Trend Surface Analysis is a global polynomial interpolation method that fits a smooth surface defined by a mathematical function (a polynomial) to the input sample points. It changes gradually and captures coarse-scale patterns in the data.

This technique is mostly used to produce a smooth surface that represents a gradual tends in the surface over the area of interest—where the lower the root mean square (RMS) error, the more closely the interpolated surface represents the input points. It is a simple way for describing large variations and its function is to find general tendencies of the sample data, rather than to model a surface precisely.

Trend interpolation creates a gradually varying surface using low-order polynomials that describe a physical process—for example, pollution and wind direction. However, the more complex the polynomial, the more difficult it is to ascribe physical meaning to it. Furthermore, the calculated surfaces are highly susceptible to outliers (extremely high and low values), especially at the edges.

There are two basic types of Trend interpolation: Linear and Logistic.

#### 1.1 Linear trend (1-D)

The LINEAR trend surface interpolator creates a floating-point raster. It uses a polynomial regression to fit a least-squares surface to the input points. This method allows you to control the order of the polynomial used to fit the surface. To understand this method, consider a first-order polynomial. A first-order linear trend surface interpolation performs a least-squares fit of a plane to the set of input points.

In linear trend (1-D): *z *varies as a linear function of *x*

**Z **= **b**0 + **b**1**x **+ **e**

- where
- Z is the interpolated parameter.
- X and Y are the coordinates of the wells.
- b coefficient is estimated from the control points.
- e: error in prediction.

Trend surface interpolation creates smooth surfaces. The surface generated seldom passes through the original data points, since it performs a best fit for the entire surface. When a polynomial order higher than one is used, the interpolator may generate a raster whose minimum and maximum exceed the minimum and maximum of the input file of the input feature data.

#### 1.2 Logistic trend (2-D)

The LOGISTIC option for generating a trend surface is appropriate for prediction of the presence or absence of certain phenomena (in the form of probability) for a given set of locations (x,y) in space. The z-value is a categorized random variable with only two possible outcomes—for example, the existence of an endangered species or the lack of existence of that species. These two z-values can be coded as one and zero, respectively. This option creates a continuous probability grid with cell values between one and zero.

In logistic trend (2-D): *z *varies as a linear function of *x *and *y*

**Z **= **b0 **+ **b1x **+ **b2y **+ **e**

- where
- Z is the interpolated parameter.
- X and Y are the coordinates of the wells.
- b coefficient is estimated from the control points.
- e: error in prediction.

A maximum likelihood estimation is used to calculate the nonlinear probability surface model without first converting the model into linear form

The aim of this method is to develop a general kind of spatial distribution of an observable fact. The surface can be modeled using a linear or trend surface. Linear trends describe only the major direction and rate of change, while the trend surface provides progressively more complex descriptions of spatial patterns.