# A New Algorithm for Simultaneous Retrieval of Aerosols and Marine Parameters
## 2 Data and Methods
### Inverse Method for Retrieval of Atmosphere-Water Parameters from TOA Radiances  
As explained above, our inverse algorithm is based on a multidimensional unconstrained nonlinear minimization MATLAB function (fminsearch) in accordance with a method due to Nelder-Mead [28,29]. The aim of the Nelder and Mead [28] method is to minimize a scalar-valued nonlinear function of several real variables using function values only. The following four scalar coefficients of reflection (\(\rho>0\)), expansion (\(x>1,\rho\)), contraction (\(0<\gamma<1\)), and shrinkage (\(0<\sigma<1\)) must be specified in order to implement the method. The Nelder-Mead search method maintains a non-degenerate simplex of \(n+1\) points for \(n\)-dimensional vectors \(x\). The algorithm starts by adding 5% to each component of the initial guess \(x_{0}\), and the new generated vectors \(x_{0}(i)\) become the elements of the new simplex in addition to \(x_{0}\). If \(x_{0}(i)=0\), then \(x_{0}(i)\) is set to 0.00025, and the algorithm then follows the procedure below until the diameter of the simplex is less than a specified tolerance.  
1. Let \(x(i)\) denote the list of points in the current simplex, \(i=1,\ldots,n+1\). Then the algorithm orders the points in the simplex from the lowest function value \(f(x(1))\) to the highest function value \(f(x(n+1))\), and at each step in the iteration the algorithm discards the current worst point \(x(n+1)\) and accepts another point in the simplex. Or, in case of step (v) below it changes all \(n\) points with values above \(f(x(1))\).
2. To generate a reflection point, the algorithm generates a point \(r=2m-x(n+1)\), where \(m=\sum x(i)/n\), \(i=1,\ldots,n\), and then calculates \(f(r)\). If \(f(x(1))\leq f(r)<f(x(n))\), then the algorithm accepts \(r\) and terminates the iteration.
3. To generate an expansion point for the case in which \(f(r)<f(x(1))\), the algorithm calculates an expansion point \(s\), given by \(s=m+2(m-x(n+1))\), and then computes \(f(s)\). If \(f(s)<f(r)\), the algorithm accepts \(s\) and terminates the iteration. Otherwise, it accepts \(r\) and returns to procedure step (ii).
4. If \(f(r)\geq f(x(n))\), the algorithm performs a contraction between \(m\) and the better of \(x(n+1)\) and \(r\). Case I: If \(f(r)<f(x(n+1))\) (i.e., \(r\) is better than \(x(n+1)\)), the algorithm calculates \(c=m+(r-m)/2\), and then computes \(f(c)\). If \(f(c)<f(r)\), it accepts \(c\) and terminates the iteration. Otherwise, it continues with step (v). Case II: If \(f(r)\geq f(x(n+1))\), it calculates \(cc=m+(x(n+1)-m)/2\) and com  
Figure 2: Comparison between radiances obtained from the RBF-NN and those obtained from a well-tested radiative transfer model (RTM) for a coupled atmosphere-ocean system (viewing angle \(\theta=0^{\circ}\)). Note that in graphs (**a–d**) both the shortest wavelength (412 nm) and the longest wavelength (866 nm) are represented by the blue color, and that the higher radiance values corresponds to 412 nm.  
putes \(f(cc)\). If \(f(cc)<f(x(n+1))\), it accepts \(cc\) and terminates the iteration. Otherwise, it continues with step (v).
* To generate a shrinkage point, the algorithm calculates the \(n\) points \(v(i)=x(1)+(x(i)-x(1))/2\) and computes \(f(v(i))\), \(i=2,\ldots,n+1\). The simplex at the next iteration is \(x(1)\), \(v(2),\ldots,v(n+1)\). The iteration continues until the set criterion is fulfilled  
In our algorithm the initial guess for the components of the vector \(x_{0}\) was the average values of the input parameters.