Abstract We focus on the nonhomogeneous backward heat problem of finding the initial temperature θ = θ ( x , y ) = u ( x , y , 0 ) {\theta=\theta(x,y)=u(x,y,0)} such that { u t - a ( t ) ( u x x + u y y ) = f ( x , y , t ) , ( x , y , t ) ∈ Ω × ( 0 , T ) , u ( x , y , t ) = 0 , ( x , y ) ∈ ∂ Ω × ( 0 , T ) , u ( x , y , T ) = h ( x , y ) , ( x , y ) ∈ Ω ¯ , \left\{\begin{aligned} \displaystyle u_{t}-a(t)(u_{xx}+u_{yy})&\displaystyle=f% (x,y,t),&\hskip 10.0pt(x,y,t)&\displaystyle\in\Omega\times(0,T),\\ \displaystyle u(x,y,t)&\displaystyle=0,&\hskip 10.0pt(x,y)&\displaystyle\in% \partial\Omega\times(0,T),\\ \displaystyle u(x,y,T)&\displaystyle=h(x,y),&\hskip 10.0pt(x,y)&\displaystyle% \in\overline{\Omega},\end{aligned}\right.\vspace*{-0.5mm} where Ω = ( 0 , π ) × ( 0 , π ) {\Omega=(0,\pi)\times(0,\pi)} . In the problem, the source f = f ( x , y , t ) {f=f(x,y,t)} and the final data h = h ( x , y ) {h=h(x,y)} are determined through random noise data g i j ( t ) {g_{ij}(t)} and d i j {d_{ij}} satisfying the regression models g i j ( t ) = f ( X i , Y j , t ) + ϑ ξ i j ( t ) , \displaystyle g_{ij}(t)=f(X_{i},Y_{j},t)+\vartheta\xi_{ij}(t), d i j = h ( X i , Y j ) + σ i j ε i j , \displaystyle d_{ij}=h(X_{i},Y_{j})+\sigma_{ij}\varepsilon_{ij}, where ( X i , Y j ) {(X_{i},Y_{j})} are grid points of Ω. The problem is severely ill-posed. To regularize the instable solution of the problem, we use the trigonometric least squares method in nonparametric regression associated with the projection method. In addition, convergence rate is also investigated numerically.