Suppose $X$ is a Markov diffusion process on $R^p$, or more generally on a manifold $N$. The diffusion variance of $X$ induces a semi-definite metric $\langle\cdot\mid\cdot\rangle$ on the cotangent bundle, a version of the Levi-Civita connection $\Gamma$ and a Laplace-Beltrami operator $\Delta$. We may treat $X$ as a diffusion on $N$ with generator $\xi + (1/2)\Delta$, where $\xi$ is a vector field. For sufficiently small $\delta > 0$, $X_\delta$ has an ``intrinsic location parameter'', defined to be the non-random initial value $V_0$ of a $\Gamma$-martingale $V$ terminating at $X_\delta$. It is obtained by solving a system of forward-backwards stochastic differential equations (FBSDE): a forward equation for $X$, and a backwards equation for $V$. This FBSDE is the stochastic equivalent of the heat equation (with drift $\xi$) for harmonic mappings, a well-known system of quasilinear PDE. Let $\(\phi_t: N \rightarrow N, t \geq 0\)$ be the flow of the vector field $\xi$, and let $x_t \equiv \phi_t(x_0) \in N$. Our main result is that $\exp^(-1(x_\delta)V_0$ can be intrinsically approximated to first order in $Tx_\delta)N$ by $$ \nabla d\phi_\delta(x_0)(\Pi_\delta) - \int_(0)^(\delta)(\phi_(\delta - t)(*)(\nabla d\phi_t(x_0))d\Pi_t $$ where $\Pi_t = \int0)^(t)(\phi_(-s)(*)\langle\cdot\mid\cdot\rangle_(x_s)ds \in T_(x_0)N \otimes T_(x_0)N$. This is computed in local coordinates. More generally, we find an intrinsic location parameter for $\Psi(X_\delta)$, if $\Psi:N \rightarrow M$ is a $C^2$ map into a Riemannian manifold $M$. These formulas have immediate application, discussed in a separate article, to the construction of an intrinsic nonlinear analog to the Kalman Filter.