u=uyx_tf(xi,eta,q,dip,nu)
R = sqrt(xi.^2 + eta.^2 + q.^2);
u = q.^2./R.^3.*cos(dip) ...
+ q.^3.*A(eta,R).*sin(dip) ...
+ J1(xi,eta,q,dip,nu,R).*sin(dip).^2;

% -----------------------------------------------------------------
function u=uyy_tf(xi,eta,q,dip,nu)
R = sqrt(xi.^2 + eta.^2 + q.^2);
db = eta.*sin(dip) - q.*cos(dip);
yb = eta.*cos(dip) + q.*sin(dip);
u = (yb.*cos(dip) - db.*sin(dip)).*q.^2.*A(xi,R) ...
- q.*sin(2*dip)./(R.*(R + xi)) ...
- (xi.*q.^2.*A(eta,R) - J2(xi,eta,q,dip,nu,R)).*sin(dip).^2;

% J... tensile fault subfunctions [equations (34) (35) p. 1146-1147]
% -----------------------------------------------------------------
function J=J1(xi,eta,q,dip,nu,R)
db = eta.*sin(dip) - q.*cos(dip);
if cos(dip) > eps
J = (1 - 2*nu) * 1./cos(dip) * (xi.^2./(R.*(R + db).^2) - 1./(R + db)) ...
- sin(dip)./cos(dip)*K3(xi,eta,q,dip,nu,R);
else
J = (1 - 2*nu)/2 * q./(R + db).^2 .* (2*xi.^2./(R.*(R + db)) - 1);
end

% -----------------------------------------------------------------
function J=J2(xi,eta,q,dip,nu,R)
db = eta.*sin(dip) - q.*cos(dip);
yb = eta.*cos(dip) + q.*sin(dip);
if cos(dip) > eps
J = (1 - 2*nu) * 1./cos(dip) * xi.*yb./(R.*(R + db).^2) ...
- sin(dip)./cos(dip)*K1(xi,eta,q,dip,nu,R);
else
J = (1 - 2*nu)/2 * xi.*sin(dip)./(R + db).^2 .* (2*q.^2./(R.*(R + db)) - 1);
end

% -----------------------------------------------------------------
function J=J3(xi,eta,q,dip,nu,R)
J = (1 - 2*nu) * -xi./(R.*(R + eta)) ...
- J2(xi,eta,q,dip,nu,R);

% -----------------------------------------------------------------
function J=J4(xi,eta,q,dip,nu,R)
J = (1 - 2*nu) * (-cos(dip)./R - q.*sin(dip)./(R.*(R + eta))) ...
- J1(xi,eta,q,dip,nu,R);